Quantum Correlations, Quantum Resource Theories
and Exclusion Game
ARCHNES
MASSACHUSETTS INSTITUTE
OF TECHNOLOLGY
by
Zi-Wen Liu
JUL 3 0 2015
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
LIBRARIES
Master of Science in Mechanical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ Massachusetts Institute of Technology 2015. All rights reserved.
Author .....
Signature
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Z-
Department of Mechanical Engineering
May 8, 2015
7/
Signature
redacted
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...0...
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Certified by....
......................
Seth Lloyd
Professor of Mechanical Engineering
Thesis Supervisor
Accepted by.
Signature redacted ...............
David E. Hardt
Ralph E. and Eloise F. Cross Professor of Mechanical Engineering
Graduate Officer, Department of Mechanical Engineering
2
Quantum Correlations, Quantum Resource Theories
and Exclusion Game
by
Zi-Wen Liu
Submitted to the Department of Mechanical Engineering
on May 8, 2015, in partial fulfillment of the
requirements for the degree of
Master of Science in Mechanical Engineering
Abstract
This thesis addresses two topics in quantum information theory. The first topic is
quantum correlations and quantum resource theory. The second is quantum communication theory.
The first part summarizes an ongoing work about quantum correlations beyond
entanglement and quantum resource theories. We systematically explain the concept
quantum correlations beyond entanglement, and introduce a unified framework of
measuring such correlations with entropic quantities. In particular, a new measure
called Diagonal Discord (DD), which is simpler to compute than discord but still
possesses several nice properties, is proposed. As an application to real physical
scenarios, we study the scaling behaviors of quantum correlations in spin lattices
with these measures. On its own, however, the theory of quantum correlations is not
yet a satisfactory quantum resource theory. Some partial results towards this goal are
introduced. Furthermore, a unified abstract structure of general quantum resource
theories and its duality is formalized.
The second part shows that there exist (one-way) communication tasks with an
infinite gap between quantum communication complexity and quantum information
complexity. We consider the exclusion game, recently introduced by Perry, Jain and
Oppenheim [80], which exhibits the property that for appropriately chosen parameters of the game, there exists an winning quantum strategy that reveals vanishingly
small amount of information as the size of the problem n increases, i.e., the quantum
(internal) information cost vanishes in the large n limit. For those parameters, we
prove the quantum communication cost (the size of quantum communication to succeed) is lower bounded by Q (log n), thereby proving an infinite gap between quantum
information and communication costs. This infinite gap is further shown to be robust
against sufficiently small error. Some other interesting features of the exclusion game
are also discovered as byproducts.
Thesis Supervisor: Seth Lloyd
3
Title: Professor of Mechanical Engineering
4
Acknowledgments
First of all, I would like to thank my academic and research advisor, Professor Seth
Lloyd, for leading me into the magnificent world of quantum information that I knew
little of prior to coming to MIT, and for all his generous support and insightful
guidance over the last two years. I can never know how blessed I am to have such an
opportunity.
I would like to thank Professors Scott Aaronson, Harry Asada, Gang Chen, Isaac
Chuang, Eddie Farhi, Aram Harrow, Mehran Kardar, Seth Lloyd, Hong Liu, Peter
Shor, Wati Taylor, Salil Vadhan, Evelyn Wang, Xiao-Gang Wen and many more,
from whom I learned priceless knowledge through courses and/or discussions.
I would like to thank my fellow graduate students and close collaborators, especially Can Gokler, Dax Koh, Kevin Thompson, Elton Zhu and Quntao Zhuang,
for all the happy time we spent together talking about everything from quarks to
universe(s).
I would also like to thank all my friends, for sharing my happiness and sorrow,
and for allowing me to do the same for them.
Thanks to all the people above, for making me believe I am doing the right thing,
at the right place.
And at last, I would like to thank my parents and my girlfriend Xiaoyu for encouraging and trusting me, unswervingly. Miraculously, I feel that you are all right
here with me, when I write down these words.
5
6
Contents
I
Quantum correlations and resource theories
14
1
Introduction
15
2
Quantum correlations beyond entanglement
19
2.1
2.2
2.3
2.4
3
. . . . . . . . . . . . . . . . . . . . . . .
19
. . . . . . . . . . . . . . . . . . .
20
. . . . . . . . . . . . . . . .
22
Entropic measures
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.1
Candidates
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.2.2
Hierarchy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.2.3
Multipartite generalization . . . . . . . . . . . . . . . . . . . .
39
Scaling behaviors in spin lattices . . . . . . . . . . . . . . . . . . . . .
42
2.3.1
Quantum correlation between two spins . . . . . . . . . . . . .
43
2.3.2
Example: 1D Heisenberg XXZ chain . . . . . . . . . . . . . .
46
2.3.3
Area laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.3.4
O utlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Diagonal Discord (DD) . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.4.1
M otivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.4.2
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2.4.3
Physical interpretation . . . . . . . . . . . . . . . . . . . . . .
58
Purely classical correlations
2.1.1
Classically correlated states
2.1.2
Creating nonclassical correlations
Quantum Resource Theories (QRTs)
61
3.1
62
Unified framework
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
3.3
3.1.1
Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.1.2
Perfect QRT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.1.3
Hierarchical structure . . . . . . . . . . . . . . . . . . . . . . .
72
3.1.4
Combining QRTs . . . . . . . . . . . . . . . . . . . . . . . . .
73
Quantum correlations as a resource . . . . . . . . . . . . . . . . . . .
76
3.2.1
Free states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
3.2.2
So
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.2.3
Promotion to S . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.2.4
Map zoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
3.2.5
Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
Dual QRTs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.3.1
General structure . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.3.2
Examples
91
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Summary and outlook
95
II
Exclusion game
97
5
Introduction
6
Preliminaries
103
6.1
General formulation of communication tasks . . . . . . . . . . . . . .
103
6.1.1
Mathematical structure . . . . . . . . . . . . . . . . . . . . . .
103
6.1.2
Exclusion game . . . . . . . . . . . . . . . . . . . . . . . . . .
104
6.2
Information and communication . . . . . . . . . . . . . . . . . . . . .
105
6.3
Classical communication complexity . . . . . . . . . . . . . . . . . . .
106
6.4
PJO strategy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
6.4.1
Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107
6.4.2
Quantum information cost . . . . . . . . . . . . . . . . . . . .
108
99
8
7
Quantum communication complexity
7.1
7.2
8
Zero error ....
.....
.......
109
...
....
...
. . . . ....
...
.
109
7.1.1
Classical encodings of quantum states . . . . . . . . . . . . . .
7.1.2
Lower bound of Qcc . . . . . . . . . ... .. . ... . .. . 111
7.1.3
G aps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
Robustness against error . . . . . . . . . . . . . . . . . . . . . . . . .
114
7.2.1
Lem m as . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
7.2.2
Quantum-classical separation of communication
. . . . . . . .
118
7.2.3
Maximum error . . . . . . . . . . . . . . . . . . . . . . . . . .
119
Concluding remarks
110
121
Appendix
123
A QD and DD of real X states
123
A.1
Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
A.2
Pairwise quantum correlation
125
. . . . . . . . . . . . . . . . . . . . . .
A.2.1
Optimal basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.2.2
Explicit calculation of pairwise QD . . . . . . . . . . . . . . .
126
A.2.3
Pairwise DD . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130
9
10
List of Figures
2-1
(Adapted from [104]) Area law.
Intuitively speaking, under locally
interacting Hamiltonians, the correlation length in noncritical phases
is finite so that sites in A and B that are separated by a distance further
than the correlation length (the shaded stripe) should not contribute
to the mutual information or correlation measures between A and B,
hence bounded by the number of sites at the boundary and therefore
scales as the boundary area. . . . . . . . . . . . . . . . . . . . . . . .
49
. .
63
3-1
Intuitive illustration of the basic content and structure of a QRT.
3-2
A hierarchical structure of maps. Columns represent correspondences,
and rows represent strict hierarchies.
3-3
. . . . . . . . . . . . . . . . . .
73
A sketch of a strategy for determining a qubit state that queries the
QRTs of coherence
and purity. The dashed circle represents the states
that has the same entropy (connected by unitary transformations), and
the solid line represents the states that are diagonal in the appointed
basis (incoherent states). For an arbitrary state p, the unitary U brings
it to one of the incoherent states, while preserving purity/entropy. . .
3-4
76
Geometrical (Bloch sphere) demonstration of the effect of a unital map
on a qubit. The unital channel keeps the two basis vectors symmetric
with respect to the center of the Bloch sphere, and the output state
can be diagonalized in the orthonormal basis corresponding to the intersections of the connecting line and the surface of the Bloch sphere.
11
82
3-5
(Adapted from [91]) The detailed hierarchy in between mixture of unitaries and unital maps. AQBP denotes "Asymptotic Quantum Birkhoff
P roperty". . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-6
86
Illustration of the dual QRT. The parts with grey fill means "free".
Dashed lines represent the pair of elements in each theory, and arrow
represent partial orders, determined by quantifiers. Note that this is
only a sketch of the idea of duality. The geometries of sets and the
partial orders in each space do not necessarily resemble this figure. . .
7-1
Exclusion game EXC,.. where m E
91
(n), 1/2 < a < I in the large n
limit. Solid arrows indicate established separations (pointing towards
the smaller one), while the dashed one indicates an unknown separation
(at most exponential).
. . . . . . . . . . . . . . . . . . . . . . . . . .
12
114
List of Tables
2.1
An LOCC state preparation protocol that generates quantum correlations. According to the outcomes of flipping their private coins, Alice
and Bob respectively apply the local operation indicated in the table.
The rightmost column shows the corresponding output states. ....
3.1
23
Level 1 descriptions of some typical QRTs. *Since NO is maximal, this
quantifier is unique. **With respect to a particular basis. . . . . . . .
13
68
Part I
Quantum correlations and resource
theories
14
Chapter 1
Introduction
Since its origination, quantum physics has been bewildering people in all sorts of
aspects. As famously said by Niels Bohr, "If quantum mechanics hasn't profoundly
shocked you, you haven't understood it yet." Even to this day, more than 100 years after its debut, people are still trying to understand more fully how quantum mechanics
works, and how to take advantage of it.
First recognized by Einstein, Podolsky and Rosen (EPR) in 1935 [35], entanglement, which Einstein termed, the "spooky action at a distance", lies at the heart of
quantum weirdness.
In essence, this concept captures a peculiar form of nonlocal
correlationthat has no classical counterpart - in principle, the output of measuring
the spin of an electron on the earth (which can be completely random) will directly
tell you what your friend will see if he (even if immediately afterwards) measures
another electron on Mars, or arbitrarily further away, as long as the two electrons are
entangled.
Due to its apparent weirdness, entanglement is one of the most investigated concepts in quantum information theory. One may naturally ask: is there any other form
of correlations besides classical correlations and entanglement? The answer turns out
to be no. In mixed states, there exist correlations that are not entanglement, yet have
no classical counterparts as well. Ollivier and Zurek first proposed the quantity Quantum Discord (QD) to quantify this type of correlations in their seminal 2001 work
177].
and started a boom of the studies along this direction for more than 10 years.
15
Since then, many papers on the topic of quantum correlations beyond entanglement
have appeared. Despite considerable efforts, however, no unified and universally accepted method for understanding quantum correlations has emerged. Two messages
can be taken home from this situation:
1. Quantum correlations are difficult to measure.
2. We probably haven't found the right way to understand quantum correlations.
These are exactly the problems that this part of the thesis is trying to tackle.
The first message can be interpreted from two perspectives. First, there are many
possible measures of quantum correlations to choose between. To simplify this choice,
I shall introduce a unified framework of entropic measures, and systematically discuss
the motivation for defining them. Second, most measures are difficult to compute.
For example, the calculation of the earliest and best-known measure, QD, is shown
to be NP-complete. Indeed, such convex optimization problems are always hard. In
a word, the field is a mess. To address the issue of computational complexity, a new
measure which is originated very naturally from both the mathematical and physical
perspectives, the Diagonal Discord (DD), is proposed. This measure is significantly
easier to compute and study. More importantly, diagonal discord can still serve its job
as a quantifier of quantum correlations beyond entanglement pretty well, as supported
by its various features.
The second message is closely related to the first one. To unify our understanding
of quantum correlations, we look at Quantum Resource Theories (QRTs). The QRT
of some quantum property x is essentially the theory of x as a physical resource. A
resource theory should be able to reveal multiple aspects of the quantity x, including
how to create it, how to quantify it, how to convert it, how to make use of it and so
on. To gain better understanding of a quantum object, establish the resource theory
of it. On its own, the theory of quantum correlations is far from a satisfactory QRT,
since the set of operations that generates/does not generate quantum correlations is
not fully characterized. One of the primary targets of this thesis work is to construct
a theory of quantum correlations that allows the natural construction of quantum
16
resource theories. Furthermore, different QRTs frequently exhibit very similar mathematical structure, which leads to a abstract framework that unifies many QRTs. I
shall discuss the details of this framework in this part, and point out an interesting
dual of it. Note that these two topics are seemingly quite self-contained, but as you
will see from the content, they are closely related everywhere.
This part is organized as follows. In Chapter 2, I systematically discuss the interpretation and creation of purely classical correlations, and motivate the concept
of quantum correlations beyond entanglement as the complement. Several candidate
entropic measures are defined in a consistent way. Their relationships and generalizations are then studied. The proposed measures are then used to study the scaling
behaviors of quantum correlations in spin models. Specifically, the new measure DD is
studied in the last section. Surprisingly, quantum correlations beyond entanglement
turn out to be the necessary condition for heat transfer, and DD measures its rate in
the infinitesimal limit. In Chapter 3, I present an abstract mathematical framework
that unifies QRTs, and discuss several interesting products.
Using this framework,
partial results towards a full QRT of quantum correlations are introduced.
final section, a dual framework of QRT is established, with examples given.
17
In the
18
Chapter 2
Quantum correlations beyond
entanglement
As already mentioned in Chapter 1, entanglement is not the only form of correlation
present in quantum systems that cannot be described classically - there is a gap
between purely classical correlations and entanglement. In this chapter, I systematically explain the concept quantum correlations beyond entanglement, and introduce
a unified framework of measuring such correlations with entropic quantities. In particular, a new measure called Diagonal Discord (DD), which is simpler to compute
than discord but still possesses several nice properties, is proposed. As an application
to real physical scenarios, we study the scaling behaviors of quantum correlations in
spin lattices with these measures.
2.1
Purely classical correlations
To understand quantum correlations, one must first understand classical correlations.
Two questions arise:
1. What kinds of correlations are purely classical?
2. What quantum states are classically correlated?
19
In this Section, I shall first answer the above questions, and then discuss ways to
prepare quantum states that are separable (unentangled), but that possess quantum
correlations, thereby illustrating the gap between purely classical correlations and
entanglement.
Note that the latter part (creation of quantum correlations) is also
closely related to the resource theory of quantum correlations, which will be discussed
in Section 3.2.
2.1.1
Classically correlated states
For the purpose of explaining the basic ideas, it is sufficient to consider bipartite
correlations for the moment. According to classical information theory, the state of
a classical system consisting of two parties respectively represented by random variables A and B can be described by a joint probability distribution {Pab}, while the
respective marginal probability distributions are denoted as {p"} and {Pb}. Consider
the case where the possible values a and b are assumed to be discrete.
To deter-
mine whether these two probability distributions are correlated or not from these
probability distributions, one can apply Shannon's theory to calculate the mutual
information
I(A : B) = H(A) + H(B) - H(AB),
(2.1)
where H(-) denotes the Shannon entropy of a probability distribution, e.g., H(AB) =
- Ea bPab log Pab. Nonzero I(A; B) means that A and B are not independent of each
other, and the value of I(A : B) quantifies the amount of information they share in
common.
Then comes the second question: Which quantum states are completely classically
correlated?
Or more specifically, how does one write down a quantum state that
registers classical correlation as introduced in the above paragraph? (Note that such
a quantum state lives in the joint Hilbert space 'RAB = RAO RB.) The answer is that
this quantum state represents the probabilistic mixture (note: different from coherent
superposition) of a set of quantum states according to {Pab}, where any pair of states
chosen from this set are perfectly distinguishable,since all events (labeled by ab) are by
20
definition distinguishable in the classical setting. As is well known, the necessary and
sufficient condition for two quantum states being reliably distinguishable is that they
are orthogonal [761. Therefore, the set of perfectly distinguishable states spans some
Hilbert space
?AB
whose dimension is the number of possible ab's, i.e., forms the
orthonormal basis of NAB. Denoting the corresponding elements of this orthonormal
set as lab), a completely classically correlated state actually represents the statistical
ensemble {Pab,
lab)}.
Definition 1 (Completely classically correlated state; CC state). A quantum state
is classically correlated, iff it takes the form
PAB
=
ZPaba)A(a 0 lb)B(b= Z
ab
Pablab)(abl,
(2.2)
ab
where EabPab = 1, (abla'b') = S5aalbb'. For reasons that will become clear later,
members of this class of states are also called classical-classical (CC) states [781.
Furthermore, one may also argue that if all possible local measurements on one
subsystem disturb the joint quantum state, that means its correlation with the other
subsystem exhibits nonclassical nature. In other words, if one subsystem is classically
correlated with the other, then there should exist a local measurement that does not
alter the joint state. The only possible local measurement basis is the Schmidt basisi
(not necessarily unique), since other choices do not preserve the local state (nonzero
off-diagonal entries will be erased).
This intuition not only provides another nice
way of understanding of the "quantumness" of correlations, but also naturally leads
to the definition of "one-way" classically correlated states by restricting the local
measurement to one side (say, A):
Definition 2 (One-way classically correlated state; CQ state). A bipartite quantum
state is called a classical-quantum (CQ) state, iff there exists a local measurement
on A that does not disturb the joint state.
Equivalently, it can be written in the
'Schmidt basis is defined as the basis with respect to which the local density matrix is diagonal
(local spectral basis).
21
following form
pA = Lpala)A(al &Pa
(2.3)
a
where Ia) is the Schmidt (local spectral) basis for A (block diagonal in A's eigenbasis),
and p' is some valid density matrix for B.
As can be directly seen, the name "CQ" comes from the form of the state: it
looks classical on A's side and quantum on B's. Likewise, we can define QC states
by placing the local measurement criterion on B. And if we require the joint state to
be undisturbed for local measurements on both sides, the definition of CC states is
easily recovered.
Note that CC is a stronger constraint than CQ or QC: CC states definitely qualify
as both CQ and QC states. I shall call the union of these three sets "-y states" from
now. Lastly, I introduce two important properties of the set of 7 states:
1. It has Lebesgue measure zero and is nowhere dense in the parameter space of
all quantum states [38], i.e., topologically negligible. That is, almost every state
possesses quantum correlations;
2. It is not convex: mixtures of -y states may possess quantum correlations.
I shall leave the explanation of the second property to Section 2.1.2.
These two
features will show great importance in later discussions.
2.1.2
Creating nonclassical correlations
Equipped by the in-depth understanding of the classical-type correlations, we shall
now explore the interesting features of those correlations that are not classical, i.e.,
quantum.
A nice first step is to think about what kind of operations are able to
generate quantum correlations between two systems.
LOCC
As has been extensively studied, a key feature of entanglement is that it cannot be
created by Local Operations and Classical Communication (LOCC) [28, 50]. (The
22
LO
Coin flip
Alice
Bob
+
+
+
-
-
+
-
-
Output
Bob
Alice
I
H
10)
X
HX
H
HX
I
X
11)
--)
Alice
1+)
Bob
+)
1-)
10)
11)
Table 2.1: An LOCC state preparation protocol that generates quantum correlations.
According to the outcomes of flipping their private coins, Alice and Bob respectively
apply the local operation indicated in the table. The rightmost column shows the
corresponding output states.
study of LOCC is actually the foundation of the resource theory of entanglement,
which will be further discussed in Chapter 3.)
That is, one can never prepare an
entangled state by operations within the scope of LOCC.
Now I present a state preparation protocol that creates quantum correlations with
the help of LOCC only. As usual, we name the two parties Alice and Bob respectively
with a classical communication channel in between, both having an unbiased private
coin, and |0) as the initial state: the initial bipartite state is simply the direct product
of two local states, i.e., I0)A 0 I0)B (completely uncorrelated).
They respectively
flip their private coins, and then communicate the (independent) outcomes via the
classical channel (say, they send each other
"+"
or "-", respectively represents heads or
tails, via text message). They then follow the instruction in Table. 2.1 to apply the
corresponding local unitary their initial states, e.g., if Alice gets heads and Bob gets
tails, Alice acts Pauli X on her state and Bob acts Pauli X followed by Hadamard on
his state. The final bipartite state will be
PAB
1
=
-(IO)A(0 ® +)B(+ + 1)A(1I 0 I-)B(-
4
+|-)A(- 0 I)B(0| + +)A(+I 0 |1)B(1I)
=
1
+ I + 1-)(1 -(I0+)(0
4
I + I - O)(-0 + I + 1)(+1).
(2.4)
This state is definitely not entangled because the initial state is an uncorrelated
product state (possess no entanglement), and the preparation procedure only involves LOCC. However, it is impossible use the forms of -y states to write
23
PAB.
More
elaborately, I will show in the Section 2.2 that the measures of quantum correlations,
which may only vanish on 'y states, are positive for
PAB.
Generally, one can always follow the similar procedure to prepare the following
state
Sep
PAR
=
piUi|0)(0|U 0
=
|0) B0O~
(2.5)
piluivi)(Uivil,
where Ui and V are unitaries acting on A and B corresponding to the classical flag
i, and I denote Ui|O) = Jui), V10) =
jvi). Or equivalently,
pR =
(2.6)
ipp,
where E pi = 1, pi and pi are arbitrary local states. All states that can be written
in the form of Eqs.
(2.5) or (2.6) are separable/unentangled.
Or in other words,
they are just probabilistic mixtures of product (uncorrelated) states. Note that the
problem of determining separability is computationally hard in general [57]. One can
also use the Perez-Horodecki Positive Partial Transpose (PPT) criterion [791 to test
the separability of this state, though it is not always a sufficient condition.
Throwing away information
Alternatively,
PAB
can be viewed as the uniform probabilistic mixture of four product
states, which exemplifies the nonconvexity of the set of y states, as mentioned in
Section 2.1.1. Nonconvexity also leads to a way to create quantum correlations by
simply throwing away information: Suppose one has two y states in hand, and he flips
a coin to determine which one to choose. By simply forgetting the outcome of the
(public) coin flip, one obtains the state which is just a mixture of these two y states,
and by nonconvexity it can be and frequently is quantumly correlated.
As will be
discussed in Chapter 3, such a scenario is not good for a reasonable quantum resource
theory since one can seemingly create quantum resources from classical correlations
24
arbitrarily - Further assumptions need to be made.
Discussions
The above example indicates that there exist unentangled states that possess quantum
correlations, which live in the gap between purely classical correlations and entanglement. The notion of entanglement is not sufficient to capture all correlations that
have no classical counterpart. In fact, people have found that unentangled states in
many cases may exhibit nonclassical physical behaviors [731. Comparing the forms
of classically correlated states Eq. (2.2) and unentangled states Eqs. (2.5) or (2.6),
it is easy to tell that the restriction on the latter class is much weaker. In fact, the
connection between separable states and -y states is very similar to the purification
of mixed states. It is shown in [61] that a bipartite state
PAB
is separable iff there
exists a CC state PAABB, (partition: AA'/BB') on an extended Hilbert space where
PAB
~ trA'B'PAA'BB'. That is, the "separable-CC" relation can be thought of as the
bipartite analogy of the "mixed-pure" relation for quantum states.
It is worth mentioning that in contrast to -y states, the set of separable states
exhibits the following features:
1. It possesses a finite volume in the set of all quantum states
155], i.e., has nonzero
measure;
2. It is convex: the mixtures of separable states are still separable.
It has been discussed that these two properties are not satisfied by -y states.
2.2
Entropic measures
In previous sections, I have discussed about the boundaries in the classical-quantumentanglement hierarchy of correlations in quantum states, in order to motivate readers
to pay attention to quantum correlations beyond entanglement. However, this characterization of quantum correlations is far from the purpose of fully understanding
and utilizing such correlations as a resource for quantum information processing tasks
25
(which is the motivation and mission for Quantum Resource Theories (QRT), as will
be discussed in detail in Chapter 3). In general, a good resource theory should both
quantify the amount of the resource associated with a quantum state, and also provide
a partial order of quantum states.
Originating from the perspectives of thermodynamics (physical) and information
theory (mathematical) quite independently, the concept of entropy, which quantifies
the uncertainty of information content, has been playing a central role in all kinds of
scientific studies. As Landauer's principle and Szilard's engine has taught us, these
two fields are closely bond together - information is physical. Indeed, correlations
between physical systems can be understood as shared information: entropy is the
most natural and powerful tool to study correlations. In this section, I shall introduce
and analyze some candidate entropic quantifiers for quantum correlations in a unified
manner. Note that some measures other than entropic quantites can be suitable for
certain scenarios, including the geometric measure [311 etc. 2 , but I shall focus on those
measures that are defined in terms of entropic quantities here.
2.2.1
Candidates
Quantum Discord (QD)
As discussed earlier, quantum correlation is simply the part of total correlation that
is not classical. This intuition leads to the original and most famous quantification of
quantum correlations, namely, Quantum Discord (QD), which was originally proposed
by Ollivier and Zurek in 2001 [77].
By definition, QD is the discrepancy between
total mutual information and the classical part, interpreted as the maximal amount
of mutual information that can be accessed locally (by measurements). I shall start by
introducing the underlying ideas of this quantity, and then proceed to other reasonable
measures. Note that without loss of generality, I assume that A is the party that does
local measurements when talking about one-way scenarios.
As introduced in Section 2.1.1, the amount of correlation between classical random
2
R.efer to
1731 for a review.
26
-
variables A and B is captured by the mutual information I(A: B) = H(A) + H(B)
H(AB) where H(X) = - E_ plogp, denotes the Shannon entropy, where X is a
classical variable with values x occuring with probability px.
On the other hand,
Bayes' rule allows us to define an equivalent form for the classical mutual information
as I'(A: B) = H(B) - H(BIA) = Ii(A: B) with the conditional entropy H(BIA) =
EZpaH(Bla), which can be understood as the information of B that can be obtained
by gaining knowledge on A. Note that for classical systems the mutual information
yielded by these two definitions are exactly the same.
Now we "quantize" the above notions by promoting random variables A and B to
quantum systems, and Shannon entropies to von Neumann entropies. Furthermore,
the conditioning on A is now realized by making measurements on the local subspace.
Then one may immediately notice that the entropic quantities will depend on the
measurement, which is indeed the case, and we should treat all knowledge of the total
correlation that can be accessed locally as classical, leading to the optimization over
all POVMs or von Neumann measurements. It can be shown that the optimal POVM
is always rank-1 and extremal [71, 431, but not necessarily orthogonal projective (by
Davies' theorem, the optimal POVM may have up to d2 elements, where d is the
dimension of the local Hilbert space), indicating that optimizing over von Neumann
measurements is not always optimal. However for certain purposes, restricting to von
Neumann measurements is good enough (indeed, simpler at least), so I shall leave it
here as a possibility. Gathering all the above considerations together, we define the
amount of classical correlation as
CA(A : B)
=
S(B) - min Z pkS(Bk),
(2.7)
where {Hek} denotes a set of measurement operators, and PB,k = trA [(lk0IA)pAB]/pk
is the reduced state of subsystem S corresponding to outcome k. On the other hand
the total mutual information is simply given by
I(A: B)
=
S(A) + S(B) - S(AB),
27
(2.8)
where S(X) = -px log(px) denotes the von Neumann entropy of a quantum state
px. Therefore we have the following definition of QD:
Definition 3 (Quantum Discord (QD)). QD is defined as the minimum amount of
mutual information that cannot be accessed via local measurements on A, or in other
words, the discrepancy between total mutual information and the maximum amount
of mutual information that can be accessed via local measurements on A:
D(A -+ B)
I(A: B) - CA(A: B)
=
S(A) + S(B) - S(AB) - S(B) + min I:pkS(Bk)
{nI} k
=
min E pkS(Bk) + S(A) - S(AB)
{nk} k
min S(BIA) - S(BIA),
{nI}
(2.9)
where the classical correlation CA(A : B) can also be interpreted as the maximum
amount of correlation in the postmeasurement state.
I emphasize again that due to Bayes' rule, D(A -+ B) always vanishes for classical
probability distributions since the classical analogs of the two conditional entropies
in the last line are exactly equivalent, but it can be positive for some separable states
(that are not CQ). Also note that QD is not necessarily a symmetric measure as its
value depends on the party chosen to carry out the measurement: it is not an ordinary
distance measure, or say, a metric.
Measurement and entropy
Now let's consider a general question: Without postselection, how do measurements
affect the entropy of a quantum system? For von Neumann measurements, it is well
known that the entropy of the state being measured cannot decrease, and it remains
unchanged when the spectral basis (in which the density matrix is diagonal) is chosen
as the measurement basis since the state itself is unchanged (Theorem 11.9 in (761).
An equivalent interpretation is that von Neumann measurements are closed operations
(uncoupled to the outside), which should never decrease the entropy of the system by
28
the second law of thermodynamics. However, POVMs (generalized measurements),
which can always be thought of as reductions of projective measurements on a larger
Hilbert space (Naimark's dilation theorem). indicating that they are essentially open
system operations, which may cause information to flow from the bath to the system,
thus decreasing the entropy.
What if one does von Neumann measurements on a
subspace? The answer is given by the following statement3 :
Theorem 1. Consider an arbitraryquantum state p living in the d-dimensional space
Rd.
Projective measurements on a d-dimensional subspace (di <; d, Rd,
1= I-td\l-d,,
Rd,
9
when d = dl: empty) cannot decrease the entropy of the state (the
postmeasurement state reads p' =
Z(HUi 0 In )p(Ui 0 I
)), i.e., S(p') > S(p), with
equality if p can be written as a CQ state with the part in Rd, being classical, and
{H} being the Schmidt basis.
Proof. By definition,
Ei H
= IHd1 , thus
ZE
Hs0
IjJ
di
=
'Rd;
and (li 0 IR-)2
d
Hi 0 Ih . Therefore
d1
S(p')
=
-tr(p log p')
=
-tr [Z(i
=
-tr
Z(fi0 Isi))PlogpI(Hi0I.t)
-tr
Z(i0M
0 Is
)p(He 0 In ) logp'
IR)plogp']
-tr(p log p'),
(2.10)
where the third equality follows from the fact that (fli 0 IH ) commutes with p', and
thus also its logarithm. Then by definition of the relative entropy distance:
S(p') - S(p) = S(pjIp') > 0,
(2.11)
with equality iff p = p', by Klein's inequality. For d, = d case, the statement simply
3
This fact is known. Variants of the same result can be found in literatures, e.g., 1631.
29
reduces to Theorem 11.9 in [76] mentioned earlier. When d, < d, there exists local
basis that doesn't disturb the state (Schmidt basis) iff it's CQ (classical over 'Nd),
and p' is the same CQ state.
Indeed, since dephasing (measurement) only happens inside the system, this kind
of operations can also be considered closed, which never decrease entropy.
Note that whenever p'
$
p, S(p') > S(p), which directly indicates that for non-CQ
states (i.e., states that possess positive one-way quantum correlation), min{rl-L} [S(p')S(p)] > 0. Treating '-t
as the space of the bath, this minimum joint entropy pro-
duction appears to be a very natural measure of the quantum part of the coupling
strength, i.e., the one-way quantum correlation.
Definition 4 (Minimum Joint Entropy Production (minJEP)). As implied by the
name, minJEP is defined as the minimum amount of entropy generation over all
possible local projective measurements on the appointed subsystem A:
M(AB) = min S(AB) - S(AB),
{InA}
(2.12)
where S(AB) and S(AB) correspond to S(p') and S(p) in Theorem 1 respectively.
Note that the local measurement is equivalent to local dephasing with respect to the
measurement basis, so this quantity can also be interpreted as the minimum amount
of information lost due to the constraint of classical communication. For pure states,
QD trivially reduces to the entanglement entropy S(A) = S(B).
This measure actually coincides with several quantities that has been defined in
different contexts, e.g., thermodynamics [109]. And by Eq. (2.11), it is easy to argue
that minJEP is also exactly equivalent to the minimum relative entropy distance
to CQ states, namely "relative entropy of discord", which has also been proposed
independently as a measure [48].
Combined with the above conclusions, for a fixed quantum state, the lower bound
of entropy change upon measurements seems to increase as the "effective" space being
30
measured.4 Independent of the current purpose of analyzing quantum correlations, it
might be interesting to study the details of this observation, e.g., bounds depending
on dimensions. (A trivial case is the maximally mixed state, whose entropy cannot
be changed by measurements.) As a side remark, the maximal set of operations that
never decreases the entropy of a quantum system is called Noisy Operations (NO),
which will be introduced in Chapter 3 as the core element of the resource theory of
purity.
In the above I only considered the entropy production of the whole system upon
local projective measurements, which may also generates entropy locally. Taking this
fact into consideration, we arrive at another important observation:
Theorem 2. Consider projective measurements on a subsystem (which do not decrease entropy both jointly and locally): the minimum difference between local and
joint entropy production reduces to QD.
Proof. As introduced in Definition 3, QD can be understood as the minimum discrepancy between pre- and postmeasurement mutual information of the two parties,
thus can be rewritten in the following form:
D(A -+ B)
=
I(A : B) - max (A : B)
=
S(A) + S(B) - S(AB) - max[S(A) - S(B) + S(AB)]
{ni}
=
min [S(AB)
{nt}
-
S(AB)]
-
[S(A)
-
S(A)]},
where the two brackets are respectively the joint and local entropy production.
(2.13)
El
Combining the above theorem with the nonnegativity of QD [73], we directly
obtain the following
Corollary. Upon local projective measurements, the joint and local entropy production
are both zero if the state is CQ. Otherwise, the former is always greater than the
latter.
4
For local projective measurements, it simply refers to 7d,, while for POVMs, it means the
extended Hilbert space by Naimark's dilation.
31
Diagonal Discord (DD): a natural simplification
QD
and minJEP both involve the optimization over all possible measurements, which
is computationally very difficult in general, i.e., unfavorable for practical use, especially when dealing with large systems. Is there a way to define a reasonable measure
that is easier to calculate? An immediate idea is to see if we can put restrictions on
the measurement. Obviously, the most natural and promising choice is the Schmidt
basis measurement due to its very special feature: does not disturb the reduced state,
which leads to the definition of Diagonal Discord (DD):
Definition 5 (Diagonal Discord (DD)). Instead of carrying out the minimization over
all local measurement bases for computing QD, we directly apply local measurement
in the Schmidt basis, and name the resulting discord value as Diagonal Discord (DD).
By Eq. (2.13), it takes the form
DD(A -+ B) = S(AB) - S(AB),
(2.14)
where the measurement is made in Schmidt basis, which does not perturb the reduced
state, thereby generating no entropy locally - It is completely equivalent to substitute
the optimization over measurements in the definition of minJEP with Schmidt basis.
Note that the diagonal basis is unique iff the decomposition is non-degenerate.
The basis choice within the degenerate subspace doesn't matter, e.g. for maximally
mixed state all bases are equivalent. As a specific example, Now recall the separable
bipartite state in Eq. (2.4). One can directly observe that the reduced state of either
party (the marginals) are maximally mixed, implying that all local measurement bases
are equivalent: QD reduces to DD. Here I calculate the exact value for illustration:
S(Ao) = S(A 1 ) = -1 log 1 -
log
A) = DD(A -+ B) = DD(B -+ A)
~ 0.81 bits, and therefore D(A -+ B) = D(B
=
log i
-+
0.31 bits, which supports our earlier
claim that the correlation exhibited in this state is not purely classical, i.e., quantum
correlations can be created by LOCC.
I shall discuss this measure in more detail in Section 2.4.
32
Criteria for candidacy
In the above, several entropic measures of quantum correlations are defined. Generally
speaking, a reasonable one-way measure must satisfy the following basic criteria:
1. It vanishes for, and only for CQ states; otherwise it's positive;
2. It is invariant under local unitaries;
All measures defined earlier do have these properties. There are some other features
that one may expect a good measure should possess, especially for certain purposes.
But these are the nonnegotiable criteria that any acceptable quantification for quantum correlations should satisfy. I shall explore some interesting common properties
in Section 2.4.
Two-way generalization
By Theorem 2, the one-way quantifiers of quantum correlations can be easily generalized to two-way, or even multipartite cases (Section 2.2.3). Specifically for bipartite
QD, the two-way version can be defined as follows:
Definition 6 (Two-way bipartite QD). The two-way QD can be defined as the minimum difference between the joint entropy production and the sum of local entropy
productions, when local measurement are made by both parties, i.e.,
D(A ++ B) =
min
[S(Ab) - S(AB)] - [S(A) - S(A)] - [S(B) - S(B)]I,
(2.15)
where {Ik}, {III} denote local projective measurements on A and B respectively.
The same quantity is known as WPM discord [1061.
Two-way generalizations of minJEP and DD can be similarly defined. Since local
entropy productions are not involved in these two measures: the only difference is
that local measurements are made on both sides.
33
2.2.2
Hierarchy
Now I analyze the magnitudes of these different entropic quantifiers, and compare
them to the entropic measures of entanglement and total correlations.
Measures for quantum correlations
In the previous subsection, I defined QD, minJEP and DD as candidate measures
for quantum correlations beyond entanglement. It is easy to show that for a certain
state, QD can never be larger than minJEP: Suppose there exists a state p such that
QD is larger than minJEP. This directly indicates that the local entropy production
is negative upon projective measurements, which contradicts Theorem 11.9 in [76j
Then one can immediately tell that DD is always larger than or equal to the other
two since it is essentially JEP without minimization. Summarizing, we have
QD < minJEP < DD.
(2.16)
As discussed earlier, this hierarchy collapses at least for states with maximally mixed
marginals, and pure states [60]. The maximal set of states that takes the equalities
is yet to be specified, and is important especially for studying DD.
However, despite that these entropic measures strictly obey the above order, it can
be shown that they determine different partial orders of states with positive quantum
correlations:
Theorem 3. For any pair of different entropic measures of quantum correlations 61
and 52, there exist two different states p and o- such that S1(p) > 62 (-),
61(a),
but 62(P) <
i.e., are ordered differently under these two measures.
Proof. The proof goes similarly as the proof of different state orderings under "good
asymptotic entanglement measures" [102]. It is known that for any pure state
IV),
61(10)) = 62(10)) =-(-4)) (property of entropic measures of quantum correlations).
Assume that 61 and 62 place the same ordering on all states. Since 6(1,0)) obviously
covers the whole range, one can always find two pure states 1@) and 10) such that
34
61 (1V)))
=
6
Si(p) + e and 61(|#))
By the equivalence of 61 and
62
i(p) -- e for any mixed state p (e > 0). Therefore
on pure states,
6i())
6 2(P)
> 61(I#)),
(2.18)
i.e.,
61(P) + f >_ 62(P) > 61(P) - .(2.19)
Taking e to zero, we see that
6
i(p) =
6
2(p)
for all p: they are the same thing. For
them to be different measures, the partial orders have to be different.
0
This theorem indicates that any entropic measure is not a monotone of another.
From the perspective of resource theory, these measures serve to quantify the usefulness of a state for some certain tasks. In this sense, they are not universal: it
only makes sense to talk about a certain measure in relation to specific information
processing tasks (giving it an operational interpretation), since more resourceful state
for this task might be less resourceful for another.
Comparison to entanglement
For general mixed states, many different measures of entanglement, including entanglement cost [121, distillable entanglement [12, 84] etc. have been proposed 5 Here
I pick one of the most widely used entropic measures of entanglement, namely Entanglement of Formation (EoF) [121, for comparison with the measures of quantum
correlations:
Definition 7 (Entanglement of Formation (EoF)). For a bipartite statae
'See 1501 for a review.
35
PAB,
EoF
is an entanglement measure defined as
EF(PAB) =
mm
piS(p ),
(2.20)
where the minimization is over all ensembles of pure states {pi, 1I0)} such that PAB
=
E pil10)(Vi1, and p' = trBI4~)4.
The following Koashi-Winter relation [591 establishes a quantitative dual relation
involving EoF and QD distributed in a tripartite system, due to the monogamy of
entanglement measures:
EF(A: B)
=
D(C -4 B) + S(A) - S(AB),
(2.21)
where S(-) denotes von Neumann entropy. Already knowing that the set of 7 states
(classically correlated states) is strictly contained in that of separable states (on which
any entanglement measure should vanish), and that LOCC can create quantum correlations but not entanglement, one may speculate that entanglement should always
be considered as only a portion of quantum correlations since measures of quantum
correlations beyond entanglement should aim at capturing all correlations that do not
exhibit classicality. However, both as entropic measures that share the same dimension (bits), EoF can be larger than QD for some states [25]: they do not obey a strict
ordering. This observation is directly supported by the so-called "quantum conservation law" [37], a further implication of the above Koashi-Winter relation, which
states that for a tripartite pure state, one can pick a particular subsystem, and the
total amount of EoF that the other two subsystems share with this one cannot be
increased without increasing the total amount of QD, by the same amount:
EF(A : B) + EF(A : C) = D(B -- A)+ D(C -+ A),
(2.22)
when we pick A. One can immediately tell that EoF and (one-way) QD for the two
pairs of subsystems (AB and AC) are either both equal, or ordered differently. Similar
dual relations for other measures of entanglement have also been studied in literature,
36
e.g., distillable entanglement [97]. In summary, further considerations are needed in
fully explaining the relationship between entanglement and quantum correlations in
mixed states.
Comparison to the total amount of correlations
Analogous to the classical case, entropy (information) shared by two quantum systems
captures the total amount of correlations between them. A formal definition is as
follows:
Definition 8 (Mutual Information (MI)). The total amount of correlations between
two subsystems of a bipartite quantum state is given by quantum Mutual Information
(MI):
I(A: B) = S(A) + S(B) - S(AB),
(2.23)
where S(-) denotes von Neumann entropy. Equivalently, we can write MI as
I(A: B) = S(pABIIpA 0 PB),
(2.24)
where S(-1.-) is the relative entropy.
It is obvious that QD < MI since the classical correlation CA(A: B) > 0; For the
other two measures of quantum correlations, S(A) + S(B)
S(AB) under any local
measurement: minJEP, DD < MI.
For entanglement measures of mixed states, it can be shown that the amount
of purely classical correlations (denoted PCC in the following) lower bounds some
entanglement meausures:
ED(A : B),
CAB( A : B)
(2.25)
where ED(.) denotes Distillable Entanglement (DE) 1100], and
CA'(A : B) < Ec(A: B),
37
(2.26)
where COB(A: B)= limT(o[CAB(pAOB)/n] is the regularized PCC, and Ec(.) denotes
Entanglement Cost (EC) [991, which is equivalent to the regularized version of EoF,
i.e., Ec(p) = EP (p) [82]. There are also indications (without general proof though,
as claimed in
1821)
that EoF is additive, meaning that
EF(p) =
EF(p). Therefore
I treat EC and EoF as a whole for the moment. Combining with the known result
that EoF is lower bounded by DE, which reflects the irreversibility of the process of
formation, we have DE < EC, EoF for the entanglement measures that I mentioned.
Note that CA(A : B)
CAB(A : B) always holds, so I do not distinguish them in
PCC for now. Summarizing, we have the following hierarchy:
PCC < DE < EC, EoF.
(2.27)
However, it turns out that the above hierarchy is not strictly upper bounded by MI,
which is discussed in the following:
Lemma 4. While D(A -+ B) < S(A) is always true [32] (note that A is the subsystem
being measured), D(A -+ B)
S(B) does not hold generally, though it is true in most
cases [621.
Theorem 5. EF(A : B)
I(A : B) (EoF < MI) does not always hold. It fails for
the same set of states such that D(A -+ B)
S(B) fails.
Proof. Combining the definition of MI and Koashi-Winter relation in Eq. (2.21), we
have
EF(A: B) - I(A: B) = D(C -+ B) - S(B).
(2.28)
By Lemma 4, the right hand side does not have a strict relation compared to zero.
Consequently, EoF < MI fails generally, but for the same set of states such that
D(A -+ B)
S(B) does not hold.
Not being able to find any related discussions in literature about this result, I may
investigate into it in more detail as future work.
38
Summary
I conclude all the above results in the following hierarchy:
QD < minJEP < DD < MI,
(2.29)
PCC < DE < EC, EoF,
(2.30)
where the bold ones are measures of quantum correlations beyond entanglement, and
the italic ones are entanglement measures. Note that the second line does not fit into
the first line since no strict relations can been established. I argue that QD and DD
are respectively the lower and upper bound.
Importantly, this hierarchy collapses
for pure states: all quantum correlations can be identified as entanglement, which is
simply measured by the marginal entropy.
2.2.3
Multipartite generalization
In the previous section, the entropic measures of quantum correlations were originally
defined for correlations in bipartite states, and I presented a unified way to view all
of them as quantities depending on joint and local entropy productions upon local
measurements. This directly leads to natural generalizations of multipartite versions
for all measures, as will be introduced in succession:
Definition 9 (Multipartite QD). Consider a quantum system consisting of n subsystems A 1 through An. We can quantify the amount of quantum correlations among
these parties by the minimum difference between joint and the sum of local entropy
productions upon local measurements on some specified subsystems labeled by
j
E [n]
(call the whole set of chosen indices J: j E J), i.e.,
Dj(PA,...An)= min
[SOA ...
A.) - S(pAi...AJ - Z[S(OAj) - S(PA,)]1,
(2.31)
where the minimization is over all local von Neumann measurements on Aj, and
PiA
...An
and
PA, are postmeasurement joint and reduced states respectively.
39
The multipartite generalization QD is previously studied in a different form in
[87] (named as global QD, or GQD). The main idea is to rewrite the bipartite QD in
terms of relative entropies and use this form to make the generalization:
DGQD(PA 1... An)-=
min S(PA 1
-...A.
(2.32)
S(PA IIA)1
-
A 1 --A
Note that in the original definition the local measurements are made on all n subsystems. For the more general case of making local measurements on Aj where
(denote this GQD as DGQD,J), we simply need to substitute E>"=
by Ej,.
jEj
It is
easy to show that our generalization of QD is completely equivalent to GQD:
Theorem 6. DJ(PA1 . . An) = DGQD,J(PA 1 ...A,).
Elements (labels of the subsystems
being measured) in J are denoted by J1 up to jj).
Proof. It is well-known that S(PAi IIfiAi) = S(fiA) - S(pA) for global projectors [761.
So I just need to show the equivalence of the first term. Similar to Theorem 1, we
have
=
Z
(nki }
Aii
~
..
(flk
®
~
17.
®
)l.Af(Ilk,
®
...
,)
(2.33)
{
A
where H's are local projectors.
Each k goes up to the dimension of its subspace.
40
Likewise,
S(fiAl...A.)
--tr
il
A... A. 109g~ . A.--,)
-
-
-tr
=
Ir~kj
)PA 1... A -(r . .(, i ) logP Ai...A.]
0 -0
--
T@
o ...
H kj
}
(HU
An>31 I
=
-tr
{njr
=
..{r.}
ni
-tr(pA
=
0Aj
(11k
PA 1 ...A,
0 Hkis)
(2.34)
1
{
(2.34)
.. ,log pA...A.),
where the third equality follows from the fact that ().
with
D
}
---
-tr
)PA...A. logpA...A.(lk,
A1
0...
0f - -- j' ) commutes
jI7|
and thus its logarithm, the fourth equality uses the cyclic property of
k ) is idempotent,
and the last equality uses
trace and the fact that ([P1A3 , (D . (D0 - Au
moetadte
ateuaiyue
5
the fact that they sum to identity. Therefore
S(fi
... A.) -
S(PA1 ... An)
-
-tr(A
=
S(PA 1...AI||VAi... A).
1
... A. log pAi...A A)
+tr(pA,...An log pA 1...A.)
(2.35)
Since corresponding terms with respect to the subsystems are equal, the overall quantity is also minimized by the same set of local measurements.
Therefore the two
El
definitions are equivalent.
For minJEP and DD, since local entropy productions are not involved in their
definitions, it is even easier to make multipartite generalizations:
41
Definition 10 (Multipartite minJEP). Using the same assumptions and notations
for defining multipartite QD, the multipartite generalization of minJEP is defined as
A) - S(pAl...A)].
M-J(PA..A) = min[S(PA 1...
(2.36)
Definition 11 (Multipartite DD). Similarly, the multipartite generalization of DD
is defined as
DDj = S( ,A...A) - S(PA...A.),
(2.37)
where all local measurements are done in the local spectral basis (with respect to
which the reduced state is diagonal).
Using the above ideas, several properties of multipartite quantum correlations can
be shown. Moreover, interesting connections may possibly be drawn to the study of
genuine multipartite correlations [13].
2.3
Scaling behaviors in spin lattices
I have already discussed how to understand and measure different classes of correlations living in quantum systems from an entropic point of view in previous sections
of this thesis, and now we can apply the knowledge to analyze real physical systems.
Since its origination, the concept of correlation has always played an essential role
in studying many-body systems, e.g., spin lattices, since it indicates how numerous
elements of the complex system may behave collectively. For quantum many-body
physics, quantum entanglement may prove crucial for studying anomalous nonclassical phenomena, such as quantum phase transitions [88].
Moreover, the scaling
behaviors of entanglement, especially in ground (zero temperature) or thermal states,
have been shown to exhibit counterintuitive properties, e.g., the entanglement entropy frequently scales as the area enclosing a subsystem, instead of its volume, in
ground states [36], i.e., obeying the so-called area law. This type of scaling behavior
connected to other aspects and topics in fundamental physics, such as black holes and
holographic principles. However, as mentioned previously, people found that nontriv-
42
ial quantum correlations also exist in certain unentangled quantum systems, leading
to nonclassical behaviors, which directly raises lots of questions, e.g., how do they
behave in certain many-body systems? And how do we characterize them?
In this section, I study the behaviors of quantum correlations in a typical model of
quantum many-body systems - spin lattices, with the aid of the results provided by
Section 2.2. First I shall introduce the calculation of measures of quantum correlations
between pairs of spins in spin-1/2 lattice models with Z2 symmetry (no Z 2 symmetry
breaking terms in the Hamiltonian, e.g., magnetic field).
As the two-site reduced
states are generally mixed, as we will see, the analysis of quantum correlations beyond
entanglement may provide new physical insights.
discussed as a specific example.
Heisenberg XXZ chain will be
Then for general dimensions, I shall discuss the
scaling behaviors of quantum correlations between a chosen part of the system with
the environment, in analogy to area law.
Note that I shall use QD and DD as
the typical measures of quantum correlations in this chapter, since they respectively
represent the tighter and looser reasonable quantifiers, as shown previously.
2.3.1
Quantum correlation between two spins
In order to calculate the correlation between two sites in a lattice, we need the joint
state of these two spins. First I shall discuss some general properties of the reduced
state of two sites in quantum spin models obeying Z2 symmetry as preliminaries for
further results.
It has been shown in [541 that in the computational basis
j
110), 111)},
(tracing out other sites) in the Z 2
-
the reduced density matrix of two sites i and
{l00), 101),
symmetric quantum spin models takes the form
pg
oo
0
0
Lo3
0
0
91
Q12
0
Q12
922
0
\03
0
0
L33
43
,
(2.38)
with only the diagonal and anti-diagonal entries being non-zero, therefore bears the
name of "X state", which is mixed in general: measures of quantum correlations do
not simply reduce to forms of entanglement entropy. Here we only need to consider
real X state, i.e., 912
*12 and 0
=
0 3 Qs,
since they can always be transformed into
Eq. (2.38) via local unitary transformations, which do not disturb the correlation.
Considering the trace constraint, we see that this density matrix actually has only
five degrees of freedom.
For some of these states, QD and DD can be explicitly evaluated.
Since the
detailed calculation is lengthy and technical, I shall place all intermediate steps in
Appendix A, and directly present the final results here.
For the following two classes of real X states, one-way QD can be explicitly calculated:
1. IV/0oo33
19121 + 10031:
P1Q221
-
D(j -+ i)
log A+,
-A+,
-
4
log A+,-
A+,
1 +GT
+ E A, log A. -
(I+ G') -
2
1-G
22log log (I
G') + 1,
a=1
(2.39)
2. (10121 + IL03)
2
(GoO - Lu)(L33 - 022):
D(j-
i)
S
=-
Ak,I
log Ak,I
{k=0,1}
It=+,-}
41
A, log A,
+
+Gz
-
2
1G
log (1 + G')
-
2
log (1 - G') + 1,
a=1
(2.40)
44
where G's denote correlation functions
Gz = (of) = tr(ou pij) =
Poo
Pu -
G- = (or) = tr(ojp~i) =
Poo -
P1 + P22
0 o-)pij] =
= (ouff) = tr[(ou
G-
-
oo - L1
= (oafuj) = tr[(ou 0 o )pij) =
Q33,
(2.41)
233,
(2.42)
(912 + 003),
9 o0j')pij] = 2 (912
G Y = (uroj) = tr[(o
G
2
P22 -
-
(2.43)
003),
(2.44)
022 + 933,
(2.45)
and
A = (1 + Gx + GYF - G ),
(2.46)
(1 - Gf - Gg - Gff),
(2.47)
S =
- GYF) 2 ),
(2.48)
(1 + Gzz - V4(Gz)2 + (GX - GY)2),
(2.49)
A+,i = A-,
A3 =
(1+Gij
4=
/+4(Gf)2
+ (G:
"t
+(Gj)2
(2.50)
A o,+ = 1(1 + Gj t Gz
Gz),
(2.51)
t Gf).
(2.52)
Ai=
=
(
(1 - G : G
On the other hand, since the computational basis is automatically the Schmidt basis,
DD of these states are given by
DD(j
=
--
i)
Ak,1
-
log Ak,1
{k=0,1}
{'=+,-}
+
A, log Aa-
1 +oz
log (1 + G') -
1
- 2
G.
log (1 - Gf) + 1,
a=1
(2.53)
using the above notations, and can be extended to all real X states that are outside
45
these two regimes.
2.3.2
Example: 1D Heisenberg XXZ chain
In this subsection, the two-site scaling behaviors of correlation measures is preliminarily illustrated via a specific example: the 1D spin-1/2 anisotropic Heisenberg XXZ
spin model, whose Hamiltonian reads
Hxxz(A) = Z(ui
+ (70ol
(2.54)
+ Ao4i ),
where the anisotropy parameter A controls the quantum phases.
model can be solved by the Bethe ansatz [261.
Note that this
For A > 1, the system is in the
antiferromagnetic N6el phase which breaks the lattice translation symmetry, and for
A < -1,
the ferromagnetic Ising phase, which breaks the spin reflection symmetry.
Both of the above phases are gapped and have two-fold degenerate groung states.
A -+ +oo and A -4 -oo
are respectively the antiferromagnetic and classical Ising
limit. The model is in the critical XY phase (i.e., gapless) when A E (-1, 1], which
is known to be described by a c = 1 conformal field theory (CFT), as the correlation
length diverges and the system becomes scale invariant
[361.
Note that the XXZ chain exhibits U(1) invariance [89], namely, [H, &
=
0,
which is even a stronger constraint over the elements of the density matrix than the
Z 2 symmetry, i.e., the two-site reduced state is an X state and
903
also vanishes,
hence we are safe to use results previous results.
Quantum entanglement in many-body systems, especially of ground states, are
important in studying the behavior of the systems, e.g., quantum phase transitions
[88]. For a pure bipartite quantum state, the entanglement entropy corresponding to
a certain partition is uniquely defined, which is very useful for indicating quantum
criticality. The volumetric entanglement entropy scaling in different regimes of the
XXZ model is discussed in [26, 2]. In the 1D critical regimes (in this model, A E
46
(-1,11), CFT yields that the (subsystem) entanglement entropy scales as
SA(1) =
6
(2.55)
log 1 + k,
where k is a model-dependent constant and c, i are holomorphic and antiholomorphic
central charges respectively [211, indicating different universality classes.
Since the pairwise correlation in gapped phases is naturally expected to decay
exponentially, I now analyze the two-site scaling of QD and DD in the critical phase
where A E (-1, 1) at zero temperature. In this gapless regime, using our notations,
the pairwise spin-correlation functions scales as [67]
Gx = G' ~, li- j,-0,
(2.56)
j-2+e2ikFliiI j -~j-,
Gf ~ i
(2.57)
with critical exponent given by
1
2
(2.58)
arcsin(-A)
7r
and e2ikFIa'I a phase factor. Notice that although the leading order term in Gf
is
not fixed, Gf is always less significant than G-, G'.
Plugging these spin correlation functions into the parametrization Eq.
(2.41)-
(2.45), we see that this state falls into the first class: the exact QD is given by Eq.
(2.39). Note that G should vanish. By Taylor expanding Eq. (2.39), up to leading
order, QD scales as
D(j -+ i) ~ (Gf) 2
-
2,
(2.59)
and it turns out that DD obeys the same scaling:
DD(j -+ i) ~ -ijl-20,
(2.60)
by expanding Eq. (2.40). Therefore in the critical XY phase at zero temperature,
measures of quantum correlations beyond entanglement decay polynomially, which
47
resembles the behavior of spin correlation function, but with different exponents.
It is also interesting to study the behavior of quantum correlations depending on
the temperature of the bath, e.g., in ID XYZ chains, QD increases as the bath
temperature grows, while entanglement is expected to decay [1031.
2.3.3
Area laws
When the interactions in quantum many-body systems are local, the ground state
entanglement entropy typically grows linearly with respect to the boundary area of
the subregion instead of the volume, in contrast with the expected extensive behavior. This kind of scaling behavior is said to obey an "area law" [36]. The general
mathematical statement if a physical quantity -1 of region A obeys
4D(A) = O(b9AI),
(2.61)
where OA denotes the boundary area of A, we say that the area law is satisfied. The
intuitive picture here is shown in Fig. 2-1. In locally interacting noncritical systems,
the correlation length is finite so that sites in A and B that are separated by a distance
further than the correlation length (the shaded stripe) should not contribute to the
mutual information or correlation measures between A and B, hence bounded by the
number of sites at the boundary and therefore scales as the boundary area.
It is worth mentioning that the area laws for entanglement entropy has deep
connections [94, 14] with the famous area dependence of the Bekenstein-Hawking
black hole entropy [9, 45], which states that the entropy of a black hole is proportional
to its horizon area A:
SBH
kA
-
42
4lp
where lp
=
~
kc3 A
h'
4Gh'
(2.62)
('
VGh/c3 is the Planck length, as they share similar scaling behaviors.
These discoveries of black hole entropy scaling laws were the driving force for several
studies of entanglement entropy scaling in quantum fields later on [46, 22].
It has
been argued that the information contained in a volume of space can be represented
48
000000004000000000
V00
00
00
0000000
.0002
00oo
00
00000000
000
000
00
000
00o
.000
c00
000
000
0
o
00
000
0000
0000
00000000000000000
Figure 2-1: (Adapted from [1041) Area law. Intuitively speaking, under locally interacting Hamiltonians, the correlation length in noncritical phases is finite so that
sites in A and B that are separated by a distance further than the correlation length
(the shaded stripe) should not contribute to the mutual information or correlation
measures between A and B, hence bounded by the number of sites at the boundary
and therefore scales as the boundary area.
by a theory which lives on the boundary of that region [98, 15], which is widely known
as the holographic principle - the information contained by a region depends on its
surface area, rather than on its volume. This insight remains one of the central topics
in theoretical physics to this day. The famous AdS/CFT correspondence [681 is a
beautiful realization of the holographic principle. At a fundamental level, all these
area laws might be related.
In this section I shall briefly discuss the area laws for entanglement entropy and
total mutual information for general spin systems, and in turn present the area law
for measures of quantum correlations in locally interacting noncritical spin systems.
Entanglement entropy
The area laws of entanglement entropy in various contexts have been been extensively
studied for years'. For quantum many-body systems on lattice (the entire system) W
where A is a subregion and B = W \ A its complement, S(A)
6
Refer to
1361 for a review.
49
=
O(IAI) implies that
the area law is satisfied in the system. We emphasize that, in fact, it is truly unusual
for a quantum state to satisfy an area law
1361 as it has been shown that the typical
entropy of a subsystem is nearly maximal [90], indicating that it should scale as the
volume instead of boundary area. For the purpose in this section it suffices to know
that for general ground states of quantum spin systems in gapped, i.e., noncritical
phases the area law is obeyed. The area law statement was first made rigorous by
Hastings in 1D [44], and recently the idea that exponential decay of correlations leads
to area law scaling of entanglement was also formally shown in ID [171 (quantum data
hiding states being obstacles of this argument in general dimensions [341). Notable
violations take place at quantum criticalities, models with Fermi surfaces etc., and
the non-trivial topological order will result in a negative term in the ground state
entanglement entropy, namely topological entanglement entropy. The analysis on 1D
Heisenberg XXZ chain earlier can serve as a nice example: in noncritical regimes,
quantum correlation between two spins decays exponentially as the distance between
them grows, indicating that the entanglement entropy fulfills the area law. While for
critical regimes, QD and DD are both shown to decay polynomially at zero temperature. Indeed, area law should fail in this case - quantum correlation is not local
(finite-ranged) anymore.
Mutual Information
As has been mentioned earlier, the total correlation in a bipartite quantum state
-
AB is given by quantum Mutual Information (MI), i.e., I(A: B) = S(A) + S(B)
S(AB), where S(.) denotes the von Neumann entropy. Now consider states in thermal
equilibrium, i.e., thermal states, which take the form PAB
-
e
H /tre-H
with inverse
temperature 3. Thermal states minimizes the free energy F(p) = tr(Hp) - S(p)/,
and F(PAB) < F(PA 0 PB), from which one can obtain the following
HB +
HA
+
Lemma 7 (Area law for MI [104]). We denote the total Hamiltonian as H
1&H
where HO collects interactions crossing the boundary. The MI of a thermal
50
state is bounded by a first order function of Ha, i.e.,
I(A : B)
/3tr[Ha(pA 0 PB
(2-63)
- PAB)] ,
the right hand side of which only depends on the size of the boundary. Therefore the
area law is satisfied.
More specifically if we only consider two-site interactions, we will have
I(A : B) < 20A max I|hijiI,
(2.64)
i,jcaA
where ||hjj|| are eigenvalues (strengths) of the two-site interaction between i and j
(across the boundary).
Notably, it is also shown that finite correlation length generally implies an area
law scaling of MI [1041.
Quantum correlations
Therefore for thermal states, the following scaling theorem for QD and DD can be
directly proven by the hierarchy in Eq. (2.29):
Theorem 8 (Area law for quantum correlations). For general quantum spin systems
in noncritical regime with local interactions, the amount of quantum correlations between a subgraph A and its complement in thermal states scales as the boundary area.
Proof. Denote the entire spin system as W and B = W \ A. As shown in Section
2.2.2, we have the hierarchy QD < DD < MI for the same state, i.e.
D(A -4 B)
DD(A -+ B) < I(A : B) </3tr[H(pA 0 PB
-
PAB)],
(2.65)
where I used Lemma 7 for the last inequality. This implies that QD and DD both
satisfy the area law, and therefore other entropic measures of quantum correlations.
51
Specifically for two-site interactions [54], by Eq. (2.64), we immediately see that
D(A -+ B) < DD(A -+ B) < I( A: B) < 2/310A max
ij EOA
Conclusively, D(A
-+
B), DD(A
-
B)
I|hjjI|,
(2.66)
|OAI,
I i.e., scales with the boundary area.
The specific argument about QD can be found in [54].
2.3.4
El
Outlook
As indicated by the holographic principle, the information of a region should depend
on its surface area instead of volume, at the fundamental level. Therefore studies on
scaling behaviors of correlation measures in many-body systems may prove fruitful for
both the fields of quantum information theory and many-body statistical physics, and
may even provide some unique insights into other aspects of physics. New findings
may emerge at this interface.
2.4
Diagonal Discord (DD)
In Section 2.2, DD, which can be thought of as the simplified version of QD by fixing
the local measurement to Schmidt basis measurement, or the joint entropy production
upon such a local measurement, was proposed as a candidate measure of quantum
correlations beyond entanglement. (The formal definitions of DD and its multipartite
generalization were given in Definitions 5 and 11).
Although this quantity has showed up in other contexts [60, 65], its properties
and physical interpretations have not been well explored yet. In this section, I shall
systematically discuss the motivations for using this quantification for quantum correlations, and some of its important features, as well as possible future directions.
2.4.1
Motivation
The quantification of quantum correlations is based on local measurements.
Since
there are infinitely many ways to measure a quantum system, a perfect quantification
52
of quantum correlations has to involve an optimization over all possible measurements,
which seems extremely difficult for evaluation. Indeed, computing QD is shown to
be an NP-complete problem [53], and this proof can be directly extended to other
entropic measures involving optimization over all possible measurements, such as
minJEP. The rigorous statement goes as follows:
Theorem 9. For some measure of quantum correlations6 that involves optimization
of an entropic quantity over all possible measurements (e.g., QD or minJEP), given a
bipartite quantum state PAB of dimension m x n and a real number b with the promise
that either (Y) S(A
-+
B) < b or (N) 6(A -+ B) > b+c where e is inverse polynomial
in m, n (E = 1/poly(m, n)), it is NP-complete to decide which is the case.
Proof. The NP-completeness of QD along with several entanglement measures including EoF and EC, is shown in [531, by reducing the problems of computing these
measures to the problem of separability (polynomially), which is known to be NPcomplete [56]. Notably, QD and EoF is connected by Koashi-Winter relation, which
indicate the polynomial reduction. Similarly, computing other entropic measures of
quantum correlations involving optimization terms can be polynomially reduced to
computing QD, thus is NP-complete.
El
Therefore, the primary advantage of DD is that it is much easier to evaluate, since
the complexity of diagonalizing a matrix is generally polynomial. Based on KoashiWinter relation (Eq. (2.21)), we can define the following entanglement measure dual
to DD:
Definition 12 (Diagonal Entanglement of Formation (DEoF)). Koashi-Winter relation establishes a duality between EoF and QD (between different parties).
By
substituting the QD term in Koashi-Winter relation by DD, an analogous dual relation can be obtained:
EF(A: B) = DD(C -+ B) + S(A) - S(AB),
(2.67)
the left hand side of which is named DEoF. Similar as DD compared to QD, DEoF is a
53
simplified version of EoF without the optimization, which is much easier to compute.
2.4.2
Properties
Due to the significant simplification, DD is not expected to capture all aspects of
quantum correlations. However, since it is easy to calculate, this quantity will be of
much more practical value if we can argue that DD satisfies the basic criteria that a
reasonable measure should possess, and still works well in various scenarios. Now I
investigate some of the properties of DD.
I shall first explicitly argue that DD satisfies the basic criteria of an acceptable
measure of quantum correlations discussed in Section 2.2.1:
1. DD is nonnegative.
Moreover, the null set of one-way DD (without loss of
generality, assume the local measurement is done on on A) is exactly the set of
CQ states, (for the two-way case, the set of CC states), i.e., the same as QD
and any other reasonable measure of quantum correlations.
Theorem 10. For any bipartite state PAB, DD is nonnegative, i.e., DD(A -+
B) > 0.
Proof. Since QD is nonnegative
[32], and QD < DD, DD is nonegative. Or
directly by Theorem 1 which states that local projective measurements can never
decrease entropy, DD(A -+ B) > 0 since DD is by definition the joint entropy
production upon local measurement in the Schmidt basis (projective).
Theorem 11. For a bipartite state PAB, DD(A -+ B)
is exactly the null set of QD. Likewise, DD(A ++ B)
-
El
0 iff PAB is CQ, which
0 iff
PAB
is CC, also
same as QD. The above null sets are also shared by minJEP.
Proof. Necessity: As has been discussed, the states with zero one-way QD is
called CQ states, which take the form PAB = EkpkIk)A(ki & pB, where {Ik)A}
is automatically the Schmidt basis, implying that DD vanishes.
54
Sufficiency: DD(A -+ B)
=
0 implies that the diagonal basis gives zero discord
value, which is minimal, i.e., exactly the value of QD, since QD is nonnegative
by Theorem 10.
The above arguments can be directly generalized to the two-way case. Since
QD
and DD are respectively lower and upper bounds of minJEP, the null sets
l
of these quantities must be the same.
As mentioned in Section 2.1.1, the set of 7 states is measure zero and nowhere
dense in the space of al quantum states: almost all quantum states have positive
QD and DD [381.
2. DD is invariant under local unitary transformations. This argument is included
in the following theorem that applies for more general cases:
Theorem 12. Suppose U and V are respectively arbitraryunitarieson the subspaces A and B of a bipartite quantum state pAB, then any entropic measures
for quantum correlations, including any DD (namely, one-way from any subsystem, two-way) of the state P'AB
=
(U 0 V)pAB(Ut 0 Vt), is the same as that
of the original state PABProof. The essence is that local unitaries can be thought of pure rotations of
basis vectors of the local Hilbert space since they preserve distances. Without
loss of generality, use the computational basis
resentation of
PAB,
and use {Uji) 0 Vjj)} for
{ ij)} to write the matrix repP'AB.
One can directly tell that
these two matrices are completely identical. Therefore all entropic quantities of
PAB and P'B, and consequently all entropic measures of quantum correlations,
are the same.
l
From the above, DD survives the basic criteria as an acceptable measure of quantum correlations beyond entanglement.
In fact, DD also exhibits several other im-
portant properties. Compared to QD and other entropic measures:
3. QD < minJEP < DD (Section 2.2.2).
55
4. The partial order of quantum states determined by DD is different from those
determined by other entropic measures, e.g., QD and minJEP, given that they
reduce to the same quantity for pure states (Theorem 3 in Section 2.2.2). There
are several equivalent statements, e.g., there must exist some states whose order
of resourcefulness given by different measures are different, or these measures
are not monotones of one another.
Independently, DD itself also exhibits the following features:
5. DD is upper bounded by the von Neumann entropy of the subsystem being
measured.
Conjecture 1. Given a bipartite quantum state
PAB,
DD(A -+ B) < S(A),
where S(-) denotes von Neumann entropy.
I currently state it as a conjecture due to the lack of rigorous proof.
Note
that if S(A) (the subsystem being measured) is replaced by S(B) (the other
subsystem), this inequality does not always hold since it is not even generally
true for QD, as discussed earlier.
6. DD is not monogamous for all states, which is a direct implication of [95].
The statement only excludes the possibility that all quantum states satisfy
monogamy of measures of quantum correlations. The complete characterization
of the states that are monogamous for different measures is an interesting open
problem which may provide insights for the nature of quantum correlations.
7. DD is invariant upon attaching pure ancilla to any local subsystem.
Theorem 13. The operation of attaching local pure ancilla on any subsystem
of a bipartite quantum state PAB preserves DD, i.e.,
DD(A -+ B),
(2.68)
DD(AC -+ B)
DD(A -+ B),
(2.69)
where PABC = PAB 0 1,0)c(VI
-
DD(A -+ BC)
56
Proof. By Theorem 12, local unitaries do not vary DD. One can always find a
unitary U acting on subsystem C such that U14')c =
I0)c. Now Eqs. (2.68)
and (2.69) are true by inspection, since the density matrix of
PCAB
takes the
form
PAB 0
... 0
0
(2.70)
UPCABUt =
0
0/
i.e., only the top-left block being nonzero
(PAB).
It is now easy to tell that all
eigenvalues involved remains invariant (only some more O's), thus the entropic
quantities as well, which completes the proof.
8. For quantum lattice systems in thermal equilibrium (at finite temperature), the
scaling of DD satisfies an area law (Theorem 8 in Section 2.3.3).
Notably, the above features are discussed under the bipartite and one-way setting for
simplicity, but they can be easily generalized to two-way or multipartite cases, as I
have done several times in this thesis.
This incomplete list includes some of the important properties that one would
naturally expect DD to exhibit as a good candidate. In these statements, "DD" can
be replaced by "well-behaved measures of quantum correlations". That is, although
the quantity is defined with a significant simplification, DD still works well as a
candidate measure, making it quite favorable in practice. Note that there does exist
a simple feature that is exhibited by QD [1081 but not DD, namely, continuity:
9. DD is not a continous function of density matrices, i.e., there exist states that
are arbitrarily close, but have non-vanishing gap in DD. Examples can be found
in [107].
Further discussions are needed for the consequences of this property.
For the moment I omitted some properties that do not seem very meaningful,
or have not been rigorously proved.
To reveal the complete nature of DD, and in
57
particular, what kind of role it should play among measures of quantum correlations
in various physical scenarios, is a fruitful research direction to pursue. Specifically,
the list of interesting open questions concerning the mathematical properties of DD
may include, but not restricted to:
" Is there a good upper bound for the difference between DD and QD?
" As several measures including QD and geometric discord etc. are known to
exhibit non-contractivity under general local operations (more specifically, nonunitary evolutions without measurements or attaching product states) while
some others do not, what about DD? (In Section 3.2 I shall elaborate more
on this issue. Locally even tracing out orthogonal flags may generate quantum
correlations from -y states, thus is forbidden for this question.)
Generally, the characterization of the sets of states where different inequalities about
DD reduce to equalities, and the comparison with those sets for QD, may provide
useful insights of DD and quantum correlations as well:
" For what states is DD equal to the entropy of the subsystem being measured?
Is this set different from that of QD?
" For what states does the monogamy of DD hold? Is this set different from that
of QD?
" For what states does the hierarchy QD < minJEP < DD collapse, i.e., the
optimal basis is exactly the Schmidt basis?
Answers to these questions may provide significant progress and insights for related
areas.
Besides its mathematical nature, the understanding and significance of DD
associated with physical phenomena will be among the topics of the following section
and Chapter 3.
2.4.3
Physical interpretation
From the discussions previously, one may already sense from several aspects that the
concept of quantum correlations is in very close relation to thermodynamics. Indeed,
58
the idea of QD was kind of originated from such a context [77, 109J.
Remarkably, it can be further shown that positive quantum correlations beyond
entanglement is the necessary and sufficient condition of generating heat transfer
between quantum systems
1641.
I shall only briefly mention the results of this work
here without going into details. The rigorous theorem goes as follows:
Theorem 14. Consider two quantum systems A and B evolving continuously from
an initialstate PAB (0) under a unitary time evolution UAB (t). If the joint state PAB (t)
remains CC for all t, then the time evolution for A and B can always be written as
(2.71)
pAB(t) = UA(t) 0 UB(t) pAB(0) Uj4(t) 0 UB(t),
i.e., they are effectively non-interacting, thus heat transfer cannot be generated.
A step further, a quantitative relation between the rate of heat transfer and generation of quantum correlations (measured by DD) over a sufficiently short period
of time At can be established.
Suppose A and B respectively starts from un-
correlated thermal states at temperature T, i.e., the initial state reads
(e-AHA/IZA)
PAB(0)
=
3 = 1/(kT), then under H = HA + HB + HAB,
0 (e -,HB/ZB) where
the rate of energy transport turns out to be directly proportional to the rate of DD
generation in the infinitesimal time limit: By Eq. (2.14), the infinitesimal generation
of DD can be written as
ADD(B -+ A)
- SAt(AB)
= St(A)
=
-trApA3 log PAs
=
tr 3 AHAAPA + trI3 BHBAPB
=
3AAEA + 3BAEB
=
(, 3B -
(2.72)
A)AEB,
where we discarded high order terms for the second line, plugged
third line, and used the fact that AEA = -AEB
PAB(0)
in for the
in the weak coupling limit for the
last line. Indeed, since thermal states are diagonal in the energy eigenbasis, DD has
59
a direct physical interpretation in terms of energy flow and entropy increase, and the
resulting "energetic discord" can be measured experimentally.
Still, it is natural to ask whether the assumption on the local measurement basis for
the measure for quantum correlations we used in establishing the above quantitative
relation can be dropped. Therefore we propose the following
Conjecture 2. In the infinitesimal limit of evolution, DD is equivalent to QD, i.e.,
the Schmidt basis can be considered the optimal basis for QD. In other words, discontinuities of optimal basis do not take place at y states.
Possible ways to prove or disprove this argument may include brute-force calculations of the behaviors of these quantities in the infinitesimal limit, or looking for
counterexamples from the class of states for which the optimal basis is already known,
e.g., the two classes of real X states discussed in Appendix A. Unfortunately, these
two classes do not share boundaries in the state space [27], so the results about them
are not useful for studying the discontinuity of optimal basis, but it might help to
keep looking in this direction.
60
Chapter 3
Quantum Resource Theories (QRTs)
A goal of quantum information theory is to make use of peculiar properties of the
quantum world to achieve effects that are inaccessible classically. The desire to find
a unified and well-structured framework in order to organize these ideas has led to
the notion of Quantum Resource Theory (QRT) [49J. The resource x is usually an
attribute of quantum states. In this sense, QRT of x essentially the theory of x as a
physical resource. A complete theory should be able to tell you basically anything you
want to know about x, including how to create it, how to quantify it, how to convert it,
how to make use of it and so on. If you want to gain full understanding of a quantum
object, a good way to do so is to study or establish the resource theory of it. Quite a
few specific QRTs that find use in a wide range of scenarios has been identified, e.g.,
entanglement [50], stabilizer quantum computation [1011, contextuality [42j, quantum
coherence [8]. Surprisingly, the ideas of QRTs have reached far beyond the scope of
quantum information theory.
Some striking examples are the resource theory of
asymmetry [701 which leads to a generalized Noether's theorem, and the resource
theory of quantum states out of thermal equilibrium [181 etc.
As can be already seen, the spirit of QRT is quite generic, and all QRTs share
a similar kind of structure. Instead of looking at scattered theories for various resources, we ought to depict the whole family of QRTs from a unified point of view.
In this chapter, I shall first discuss the unified framework of traditionalQRTs where
the resources are associated with quantum states, so that the theories of specific re-
61
sources can easily fit into this structure, and work as sub-theories. As mentioned,
people have established lots of QRTs within this framework, among which the QRT
of entanglement, LOCC being the corresponding set of free operations, is the most
well-understood and famous one. Entanglement has long been considered as the most
important resource for various kinds of quantum advantages. However, people found
that quantum correlations beyond entanglement, which is the topic of Chapter 2, are
actually responsible for the triumph of "quantum" in several contexts [33, 30, 81].
Nonetheless, a satisfactory QRT for quantum correlations has not been fully characterized.
Some discussions on this problem will be carried out in Section 3.2. In
fact, analogous to the studies on traditional QRTs, quantum operations can also be
treated as resources, thus a dual framework can be formulated. In Section 3.3, I shall
introduce the ideas of dual QRTs. As will be seen, this duality is quite natural from
a mathematical point of view, and an even more general category-theoretic structure
might be identified.
3.1
Unified framework
The purpose of this section is to establish an underlying unified framework such that
specific QRTs can fit into this structure as a sub-theory.
I shall start from very
intuitive analysis on the whole structure, and then proceed to establishing a more
rigorous mathematical framework.
3.1.1
Elements
To begin with, let's review the general logic of QRTs by considering the basic ingredients that a well-defined QRT should contain. First of all, a particular resource
associated with quantum states serves to label the QRT, along with which there is
a quantifier that determines the "resourcefulness" of a particular state. Correspondingly, a set of free operations which cannot increase the amount of resource should
also be identified. Or in the other way, one can put particular restrictions on the
allowed operations, and then those states that can never be prepared with these op-
62
Free Operations
* 9 CPTP maps
- Cannot increase the amount of resource
-
-+ Set of free states: closed
+
COMPLEMENTARY
Resource
- Usually peculiar quantum properties
- Cannot be created by free operations
- Monotones: quantification
4f
Larger set of accessible operations,
states etc.:
additional power, universality...
Figure 3-1: Intuitive illustration of the basic content and structure of a QRT.
erations become resources.
With the aid of these operations, it is also impossible
to create more resource than before, as measured by the quantifier. The power of
free operations (which can only prepare free states) may be very limited, but with
resources, we are enabled to achieve a certain kind of universality. One should have
already noticed the complementarity between these ingredients.
Certainly, these respective components also have to satisfy some basic properties. For example, the allowed operations should be a strict but not empty subset
of all possible quantum maps/channels (the theory is trivial if all or none operations
are free). Moreover, one would expect that putting garbage together will not create
resource, and a series of free states should not converge to a resourceful one. Mathematically, the set of non-resourceful states should be closed under tensor product,
and is a closed set.
In summary, the basic structure and ingredients of QRTs are illustrated in Fig.
3-1. Note that as expected, an abstract framework that unifies the ideas of QRTs can
be constructed using category theory [29].
63
With the above basic understanding of how QRTs work, we can now give more
rigorous and reliable descriptions of the unified framework for QRTs. Every QRT is
consisted of the following three fundamental pieces of elements, respectively satisfying
several basic postulates:
1. Free operations.
Denote the set of free operations as A 1 . It satisfies:
(a) A1 is the subset of all possible operations (CPTP/CP maps, depending on
the context).
(b) A1 can never increase the amount of resource, as measured by a particular
quantifier.
2. Resourceful states.
Denote the set of resourceful states as R, and its complement, i.e., the set of
free states, as F. F needs to satisfy:
(a) F is closed upon doing tensor product.
(b) F is a closed set.
Note that one can also define the "maximally resourceful states" as those states
that can be deterministically converted to any other state by free operations.
3. Resource monotone/quantifier.
The quantifier is a continuous function of quantum states (denoted by m(p))
that quantify its resourcefulness (ratio determines conversion rate). It determines the partial order of states with nonzero resource. The properties that m
has to satisfy are:
(a) m(p) > 0: always nonnegative.
(b) m(p) = 0 iff p E F.
(c) m(p) is finite (resource should be limited).
64
We have to be very careful in explaining the relationships between these elements.
Representing different objects, these components do possess a certain extent of independence, but are indeed all tied together. The set of free operations Af need not
be maximal. The quantifier is associated with the state, but as will be discussed in
Section 3.1.2, given that Af is not maximal, the quantifier may not be unique: there
can be various measures corresponding to different information processing tasks. For
instance, the entropic measures of entanglement/quantum correlations even give different orderings for entangled/non-y states (Theorem 3). In this case, as mentioned in
Chapter 2, it only makes sense to talk about resource quantifiers in relation to specific
operational contexts involving conversions between resources. This is the reason why
I suggest the monotone/quantifier as an element instead of the resource, since when
the quantifier is not unique, a resource actually corresponds to a class of theories.
Since the second postulate on Af heavily depends on the quantifier, so for this class
of theories the free operation sets and quantifiers are paired up.
Therefore we have the following hierarchy of QRTs:
Level 1 :
{., R, .}
Level 2 :
{Af, R, m}
The "resource" labels the Level 1 QRT, where {Af, m} may be non-unique. This is
the level where usual notions of QRTs are at, e.g., entanglement theory. For Level 2
QRTs, all elements are fixed.
Note that a very recent work [16] introduced a general structure of QRTs in a
slightly different way. The basic ingredients of QRTs as they suggested are i) the
resources (e.g., entanglement); ii) the non-resources or free states (e.g., separable
states); iii) the restricted set of free (or allowed) operations (e.g., LOCC). Here the
ingredient "resources" are essentially the labels of level 1 QRTs (corresponding to a
class of resource quantifiers) in this framework, and the set of free states is the complement of the set of resourceful states. As I discussed in the last paragraph, a resource
may correspond to various theories by my definition. Moreover, [16] proposed some
65
more postulates on F including convexity and closeness under tracing out spatially
separated systems and party swap (which seems redundant), which I do not regard
as basic requirements. But they can be set as restrictions in specific QRTs.
Here are some quintessential examples described by Level 1 language:
1. Entanglement: Af = LOCC.
The resource theory of quantum entanglement has been widely studied and recognized [51, 101. If two or more parties are only allowed to do local operations
and communicate with one another via classical channels (LOCC), then entanglement becomes a corresponding resource in the sense that it can never be
created using LOCC, and furthermore, allows us to perform tasks that are impossible with LOCC only, e.g., quantum teleportation [11], and therefore serves
as the essential element of quantum communication. Entanglement is also identified the indispensable resource for lots of computational and cryptographic
tasks (I shall not delve into details here).
Note that this is a typical example of Level 1 QRT. There are various measures
of mixed state entanglement (note that for pure states, quantum correlations
fully reduce to entanglement, and all entropic measures of entanglement become
equivalent), in relation to different physical scenarios [511, such as EoF, EC,
DE as defined in Chapter 2, each determining the conversion rate under some
specific settings (corresponding to a Level 2 QRT), e.g., distillation and dilution
[75, 10]. The fundamental reason for this non-uniqueness is that LOCC is not
the maximal set of operations that cannot create entanglement (LOCC is a
strict subset of all non-entangling operations), as will be further illustrated in
Section 3.1.2.
Interestingly, if we restrict to pure state entanglement, LOCC
becomes maximal, and the different measures all reduce to a unique one - the
entanglement entropy.
2. Purity: Af = NO.
Purity, which represents knowledge/information of the state, is defined as log dS(p), where S(-) denotes von Neumann entropy (which is automatically the
66
quantifier). Indeed, purity is zero for maximally mixed states (zero knowledge),
and maximal for pure states (full knowledge). The corresponding free operation
set is characterized as Noisy Operations (NO) consisting of
(a) attaching maximally mixed (I/d) ancilla;
(b) doing partial trace (PT);
(c) unitaries (U),
which is the maximal set of operations that can never decrease the entropy: By
Theorem 16, this theory is reversible, thus a perfect QRT (refer to Section 3.1.2).
NO that preserves dimension reduces to Exactly Factorizable maps, which will
be discussed later.
Surprisingly, people have discovered connections between
the resource theory of purity and the generalizations of Birkhoff statements,
which are essential elements in the hierarchy of operations that I shall discuss
in Sec. 3.1.3. The idea of NO is adopted to disprove the asymptotic Birkhoff
conjecture, which claims that the n-fold tensor product of a doubly stochastic
map can always be approximated by mixtures of unitaries in the limit of large
n 192, 91]. The resource theory of purity will also play an important role in that
of quantum correlations beyond entanglement.
3. Magic: Af is stabilizer/Clifford operations.
Natural connections can be drawn between QRTs and universal quantum computation. According to the well-known Gottesman-Knill theorem [41, 75] and
Aaronson's improvement [11, quantum computation based on stabilizer operations (generated by Clifford group elements) can be efficiently simulated on a
classical computer (in classical complexity class EL [11). This seems to have
closed the gap of computational power between quantum and classical computers for some cases, however the point is that with only Clifford operations
we cannot obtain universal fault-tolerant quantum computation. The resource
needed to achieve this universality is then non-stabilizer states, or so-called
magic states [19, 31. In this sense, the resource is the "magic", or negativity
67
Resource
Entanglement
Af
LOCC
F
Separable states o,,
Purity
NO
Maximally mixed state I/d
Magic
Coherence**
Clifford
Stabilizer states -stab
p s.t. Vi
j, Pi= 0
[81
Example m,
infsP R(pI Usep)
R(pj lI/d)= logd -S(p)*
inf,,,tb R(plI Ustab)
Z
piI
Table 3.1: Level 1 descriptions of some typical QRTs. *Since NO is maximal, this
quantifier is unique. **With respect to a particular basis.
of Wigner function. It is worth mentioning that one step further, it has been
shown that contextuality is the actual resource of this magic state model [93, 69].
Similar to entanglement, "magic" can also be quantified by'suitable monotones,
and distilled using the idea of quantum codes
119].
4. Coherence.
The QRT of coherence is systematically discussed in [8]. The authors established a framework for quantifying quantum coherence essentially by identifying
the set of incoherent states (diagonal in the appointed basis) and operations,
with respect to a certain basis. The 11 -norm of coherence, i.e., the sum of absolute values of off-diagonal entries in the density matrix, is suggested as a good
quantifier of coherence.
The elements of the above example QRTs are summarized in Table 3.1.1.
3.1.2
Perfect QRT
According to the discussions above, the theory is not necessarily unique when we only
specify the resource (Level 1), which often makes related topics quite messy, such as
mixed state entanglement. Despite of some practical situations where we have to deal
with those technicalities, we would still want to see a cleaner theoretical framework.
This is the purpose of this subsection.
Intuitively, legitimate monotones should vanish for the free set, and the farther a
state is from free states, the more valuable it should be. In this sense proper distance
measures naturally become candidates of a resource monotone. The quantum relative
68
entropy, which is actually the quantum analog of Kullback-Leibler divergence for two
classical probability distributions, becomes a natural choice.
Definition 13 (Relative entropy). The relative entropy of two density operators is
defined by
S(p lo-) = tr(p log p) - tr(p log o-)
(3.1)
By Klein's inequality, this quantity is nonnegative. Actually the relative entropy
distance is not strictly a distance measure since it is not necessarily symmetric under
party swap, whereas it still only vanishes iff two states are exactly the same. Then
we define the regularized relative entropy distance of a state p to a set of states Q as
Definition 14 (Regularized relative entropy distance). The relative entropy distance
of a state p to a set of states Q is the minimal relative entropy distance between p
and all members of Q:
7(p, Q) = inf S(p||j-).
(3.2)
Then the regularization of this quantity simply reads
q'(p, Q) = lim i
n-+oo
n
(3.3)
A general theorem introduced in [471 indicates that the regularized relative entropy
distance to the free set, which is defined as following, is the unique monotone, as long
as the QRT is reversible. The theorem goes as follows:
Theorem 15 ([471). In nontrivial QRTs (some, but not all states are resourceful),
for two arbitrary states p and a with some amount of resource, i.e., outside F, the
asymptotic conversion rate is given by the ratio of their respective regularized relative
entropy distance to the F:
R(p
-+
-) =
7(O-,
r, )o
(3.4)
given that this theory is asymptotically reversible.
Qualitatively speaking, "reversible" simply means that the conversion between
resources can be two-way, i.e., with the aid of the set of operations specified by the
69
QRT, as long as p*"f can be converted into o-",
U"
can be transformed back into
p "', without loss. The formal definition is as follows:
Definition 15 (Asymptotically reversible QRT [47]). A QRT is asymptotically reversible iff for any two states p and
- we have 0 < R(p -+
R(p -÷
where R(-
-)R(a
p)
-) < 00. and
= 1,
(3.5)
-) denotes the asymptotic conversion rate defined by Definition 3 in [471.
A direct corollary of this theorem is that
Corollary. The regularized relative entropy distance to the free set F, r; (p, F), is
the unique measure of the resourcefulness of a quantum state p in reversible QRTs.
Note that concerning the resource theory of quantum correlations beyond entanglement, 77(p, F) where F is the set of CQ states is shown to be equivalent to one-way
QD [48], which indicates that the QRT of QD as the chosen measure has special
importance.
Remarkably, it is recently shown in [16] that under a similar set of postulates, the
maximality of free operation set indicates reversibility, for a QRT:
Theorem 16 ([16]). A well-defined (satisfying the set of postulates suggested in [16])
QRT is asymptotically reversible, if the set of free operations is maximal, i.e., includes
all operations that cannot increase the amount of resource.
I shall now formally define a special class of QRTs:
Definition 16 (Perfect QRT). The set of free operations of a perfect QRT is maximal
for the resource x, i.e., it contains all possible operations that do not increase x.
It is straightforward to tell that among any Level 1 QRTs there is a single perfect
Level 2 QRT. Note that among the examples I provided in that last section (summarized in Table 3.1.1), only the theory of purity is a perfect QRT. The theory of
entanglement, for example, is not a perfect one, in the sense that LOCC is known to
70
be strictly smaller than the set of all non-entangling operations, leading to various
measures in relation to different conversion tasks. As noted earlier though, the QRT
for pure state entanglement satisfies the requirements for a perfect QRT.
It can be shown that perfect QRTs are unique given knowledge on any of the basic
elements:
Theorem 17. Knowing any of the three basic elements of a QRT gives you a unique
,
perfect one. More specifically, the characterizationof either of the basic elements A 1
R (the resource) or the partialorder of quantum states uniquely determines a perfect
QRT.
Proof. Note that F is simply the complement of R, thus the knowledge of either one
implies the other. Combining Theorem 16 and the above corollary of Theorem 15,
we know that the unique resource monotone is
(p, F) for a perfect QRT. Therefore
full knowledge about this monotone is equivalent to knowing F or R.
If R is specified, then the monotone is directly given by 71'(p, F), and Af of the
corresponding perfect QRT is simply the set that contains all operations A such that
T
(A(p), F) <; q' (p, F),
(3.6)
for all p, which is unique.
If Af is specified, then F contains the states that can be prepared by operations
in A 1 , and the monotone is given by r'(p, F).
It would be interesting to find out the relation between the set of operations that
cannot create resource out of a free state (denoted as A 0 1 ), and the set that cannot
generate more resource from any state (A 1 ). Obviously, Af 9 Ao 1 since the constraint
on Af is stronger. More formally, if any operation A
C Af satisfies
Vp, y' (A (p), F) _< r1' (p, F),
(3.7)
Vpo, y' (A(po), F) = 0,
(3.8)
then it must satisfy
71
where po E F. In the context of perfect QRTs, a clean answer for this question wou;d
significantly simplify the study of QRTs.
3.1.3
Hierarchical structure
Intuitively speaking, by placing stronger and stronger restrictions on the allowed set of
operations, the set of free states (those can be prepared by the free operations) should
also become smaller and smaller. Consider two extremes: if almost all operations are
allowed, then almost all states are free since they can easily be prepared from anything;
if almost no operations are allowed, then almost everything becomes a resource. A
more formal statement of perfect QRTs goes as follows:
,
Theorem 18. Consider two perfect QRTs, whose free sets of states are F and F
2
and the set of allowed operations are Af,1 and AJ,2, respectively. Af,
Af,2 implies
F1 C F2 , and vice versa.
Proof. Consider a state p such that p E F1 , i.e., it can be prepared by operations
in Af,1. Given that Af,1 g Af,2, p can certainly be prepared by operations in Af,2,
indicating that p E F2 . Therefore, there does not exist any state that is free for theory
1 but not free for theory 2, i.e., F1 C F2 . By Theorem 17. the other direction is also
true.
El
Note that for general QRTs this conclusion does not necessarily hold.
By strengthening the constraints in different directions, hierarchical networks can
be established to organize the behaviors of QRTs, in analogy to the "zoo" of computational complexity classes. Fig. (3-2) roughly depicts an example of such a hierarchical
structure of operations, coming from the discussion of QRT of quantum correlations
in Section 3.2, which I shall come back to later.
Interesting connections can be drawn between map classes and the computational
complexity classes.
For example, the computational power of Stabilizer (Clifford
group) operations is not better than classical Turing machines (corresponding to the
complexity GL). But with the aid of magic states as resources (refer to example
72
Local
Local unitaries
quantum?
(LU)
Mixtures of
-
Local unital?
local unitaries
C
in
in
in
Quantum
Unitaries (U)
G
Classical
Permutations
C
No Q. Birkhoff
Mixtures of
ua
LOCC maps?
C
s
Birikoff
Mixtures of
permnutations
Figure 3-2: A hierarchical structure of maps.
and rows represent strict hierarchies.
Unital CPTP
r-maps
Doubly
stochastic maps
C
CPTP maps
C
Stochastic maps
Columns represent correspondences,
3 in Section 3.1.1: the QRT of magic), we now have access to universal quantum
computation.
If we allow less or more operations, analogous connections may be
established to other computational complexity classes.
3.1.4
Combining QRTs
To this point, the framework of QRTs has already revealed its essence as a unified
mathematical structure to some extent, which guides us to consider the QRTs as
abstract mathematical objects. Then a natural question comes into mind: Can we
do "algebra" on QRTs? To illustrate the idea, let's ask the following specific question,
which is the simplest case:
Suppose we have two perfect QRTs, respectively labeled by free sets F1 and
.
F2 . The correspondingfree sets of operations are respectively Af,1 and Af, 2
Now define a new perfect QRT with F3 = F1 U F2 . What can be concluded
about the free set of operationsAf, 3 of this theory?
First of all, by Theorem 17, any element equivalently labels a perfect QRT, so this
question is well-defined. We then obtain:
,
Theorem 19. Consider two perfect QRTs, whose free sets of states are F1 and F2
and the set of allowed operations are Af,1 and Af, 2, respectively. For a new perfect
QRT with F=
F1 U F2 , (Af,1 n Af, 2 ) C Af, 3 , i.e., the operations that belong to both
73
.
Af,1 and Af, 2 is also in Af,3
Proof. Denote the unique resource monotone, the regularized relative entropy distance, as r7' as earlier. A is an operation. The members of Af,1 satisfy
Vp, q' (A (p), F1) < qr (p, F1),
(3.9)
while the members of Af, 2 satisfy
Vp, r (A(p), F2 ) < r
(p, F2 ).
(3.10)
min{r (p, F1 ), j7 (p, F2 )},
(3.11)
By inspection,
r
(p, F U F2 )
=
(A(p), F1 ), r 0 (A(p), F2)}.
0 (A(p), F U F2 ) = min{r
If an operation A satisfies both Eqs.
(3.9) and (3.10), i.e., A E (Af,1
(3.12)
n Af, 2 ), the
following relation always holds:
Vp, minfr
(A(p), F1), ry (A(p), F2)}I
min fr/j (p, F1), n' (p, F2)},
(3.13)
By Eqs. (3.11) and (3.12),
Vp, q' (A(p), F U F2)
r
(p, F U F2 ),
(3.14)
i.e.,
Vp, r7' (A(p), F3) < r (p, F3 ).
That is, A E Af, 3
(3.15)
.
El
Obviously, this result can be easily extended to the more general case where more
QRTs are involved.
There is another interesting way to think about the combination of QRTs. Con-
74
sider a task where you want to determine a quantum state by querying QRTs, which
serve as a black box that can always tell you the resourcefulness of this state, and the
state after some known operations. How would you devise the strategy?
Obviously, one single QRT is not enough, simply because generally there can be a
family of states possessing the same amount of resource. Then how many QRTs are
informationally complete for determining a quantum state? Or even probabilistically?
I am still working on obtaining more general results for this question, but a simple
and intuitive example can be given as follows:
Theorem 20. By unitary transformation and the QRTs of coherence and purity as
oracles respectively, one can correctly determine an arbitrary qubit p with probability
at least 1/2.
Proof. Given a qubit p, one can always find unitaries U (preserves entropy/purity)
such that p' = UpUt is incoherent in the appointed basis for the QRT of coherence. If
the QRT of purity tell you the purity of p' is 0, then the game is over since p'
1/2,
thus the answer is simply p = 1/2. Otherwise if the purity of p' is a (0 < a ; 1), i.e.,
S(p') = 1 - a, we can obtain a unique solution of the two real eigenvalues m and n
such that
-mlogm - nlogn
1 - a,
(3.16)
m + n = 1,
(3.17)
m,n > 0.
(3.18)
=
Note that there is no way to distinguish the two possible orderings of m and n in the
diagonal matrix p', so p is either
Ut
M 0
0 n
U
(3.19)
Ut
n
0
U.
(3.20)
(0
M)
or
75
U
P
State space
Figure 3-3: A sketch of a strategy for determining a qubit state that queries the QRTs
of coherence and purity. The dashed circle represents the states that has the same
entropy (connected by unitary transformations), and the solid line represents the
states that are diagonal in the appointed basis (incoherent states). For an arbitrary
state p, the unitary U brings it to one of the incoherent states, while preserving
purity/entropy.
That is, the QRTs of coherence and purity can "cooperate" to give a strategy for this
simple task. A sketch of the idea of this strategy is given in Fig. 3-3.
11
This type of task can be generalized in many ways. For example, the player is
only allowed to query a limited number of times, or some error is allowed. This is an
interesting topic to explore, since it might provide useful techniques or insights for
practical quantum tasks.
3.2
Quantum correlations as a resource
Entanglement has long been recognized as the most important and indispensable
quantum resource for computational and cryptographic tasks. Indeed, entanglement
76
seems to play the central role in all sorts of famous quantum protocols, such as quantum teleportation [11] and Shor's factoring algorithm [91].
However, the ultimate
resource for all quantum advantages remains a concept that no one can really identify. It is possible to achieve an exponential speedup over classical algorithms using
mixed states with vanishingly amount of entanglement [331 in a quantum computational model called DQC1 [581. The resource for this task is recognized as quantum
correlations measured by QD. In recent years, quantum correlations have been identified to be the resource responsible for many other useful quantum protocols, including
remote state preparation [301, quantum cryptography [811 etc. And the quantity QD
has also been bestowed an operational interpretation in terms of entanglement consumption in an extended quantum-statemerging protocol [231. However, the theory
of quantum correlations is not yet a satisfactory QRT at the moment, since the class
of operations that do not increase the amount of quantum correlations (which in
general depends on the measure we choose) has not been fully characterized. In this
section I introduce some efforts towards the full theory. I denote the whole set of free
operations as S, and the set of operations that cannot create quantum correlations
from a -y state as So. Obviously, S C So.
Before going into detailed discussions, let's first elucidate the motivation. A welldefined resource theory of quantum correlations will, in the most straightforward
sense, tell us how nonclassicality is created and how to make better use of this special
type of correlations beyond entanglement, and therefore provide valuable insights of
the foundations quantum theory. And of course, if such a QRT is developed, it will
be the theory that people refer to for discovering the uses of quantum correlations as
a physical resource in practical quantum information processing tasks.
As the key element of a QRT of quantum correlations, the candidate measures
discussed in Chapter 2 can all be possible resource monotones in relation to different
tasks. However, worth mentioning, the partial order of quantum states determined by
different entropic measures (e.g., QD, minJEP, DD) have to be different, by Theorem
3.
For the perfect theory, the measure should be the regularized relative entropy
distance to the free set.
77
3.2.1
Free states
For the theory of quantum correlations, the free states are obviously -y states, as
defined in Section 2.1. Note that in this section I do not explicitly distinguish CC
and CQ since conclusions about them can be easily generalized to the other.
The properties and interpretations of 7 states were studied in Section 2.1, but
here let's review some very important points. First of all, recall the mathematical
forms of 'y states. A state with no one-way quantum correlations (without loss of
generality, from A) is called a CQ state, which takes the form
PA
Pa&aAaI 0 p,
=
(3.21)
a
where Ia) is the Schmidt (local spectral) basis for A (block diagonal in A's eigenbasis),
and pa is some valid density matrix for B. And analogously we have QC (omitted)
and CC states, which takes the form
p
= ZBPabja)A(aI 0 |b)B(b = ZPabjab)(ab|,
ab
where
ZabPab
(3.22)
ab
= 1, (abja'b') = 6aa'3 Ikl. The common feature of these states is that
there exists some local measurement that doesn't perturb the state at all, or say the
post- and premeasurement states are exactly the same.
For constructing a QRT of quantum correlations, a crucial property of the set
of 7 states is its nonconvexity, i.e., as shown in Section 2.1.2, this set is not closed
under probabilistic mixing. States with positive quantum correlations can be created
from 7 states simply by throwing away information. The authors of [161 even list the
convexity of the set of free states as one of the postulates that a QRT has to satisfy.
To overcome this difficulty, the operation of throwing away information is deemed
to be not free (which makes sense, since it creates entropy). An alternative way to
think about this assumption is to attach a classical (orthogonal) flag to each 7 state
that we work with. Taking the example from in Section 2.1.2 (Eq. 2.4), which can
be thought of as the uniform mixture of four product state: 10+),
78
11-), 1 - 0) and
I + 1). Now we attach an ancilla for each subsystem of these for states. Denote the
extra subsystems A' and B' respectively. Let the four ancillae for each side (denoted
by 10), 11), 12) and 13)) be mutually orthogonal, and the state after the same mixing
reads
1
PAA',BB'
=
-(100)(001
4
9
+ 0)(+01
+ 111)(11 &I
-
1)(-1
+1 - 2)(-210 102)(021 + I + 3)(+310 113)(131),
(3.23)
111), 1 - 2) and
I + 3) are
which is now CC, since e.g., for subspace AA', 100),
mutually orthogonal (same for BB'). That is, the creation of quantum correlations
(the problem of nonconvexity) can be prevented by attaching orthogonal flags.
3.2.2
So
In Section 2.1.2, I showed that quantumly correlated state can be created by LOCC.
A direct implication is that So must be a strict subset of LOCC, i.e.,
So C LOCC,
(3.24)
since the set of 7 states is a strict subset of separable states, which implies that any
operation outside LOCC (entangling operations) can definitely generate quantum
correlations.
Actually, I have already mentioned a nice observation in Section 2.1
that the "separable-CC" relation is very analogous to the "mixed-pure" relation. By
this result, we arrive at the following conclusion:
Theorem 21. Suppose as introduced earlier, we forbid the creation of quantum correlations by mixing (throwing away information). Denote the set So under this assumption as So,g. Then So,g = LOCC ("9" means that mixing is forbidden).
Proof. As discussed, we assume that the probabilistic mixing is prevented by attaching mutually orthogonal ancillae (flags) to all states in the ensemble.
of the flag-attaching protocol is Eq.
An example
(3.23), where it can be seen that the LOCC
protocol used to create non-7 states from y states is invalidated. Obviously, LOCC
79
keeps separable states separable, and due to the orthogonal flags (actually analogous
to purification), y states are kept -y. Thus by Eq. (3.24),
S0 , = LOCC,
(3.25)
El
under this flag-attaching protocol.
Now let's consider the things we can still do if mixing is allowed. More specifically,
flags are not attached, and classical communications, which actually allows for global
mixing of states using analogous protocols as the coin flip introduced previously,
is allowed.
(To distinguish with the set of CC states introduced before we denote
classical communication as "CComm" from now on.)
In this case, Local Unitaries (LU), which transform local basis, cannot be done
arbitrarily because the output states of any operations must be orthonormal to one
another. In other words, quantum correlations can simply be created by preparing
lots of copies, apply certain LUs on them, and mix them up. I believe this is one of
the fundamental reasons why the authors of [16] directly argue that acceptable QRTs
should not have a nonconvex free set. In our case, LU certainly should not be able
to create any sort of "correlations", but the nonconvexity introduces such a problem.
The set of local channels that can still qualify certainly includes the Semi-Classical
(SC) channel in a fixed basis (LSC), which can be decomposed into local measurements with respect to a fixed basis (local dephasing/decoherence, denoted by LM)
plus manipulation on the probability distributions, which can be achieved by generalized depolarizations on certain subspaces (GD).
Definition 17 (Semiclassical (SC) channel Asc [96]). A semiclassical (SC) channel
maps arbitrary states onto output states that are diagonal in the same orthonormal
basis, i.e., Asc (p) = E PA(P)Ik)(k1. Note that by definition SC channels may alter the
uniform probability distribution of a maximally mixed state, i.e., are not necessarily
unital. So SC and Unital are essentially two independent sets of maps that have a
non-empty intersection.
80
Definition 18 (Generalized Depolarizing (GD) channel
AGD;
white noise). A quan-
tum depolarizing channel is a model for noise in quantum systems, which maps a
quantum state p
H-+
Ap + (1 - A)I/d. Note that the CPTP requirement bounds the
parameter: A E [-1/(d 2
-
1), 1], but the point is that A can be negative.
Here if Markovianity of the map is considered we can restrict A > 0 (information
can only flow outwards) and consider transitions from low to high entropy (high to
low purity) states without loss of generality. Then for this case:
So,g - L(NI +
D) + CComm = LSC + CComm.
(3.26)
But in this case, as mentioned, since CComm allows for mixing, which disqualifies
arbitrary LU due to the nonconvexity of the set of 7 states (which does not really
make sense), we need to be careful in using this result. The basic interpretation is
that the states involved have to be locally classical with respect to the same basis.
For qubit systems the answer is clearer. It is shown in [96] that unital channels
are also not able to create quantum correlations (note that this is not true for higher
dimensions):
Definition 19 (Unital channel AUnitai). A unital channel maps maximally mixed
state to itself, i.e., AUnital(I/d)
=
I/d.
The rigorous theorem combining results from [96] goes as follows:
Theorem 22. For qubit systems, the maximal set of local channels that cannot create
quantum correlationsis S0
=
L(Unital + SC) (a fixed basis is appointedfor SC), while
CComm can be allowed.
Proof. First we show that the set of classically correlated two-qubit states, which
can be written in the form pcc 2 =
6
wi and (jj')
=
i=OE=OPijji)A(i 0 Ij)B(ji where (ili') =
j,, is closed under L(Unital + SC), i.e., the sufficiency.
Using
Bloch sphere representation, the orthonormal basis of each party can be generically
expressed as |0)(0|
=
(I + ' - o)/2 and 11)(11
81
=
(I - d - 6)/2, where a E ]R3 is the
Unital
'1/2
Figure 3-4: Geometrical (Bloch sphere) demonstration of the effect of a unital map
on a qubit. The unital channel keeps the two basis vectors symmetric with respect
to the center of the Bloch sphere, and the output state can be diagonalized in the
orthonormal basis corresponding to the intersections of the connecting line and the
surface of the Bloch sphere.
Bloch vector and ' are Pauli group elements. Obviously the local SC channel keeps
the reduced density matrix diagonal in some orthonormal basis, i.e., after the action
of SC the state is still classically correlated. Now notice that the local unital maps
are linear, and by definition they map 1/2 onto itself. So local unital channels map
the original basis states onto Aunitai(I0)(0) = (I + d - 6')/2 and Avnitai(I1)(1I) =
(I - S-
')/2 where the elements of a' are mapped from the corresponding elements of
a, which are still symmetric with respect to the center point 1/2 of the Bloch sphere.
Therefore the reduced density matrix of the output state can be diagonalized in the
orthonormal basis, whose two basis vectors are just the pure states corresponding
to the intersections of the line crossing Avnita(10)(0j) and Aunitai(I1)(1), and the
surface of the Bloch sphere (See Fig. 3-4 for a geometrical illustration). Then we
immediately conclude that L(Unital + SC) is the sufficient condition of not creating
quantum correlations.
Note that this is not generally true for higher-dimensional
systems.
Next we prove the necessity: any local channel that cannot be decomposed into local unital and SC channels (denoted as A,) will create quantum correlations. Nonunitality implies A.(I/2) = (I+ b - U)/2 where 3bi $ 0, and the channel being not SC
implies ]p, = (I + '. a')/2 such that A.(ps) = (I + d- U)/2 where b and d are lin82
early independent, which ensures that An(I/2) and A,(ps) are not diagonal in the
same basis. According to the proof of sufficiency, the following state is classically
correlated:
I + C-
Pcc2 = I
2
0 10)(01+
I.01 -
-
67
(3.27)
1)(1I.
2
Now if we put this state through the previously defined channel and define An (C-)
=
e 6 where ]ei -$0, the output state reads
I+(b
An(cc 2 ) = I+(bIO)(0+
2
where b - e and b -
2
e)I
1)(11,
(3.28)
' are linearly independent since b and d = b+ ' are linearly
independent, implying that An(ficc 2 ) is not classically correlated since 1 + (b+ e) - 5
and 1 + (b-)
)-
a are not diagonal in the same basis. This completes the proof.
Similar statements can be found in [96].
l
Therefore for two-qubit states
O g = L(Unital + SC) + CComm.
(3.29)
As shown in Eq. (3.26), unital channels are not generally allowed for higher dimensions: there exist unital channels that are not in SO,g. Eq. (3.29) is a particular case
of So,g that is only valid for qubits.
3.2.3
Promotion to S
The above discussions are about the creation of quantum correlations from -y states,
which do not depend on the specific quantifiers. However as the general element for
a complete QRT, members of the set S should not be able to increase the amount of
resource, as measured by some quantifier m(p), which is generally a stronger requirement than
So. Furthermore, the analysis on S will depend on the measure chosen
(denoted as Sm(P) where m is the measure), generally (as the same theory at Level 1,
83
at most one of them can be perfect). Thus the general conclusion is that
Sm(p) C So.
(3.30)
For the theory of quantum correlations, it is possible to show that
Theorem 23. For the QRT of quantum correlations, the set S for any non-perfect
theory is strictly contained by that of the perfect theory, i.e., Sm(p) C SO (p,Y).
Proof. The theory with q*(p, 7) as the resource monotone is perfect, i.e., SI(P)
C
So (P,-). It can be shown that there must exists operations in SiO (P,) that are not in
Sm(p), as long as m(p) is an entropic measure of quantum correlations different from
77 00(p, -Y).
By Theorem 3, there must exist two states p1 and P2 such that
m(p1) > m(p 2),
(3.31)
'(P, 7) < q* (P2, 7)
(3.32)
By Theorem 15, the latter monotone q00(p, -y) has clear practical meanings: the ratio
represents the asymptotic conversion rate (reversible), given that the operation set
is maximal (SNO(P,'Y)) [16]. By Eq. (3.32), there exists a process using operations in
S7 (P,-) that transforms p" to pof2,
where ni > n 2 , asymptotically. But for the QRT
with measure m(p), since Eq. (3.31) essentially says that pi is more valuable than P2,
the same transition is not possible with operations in Sr(p), which implies that the
above transision process must involve operations that is not in Sm(p). Consequently,
SM(P) C sno(p,,).
(3.33)
This theorem indicates that the set of free operations for the QRT of DD (or any
measure that is different from q(p, -)) is not maximal.
84
Now we know that Sm(P) C Snoo (P,) C So.
The work left is to identify what
operations in So may increase QD for general scenarios and exclude them.
For the case where mixing is still forbidden (by attaching flags to every state),
due to the resemblance of to entanglement theory, I conjecture that
Conjecture 3. Under the assumption that mixing is not allowed, Sc
=
LOCC.
When mixing is not mandatorily banned, the general behavior of quantum correlations under SC channels seems complicated.
A simple observation is that the
depolarizing (A > 0) the whole space of local subsystems will not increase the amount
of quantum correlations under any measure. The complete characterization will be
left for future work.
3.2.4
Map zoo
Here I introduce some interesting connections between our results and hierarchies
of maps that have been studied before. Under the assumption that mixing is not
acceptable, we have the following hierarchy (some detailed results are not completely
shown):
LU
C Sm(P) C
S77(,") C So,c C LUnital C So,g = LOCC.
(3.34)
We can immediately notice several interesting analogies of some important classes of
maps with quantum and classical map hierarchies, as the structure shown in Fig. 3-2,
where the second and third row present the analogy between quantum and classical
hierarchies. Here a possible generalization of such hierarchies to local quantum maps
shown can be proposed (the first row of Fig. 3-2). According to Eq. (3.34), Sm(P)
C
SUto(,-) C So,: lives in between LU and LUnital on this row, and SO,g overlaps with
LOCC. Such intuitive hierarchy structures may lead us to clearer understandings and
important insights.
An interesting fact to mention is that in the classical hierarchy, Birkhoff's theorem
states that doubly stochastic maps can always be written in terms of mixtures of
permutations. However this is not true for their quantum counterparts in the sense
that not all unital maps can be decomposed into mixtures of unitaries, i.e., the latter
85
set is a strict subset of the former one [91]. The ideas from the resource theory of
purity, which is very closely related to that of quantum correlations, has been used
to disprove the asymptotic version of quantum Birkhoff's conjecture, and more deep
connections between these topics are expected. To see this, let's expand the hierarchy
in between mixture of unitaries and unital maps shown in Fig. 3-5. The class of maps
Unital
UF
Factonizable
U'
Strongly Factorizable
U'
Exactly Factorizable
U'
AQBP
Ut
Mixtures of Unitaries
Figure 3-5: (Adapted from [91]) The detailed hierarchy in between mixture of unitaries
and unital maps. AQBP denotes "Asymptotic Quantum Birkhoff Property".
called Exactly Factorizable (EF) maps, which is essentially those that can be written
in the following form:
AEF(P) - trB[U(p&
)U0],
is a special case of NO, which preserves the input space.
(3.35)
And obviously, all NOs
are unital. That is, i.e., the free operations of the resource theory of purity locates
between EF maps and unital maps. Certainly, more and more results and insights
86
can be found in this direction.
As a concluding remark of this subsection, this hierarchical network can be extended in analogy to the "complexity zoo", which I name, the "map zoo". This is the
land where QRTs find there own place to live. For example, in Fig. 3-2, the entanglement theory lives on the top-right corner, and some important sets of the theory of
quantum correlations live on its left. As discussed in Section 3.1.3, there are indeed
interesting connections between these two zoos, and hopefully more and more will be
established!
3.2.5
Remarks
Another possible way to consider the QRT for quantum correlations is through the
combination of QRTs introduced in Section 3.1.4, i.e., the hybrid theory. The authors
of [47] pointed out that the QRT of quantum correlations is very closely related to
the theory of local purity, and may possibly be the hybrid theory of entanglement and
purity. Furthermore, the theory may also be a similar theory to the QRT of coherence
(consider the nature of SC channels). It will be very interesting and helpful to give
rigorous mathematical formulations of such hybrid theories.
3.3
Dual QRTs
Due to the complementarity of the elements of QRTs, as has been extensively studied
in previous sections, it is natural to expect that the theoretical framework can be
symmetrized. Ordinary QRTs specify the set of allowed operations, the corresponding
set of resourceful states, and quantify how resourceful they are. In analogy, we can
also quantify how useful an operation is, in terms of the capability of this operation
to create resource. Based on the above ideas, I shall introduce a structure of theories
which is dual to that of the QRTs in this section, as discuss possible interpretations
of this new framework. The results presented here are preliminary.
87
General structure
3.3.1
Now I propose the dual (symmetrized) framework of the ordinary QRTs, namely dual
QRTs. In this class of theories, some operations become resources, and the argument
of the quantifier becomes operations instead of states. Analogous to ordinary QRTs,
the three basic building-blocks of dual QRTs are:
1. Free states.
Denote the set of free states as F. Similarly, it satisfies:
(a) F is closed upon doing tensor product.
(b) F is a closed set.
2. Resourceful operations.
Denote the set of resourceful operations as A,. It should satisfy
(a) A, is the subset of all possible operations.
(b) Under operations that are not in Ar, F is always closed.
3. Quantification of the resourcefulness of operations.
The quantifier is a continuous function of quantum operations (denoted by K(A))
that quantify its resourcefulness. It determines the partial order of operations
with nonzero resource. The properties that , has to satisfy are:
(a) K(p) > 0: always nonnegative.
.
(b) K(p) = 0 iff p E Af
Indeed, the quantification of the value of operations/channels seems tricky, since it is
not natural to generally define the conversion between channels. On the other hand,
in ordinary QRTs, the basic mathematical process is mapping a quantum state to
another by quantum operations:
pI+p',
88
(3.36)
where a quantum channel is indeed the map, as defined. However in dual QRTs, the
dual process becomes:
A 4 A',
(3.37)
where a quantum state maps a CP map to another, which is not as natural to give a
physical interpretation. However, the mathematical structure is well-defined. There
are also some other subtleties, such as the closeness of several sets. Details remain to
be worked out in subsquent work.
All in all, the idea about the quantification is that the more resource an operation
can possibly generate, the more powerful it is, and a dual QRT serves to quantify the
power. Here I suggest the following function as a possible quantifier of the resourcefulness of operations A, of the dual QRT with free set F:
R(A, F) = sup {rf
P
(A(p), F) - r/ (p, F)}.
(3.38)
Indeed, this quantity satisfies the postulates for a legitimate K.
Let's consider the following intuitive example, the dual QRT of entangling operations. Obviously, LOCC is not a resource, since it cannot create entanglement. But
for two-qubit states, the CNOT gate is obviously very powerful since it can directly
bring a product state (which is not correlated at all) to a maximally entangled Bell
state:
CNOT(l+) 0 0))
=
100),Ii)
(3.39)
where the control qubit is assumed to be the first one. In any reasonable quantification
scheme for dual QRT of entanglement, CNOT should be maximally resourceful for
two-qubit states. Using the quantifier suggested in Eq. (3.38), the "entangling power"
of CNOT is given by
R(CNOT, usep)
89
1.
(3.40)
Similarly, we have the following hierarchy for the definition of dual QRTs:
Level 1 dual:
{-, Ar, }
Level 2 dual:
{F, Ar, K
The resource (of operations) labels a Level 1 dual theory, and the three elements
together labels a Level 2 dual theory.
Then one may naturally ask: what is the
dual of a particular QRT? As showed in Section 3.1.2, for each Level 1 QRT there
is a single perfect one that can serve as the representative. It can be directly shown
that for such a perfect QRT there exists a perfect dual, which can also be the sole
representative of Level 1 dual QRTs.
Theorem 24. There always exists a unique dual for a perfect QRT.
Proof. By Theorem 17, a perfect QRT can be uniquely labeled by F or Af.
Let F
and Ar (the complement of Af) be the labels of the dual theory. Since by definition of
the perfect QRT, Af is maximal, i.e., any operation A E Ar is definitely resourceful,
in the sense that
-p, n' (A(p), F) > ij'(p, F),
(3.41)
R(A) > 0.
(3.42)
i.e.,
Therefore, the dual theory is legitimate. That is, the dual of a certain perfect QRT
can be obtained by taking the complements of R and Af as the elements.
In fact, the elements (state and operation sets) of the original and dual perfect
QRTs are exactly complement of each other in the respective space, as illustrated in
Fig. 3-6.
This idea of the dual structure of QRTs is actually the underlying mathematical framework of a diverse set of studies, and may establish interesting connections
between some quite independent subfields.
90
Dual
QRT ....... ....................
R.
QRT
.............
Operations
States
Figure 3-6: Illustration of the dual QRT. The parts with grey fill means "free". Dashed
lines represent the pair of elements in each theory, and arrow represent partial orders,
determined by quantifiers. Note that this is only a sketch of the idea of duality. The
geometries of sets and the partial orders in each space do not necessarily resemble
this figure.
3.3.2
Examples
Although the generalized framework is new, there are already some studies that actually applied the similar idea: quantifying the strength of some specific properties
of quantum operations/maps/channels.
Here I shall briefly introduce some exam-
ples. As claimed at the end of last subsection, this unified framework may provide
connections between some quite independent and scattered topics.
Non-Markovianity
Quantum non-Markovianity has been a central theme in the theory of open quantum systems for quite a long time.
It essentially characterizes the memory effect
in quantum dynamics. Obviously, all the studies on this topic surround two central
questions:
1. Characterization: What kind of processes (dynamics) are non-Markovian?
2. Quantification: How do we measure the extent of non-Markovianity?
91
My point is automatically made clear: the whole theory of quantum non-Markovianity
[85] can be formalized using the language from dual QRTs. There are various measures
of non-Markovianity based on different ideas, such as the the size of memory space
needed, the distance to a convex Markovian set 11051 etc. However, the point I want
to emphasize here is that the majority of results on measuring non-Markovianity
is based on the idea of "witness": for systems undergoing Markovian dynamics, the
witness monotonically decreases as time goes. Therefore if the growth of the witness in
the process signals non-Markovianity, and (roughly speaking) the amplitude of such a
growth can serve as a measure of how much the dynamics deviates from Markovianity.
Several quantities can be such a witness, including entanglement which leads to the
most famous RHP measure [86], QD [51, and MI [66]. Then these QRTs are dual to
the theory of non-Markovianity in the sense that non-Markovian process can increase
the amount of these quantities. Via the duality of QRTs, the studies of correlations
(Chapter 2) and non-Markovianity can be naturally connected!
Nonclassicality
The resource theory of the nonclassicality of maps is very closely related to the theory
of quantum correlations beyond entanglement discussed in Chapter 2, and the QRT
of it in Section 3.2.
A recent work [72] characterizes the nonclassicality of CPTP map Q by the extent
of its non-commutativity with a complete dephasing channel F:
W(Q)
=
supS(Qo F(p)||Fo2(p))
=
sup(S(FoQoF(p)IFoQ(p)) +S(QoF(p)IIFoQoF(p))).
(3.43)
Since the supremum is over all states, the choice of the dephasing basis can be arbitrary. Actually this quantification resembles Eq. (3.38) in many ways. A possible
future work is to figure out the exact relation between them. This relation can be used
to specify genuinely classical operations Q, among all CPTP maps by W(Qc) = 0. As
noted in [72], there is a natural complementarity between quantumness of operations
92
and quantumness of states. At Level 1, this theory is dual to the QRT of quantum
correlations beyond entanglement.
93
94
Chapter 4
Summary and outlook
As proposed in the introduction, the ultimate goal of this work is resolving the following two difficulties:
1. Quantum correlations are indeed very difficult to measure.
2. We probably haven't found the right way to understand quantum correlations.
The first part of the thesis introduces some progress towards this goal. There are
still lots of interesting and important open problems waiting to be solved under this
huge framework. Besides the conjectures and questions already stated in the text,
the following problems or messages are of special importance:
1. Study the properties of DD more comprehensively, in order to find out if it
should be recommended as the best overall measure of quantum correlations.
2. Connect DD to real quantum information processing tasks: endow DD operational interpretation.
3. Establish a satisfactory resource theory of quantum correlations by studying
the S of DD (easy) or the perfect theory (unique).
4. Obtain more general results on the manipulation of abstract QRTs, such as
combination.
5. Establish a "map zoo" (actually, QRT zoo).
95
These questions will be the major targets of future work in this direction.
96
Part II
Exclusion game
97
98
Chapter 5
Introduction
Quantum communication, the area of study that essentially deals with the transfer
and manipulation of quantum information, is one of the central topics in the large
field of quantum information science. The basic setting of the theory of communication is very simple: two players, usually called Alice and Bob, try to perform some
cooperative tasks by exchanging information. For example, Alice and Bob each has a
completely random n-bit string, and they would like to know the sum of their strings.
Obviously, the only way to succeed is that one of them send the whole private string
to the other. Roughly speaking, all scenarios involving communication can be reduced
to some simple form like this, in principle.
The origin of the mathematical foundations of communication can probably be
traced back to the 1948 work "A Mathematical Theory of Communication" by Shannon.
Obviously, the playground of this whole theory can be completely classical.
Players have classical information, can they exchange classical bits. However, it turns
out that quantum physics can again provide us with surprising extra power, just as
for computation. By exchanging quantum states, which encodes quantum information, some tasks can be achieved far more efficiently, e.g., in terms of the amount of
information that has to be transferred for any successful protocol, which is measured
by the communication complexity of the task.
The notion of quantum communication complexity, which measures the minimum
size of quantum message that has to be sent to achieve a certain task, was first in-
99
troduce by Yao in 1979.
From Holevo's theorem, we learned that n quantum bits
(qubits) can represent no more than n bits of classical information without entanglement for communication.
Even when the two parties share maximal entanglement,
n qubits is still only able to carry at most 2n bits of classical information. However,
surprisingly, quantum protocols are indeed much more efficient than classical ones,
for many communication tasks. For example, there exists some partial functions f
with an exponential quantum-classical separation in communication complexity, i.e.,
as the problem size n goes to infinity, there exists a winning quantum strategy that
only requires O(log n) qubits to be sent, while if restricted to classical strategies, at
least Q(n) bits of communication is needed [40].
In this part of the thesis, I shall focus on a communication task called the exclusion game, recently introduced by Perry, Jain and Oppenheim [80]. The game, which
involves two players Alice and Bob who have infinite computational power, may be
described as follows: Alice and Bob randomly draw an n-bit string x and some subset
y C [n], where
IyI
= m, respectively. They win the game if Bob is able to output
a string z that is different from x restricted to the bits specified by y, subject to
the constraint that the only allowed communication, whether classical or quantum, is
from Alice to Bob. Strikingly, the quantum strategy is infinitely better than any classical one for this task, in terms of the amount of information that has to be revealed,
namely, the information cost. It can also be shown that although for appropriately
chosen parameters of the game, there exists an winning quantum strategy that reveals vanishingly small amount of information as the size of the problem n increases,
i.e., the quantum (internal) information cost vanishes in the large n limit, the quantum communication cost (the size of quantum communication to succeed) is lower
bounded by Q(log n) for those parameters. That is, there is an infinite gap between
quantum communication complexity and quantum information complexity: although
for Alice, almost no information of her string needs to be revealed to Bob to succeed
the task, she still has to send a very large message via the channel. This infinite gap
is further shown to be robust against sufficiently small probability of error: it holds
not only for the original zero-error protocols, but also for the generalized case which
100
admits some tiny error. The significance of this result may be compared with its
classical counterpart, where only an exponential separation was shown between classical information and communication costs [391. In conclusion, the relations among
four important quantities, the quantum/classical information costs and the quantum/classical communication costs, are carefully studied in different regimes of the
exclusion game.
This part is organized as follows. In Chapter 6, I shall first formally introduce the
generalized mathematical structure of communication tasks and the exclusion game,
and rigorously define the informational quantities that I shall repeatedly refer to as
preliminaries of the following discussions, and then present the ideas and detailed
procedures of a winning quantum strategy for zero-error exclusion game proposed in
[80] with vanishingly small quantum information cost. In Chapter 7, I shall carefully
consider the quantum communication complexity of the exclusion game, and prove
lower bounds for it in various regimes, to establish the infinite gap between quantum
information and communication costs. At last, Chapter 8 contains some concluding
remarks and interesting open problems.
101
102
Chapter 6
Preliminaries
In this chapter, I shall introduce the formal mathematical definition of the exclusion
game, and the winning quantum strategy proposed by Perry, Jain and Oppenheim
with vanishingly small quantum information cost [801. These results are the premises
of the infinite gap between quantum information and communication costs that we
shall establish later.
6.1
General formulation of communication tasks
The goal of usual communication tasks is to compute a function (which can be total
or partial) of Alice's and Bob's private strings. However, the output of the exclusion
game is not a "function" of these two strings, basically because there can be multiple
acceptable answers. In this section I shall generalize the traditional mathematical
formulation of communication tasks, and use it to formally define the exclusion game.
6.1.1
Mathematical structure
The task of a typical one-way communication is to compute such a function
f
de-
pending on both players' private strings, by transferring information from Alice to
Bob:
f : {0, 1}a x
{o,
103
1}b -+
{0, 1}.
(6.1)
The task may be described as follows: Alice draws some string x E {0,
I}a
and Bob
draws some string y E {0, 1}'. Alice sends a message to Bob and then Bob outputs
a string z. They win the game if z = f(x, y).
The exclusion game introduced in [801 requires a more general framework, which
we introduce here. We shall describe a general one-way communication task by the
function
F : {0, I}a x
{0, I}b
-+ p({0, }*),
(6.2)
where P(S) denotes the power set of S. As before, Alice and Bob draw some strings
E
{0,
I}a
and y E {0, 1}b respectively. Then, Alice sends a message to Bob and
Bob outputs a string z. The winning condition is made more general though. We say
they win the game if z E F(x, y).
It is clear that the more general framework reduces to the typical framework when
F(x, y) contains exactly one element, say f(x, y) . In this case, the only way in which
they can win the game is if z =
6.1.2
f(x, y).
Exclusion game
We shall now define the exclusion game. Let
M : y~nm
X {0, 1}" -+0, 1}m
(y, X)
M" W)
M
where Y(n,m) is the set of all subsets of {1,... , n} of size m, and My(x) is the m-bit
string formed by restricting the string x to the bits specified by y. We also add another
parameter -y to label the allowed probability of error. For the original exclusion game,
=
0. The exclusion game EXCn,r is then defined as
EXCn,my : {0, 1}s(m) x
{0,
i}
(Lyl,x)
104
-+
{0, 1} m
{zlz # My(x)},
where -y represents the allowed probability of error,
LyJ
is the binary representation
of y, and s(m) is the number of bits needed to specify a subset y of size m. The
winning condition may then be stated as follows: Alice and Bob win if for given x
and y, Bob outputs a string z such that z = My(x).
6.2
Information and communication
There are two informational quantities that are very important for the communication
task, namely the information cost and the communication cost. The information cost
of a protocol measures the amount of classical or quantum information that is actually
revealed, whereas the communication cost measures the number of bits or qubits that
has to be exchanged in order for the protocol to succeed.
We now formally define the communication complexity and information complexity for communication tasks. For a protocol H that wins the game defined by the
function F, we denote the information cost of a A-protocol (where A
sical) or
=
Q
=
C (clas-
(quantum)) by ACC(I), and the corresponding communication cost
by RF(H). Then A(C(H) is defined to be the number of bits or qubits exchanged
throughout the protocol, and A'(H) is defined as follows:
AC(1) = I(X : 1lY) + I(Y : 1IX),
(6.3)
I(S : TIU) = H(SU) + H(TU) - H(STU) - H(U)
(6.4)
where
measures the mutual information between S and T given U.
The A-information complexity of a game is then defined to be
A 1c(F) = inf AFc(f).
ri im
The A-communication complexity is defined similarly.
105
(6.5)
6.3
Classical communication complexity
For the regime of the exclusion game defined in 6.1.2 that is relevant for later discussions, it can be shown that the communication cost of any winning classical strtategy
is lower bounded by Q(n), i.e., the number of bits that Alice has to send to Bob for a
classical protocol to work scales linearly as n, asymptotically. The rigorous statement
goes as follows:
Theorem 25. Let w(fIn) K m < an, where 0 < a < . is a constant. Any classical
strategy that wins the exclusion game EXCn,
in the large n limit with certainty,
requires that the number of bits sent from Alice to Bob be of order Q(n), i.e., for all
strategies H, CC n",n (H) G Q(n) where m is within the above regime.
Proof. In Theorem 2 of [801, it was shown that
For w(n2)
to C C
m < an, where 0 < a < -,
ICn'(H) > n
-
log 2 (EiL
())-
the lower bound could be simplified
'"'(H) E Q(n) (see Appendix B2 of [80]). Since the amount of information
revealed is bounded above by the communication cost, i.e., CIc 5 Ccc for any
communication protocol, it follows that Ccc E Q(n). As Alice can always send the
EXCn~m i)E
whole string to Bob in order to win, this bound is tight, and we obtain CE
Q(n) in the specified regime of m, as desired.
6.4
'
'')
E
El
PJO strategy
Perry, Jain and Oppenheim (PJO) shows the first example of a communication task
for which there is an infinite separation in information costs between classical and
quantum strategies by devising a winning quantum protocol for a certain regime of
the exclusion game, such that its information cost tends to zero in the large n limit,
thus upper bounding the information complexity, and prove a Q(n) lower bound for
its classical counterpart [80]. Now we introduce this PJO strategy.
106
6.4.1
Protocol
The PJO strategy runs as follows. For given strings x and Lyi that Alice and Bob,
respectively, draw, we first describe how Alice encodes the string x in her quantum
message: she encodes each classical bit xi using the state
I4'xi(0)) = cos
(
0) + (-1)X sin
where 6 = 2 tan-'(2 1/m - 1). Her n-bit string Y
joint state
kbi (0)),
=
(0)
1i),
x.-
is then encoded as the
(6.6)
r
IIy(o))
(6.7)
1
=
i=1
which she sends to Bob via the quantum channel.
Upon receiving the state from Alice, Bob is now able to perform a global measurement whose outcome indicates the string z he is going to output. The measurement
technique used may be described as a conclusive-exclusion measurement, which was
first introduced by [24] and subsequently used to prove the PBR theorem [83], a result in the field of quantum foundations that rules out a certain class of V)-epistemic
models of quantum mechanics.
projective measurement
In [80, 71, it was shown that if Bob performs the
{(z)}zE{o,im on the subspace designated by y where
1(Z)
1
(10) -
(-1)z' Is)
.
(6.8)
It is easy to see that
(X)(0)
(X)= 0,
(6.9)
i.e., the probability that the indicated output z = My(x) is zero, and hence, by
outputting the result corresponding to the projection |(2), they win the game with
certainty.
107
6.4.2
Quantum information cost
As shown in [801, this protocol exhibits a striking property: asymptotically (in the
limit of large m), the amount of information it actually reveals (the internal information cost) tends to zero as n increases in the m E w( l/-) regime, while the
players are always guaranteed to accomplish the task. Indeed, the quantum information can be calculated to be given by
Q1 C
"'(H) < 2S(MQ) E o(nm- 2 ), where
S(MQ) E o(n-i-2) is the von Neumann entropy of the quantum message MQ that
Alice sends to Bob. Hence, it vanishes in the large n limit, when m E w(fd).
108
Chapter 7
Quantum communication complexity
The PJO protocol, however, requires that n qubits be sent from Alice to Bob. Therefore, even though the information cost becomes vanishingly small with increasing n,
the communication cost of the protocol grows with n. An interesting question that
then arises is whether it is possible to succeed in the game with a smaller communication cost, i.e. can Alice and Bob still succeed if Alice compresses her quantum
message? In particular, is it possible for the message to be compressed so that the
communication cost also goes to zero with increasing n? In this chapter, we answer
this question in the negative. It can actually be shown that for certain parameters of
the exclusion game, the quantum communication complexity is at least Q(log n), i.e.,
a Q(logn)-size quantum message is always necessary to accomplish the task, when
no error is allowed. Then one may naturally wonder whether the results are robust
against error. We then present a variant of the zero-error protocol that naturally fits
into this picture. As it can be shown, the lower bound and thus the infinite gap is
robust against sufficiently small error.
7.1
Zero error
In this section we explicitly present the proof of the lower bound of Qcc for the zeroerror exclusion game EXCn,m,o where m E o(n). The main idea of the proof is to devise
a classical protocol that approximately simulates the imaginary winning quantum
109
protocol. We show that if the size of quantum communication in the winning quantum
protocol is too small, then we can also have a classical strategy with communication
cost less that linear, thus contradicting the classical lower bound (Theorem 25 in
Section 6.3).
7.1.1
Classical encodings of quantum states
Before proving the main theorem, we shall introduce the following
Lemma 26. A q-qubit quantum state can be classically described by a set of real
numbers encoding the real and imaginary parts of all amplitudes to accuracy e using
o
log(1/c)) bits.
2q
Proof. Generically, a q-qubit pure state
where a E C, and
IQ)
can be written as I'Oq)
{Ii)} is a complete orthonormal basis set containing
= Eqi
2
q
aji),
elements.
We express all complex amplitudes as ai = bi + ici where bi, c, E R, satisfying
ZIil
JaiI 2
=
f
1
(bi
+ ci) = 1. Thus, 0 < Ibil, Icil < 1. To approximate each
of these real numbers to accuracy e = 2-, we keep the first r bits after the binary
point, and use one extra bit to indicate its sign, i.e., we can find an (r+1)-bit classical
string that encodes an approximation bi of each bi such that for all i,
Abi = |bi - bil < e,
Aci = Jai - cil < e.
Notice that there are 2 -2 q such numbers in total, thus only 2q+l(r+1) = 0 ( 2 q log(1/e))
bits are needed to encode
I bq) such that we have specified the real and imaginary parts
of all amplitudes to accuracy e.
This lemma essentially says that generally, a quantum state can be approximately
encoded by exponentially many classical bits by registering its amplitudes to some
predetermined accuracy. This result play an important role in the classical simulation
protocol, thus the proof of the main theorem as well.
110
7.1.2
Lower bound of Qcc
Now we prove the main theorem: any winning quantum strategy for a certain regime
of the zero-error exclusion game requires at least Q(log n) qubits to be sent.
Theorem 27. QCC(EXCn,m,O) ;>
(log n), where m E o(n). That is, the size of quan-
tum communication in any winning protocol for the specified regime of the exclusion
game is at least Q(logn).
Proof. Suppose that for the specified regime of the exclusion game, there exists a
winning quantum strategy HQ such that Qcc(Hq) = q
=
o(log n). Then based on
HQ, we can devise a classical strategy Etc with o(n) bits of communication, which
would contradict Lemma 1 (classical lower bound), therefore negating the existence
of HQ.
Most generally, HQ can be divided into three steps: i) based on her n-bit string
x, Alice prepares a quantum message lox) of size o(logn) (defined by log JRI, R
being the Hilbert space); ii) Alice sends
I4x)
to Bob; iii) Bob feeds |ox) into his local
quantum computation, and obtains an m-bit string z such that z 7 My(x) according
to the output (measurement outcome). (Without loss of generality, we can assume
that the quantum message is pure, since any mixed state can always be purified by an
ancilla space of the same dimension: the asymptotic scaling of Qcc is not affected. In
addition, both players agree on a fixed basis for the matrix representation of operators
and amplitudes of state vectors beforehand.
The essence of Hc is to classically simulate all steps of HQ.
The basic proce-
dure is the following. First, Alice prepares a classical message C(I'x)) that describes
l)=x)
=
zl7I
ajIi)
=
E
(bi + icj)Ij) ({ij)} is the appointed basis.), by registering
the real (bj) and imaginary parts (cj) of all amplitudes (as) to some desired accuracy
i (we denote the approximations by b and Ej), and then send it to Bob. Note that the
size of
C(oI)),
i.e., the communication cost of 11c, depends on the desired accuracy
of the description. (For example, the message needs to be infinitely long to achieve arbitrary precision.) In IIQ, Bob's local strategy can always be modeled as a completely
positive trace-preserving (CPTP) map followed by a generalized measurement, which
111
is altogether equivalent to some POVM. Note that although Bob can use an infinite
amount of ancillas for his local quantum computation, i.e., the original POVM {Pi}
may contain an arbitrary number of elements, there are only
he can eventually output: g(P)
=
2
"' possible strings that
z, where z is an m-bit string. We can therefore
combine all Pi's corresponding to the same z as an element P' of a new POVM set
{P'} by
P' =
(7.2)
g(P)=z
or in the continuum limit where the elements are labeled by a continuous variable p,
P' = (
d pP([t).
(7.3)
Jg(PGL))=Z
Due to the convexity of the set of all non-negative Hermitian operators (valid POVM
elements), {P'} forms a discrete effective POVM with 2' elements labeled by z.
A subtlety here is that the amplitude vector encoded in C(1,,)) is not necessarily
normalized, i.e., the probabilities do not necessarily sum to one.
Bob first nor-
malizes the amplitude vector by dividing each component with the 2-norm v
(EyW 1 (bj +
2)) 1/2, and then applies Born's rule to compute the approximate prob-
ability of obtaining each z:
P
2>3 (bjbk
-
iajbk + ijk+
ajak)- Jk,
(7.4)
j,k=l
where p,,
is the
(j,k)-th entry of P'. The requirement that HQ never fails indicates
that the probability of outputting the POVM elements corresponding to a wrong
answer is exactly zero. As shown in Lemma 29, the whole approximate distribution
{p'} remains very close to the true one (denoted by {pz})
when a is small, so the
classically calculated probability corresponding to the wrong answer My(x) is well
bounded. Therefore Bob simply sets an appropriate threshold value J(i) that p'
cannot exceed, and refuses to output any z with p' <a(2). As long as there exists an
answer above this threshold, the protocol is guaranteed to be successful.
112
Finally, we determine the values of K and 6 in the above protocol 1c. In order
to guarantee that there always exists a valid output, it is sufficient that the upper
bound of perturbation on all z's, 6, satisfies
6
sup 1p, - p'I < 2-.
(7.5)
z
Then we can simply set the threshold value
6
=
(7.6)
2-,
i.e., Bob only outputs a z with p' > 2-' (it always exists). By Lemma 29, 6 < 202/2
Then according to Eq. (7.6), we can set
=
so that 6 < 6.
20
(77)
2-(m+q/2)
In summary, Hc runs as introduced with the e and 6 respectively
specified by Eqs. (7.7) and (7.6).
By Lemma 26, Ccc(Flc) with the above accuracy scales as 0 ((m + q/2)2q) =
O(m2q) = Q(n2q) since m = E(na) where 1/2 < a < 1 and q = o(logn). Since
2q = o(nfi) for any positive constant /,
we simply set 3 = 1 - a, and it can be directly
seen that Ccc(flc) = o(n+O) = o(n). Hence, we have reached a contradiction by
constructing a classical protocol with o(n) communication cost for the specified regime
of EXC.n,m,O, where the classical communication complexity is Q(n).
7.1.3
0
Gaps
Based on this lower bound on quantum communication cost, we can directly arrive
at the following gaps:
Corollary. For EXC,m,o with m C
e(na) where 1/2 < a < 1, the gap between
1. Qic and QcC: O(nm- 2 ) vs. Q(logn). Note that O(nm- 2 )
infinitely larger than Qic;
113
=
o(1), i.e., Qcc is
Q(n)
C cc
CC
Q
00
Q(log n)
QCC
Q/C
-+ 0
Figure 7-1: Exclusion game EXC., where m E 0(0), 1/2 < a < 1 in the large
n limit. Solid arrows indicate established separations (pointing towards the smaller
one), while the dashed one indicates an unknown separation (at most exponential).
2. QcC and Ccc: Q(log n) vs. Q(n), i.e., at most exponential.
The first statement implies the "infinite" separation between quantum information
and communication costs.
In contrast to the infinite separation of quantum and
classical information costs, the second statement upper bounds the quantum-classical
separation of communication costs to be exponential. The results about the above
separations are illustrated in Fig. 7-1.
7.2
Robustness against error
In this section, we investigate if the above results still hold when some probability of
error is allowed, using a slightly different protocol.
7.2.1
Lemmas
Same as the last section, we introduce some useful lemmas before going into details
of the main argument. We first rigorously prove a intuitive conclusion: Suppose that
the error allowed in the classical encoding of a quantum state (as introduced in the
last section) is very small. Then if we use the approximate amplitudes registered in
this classical encoding to reconstruct a quantum state, this state is very close to the
original one. The rigorous statement goes as follows:
114
Lemma 28. W is a Hilbert space of dimension
aIi) =Z_ 1(bj +icj)lj). Suppose
{ll),
we have
---
,
l
4') E)}.
W with 14')
IR| = 1, with orthonormal basis
I
=
{bj,6 } such that Vj, |bj -b|, cj - ajI
E < (6V 1)-1 (which is always satisfied in this
&ji) = Z_ 1 (b+iaj)|j)/vwhere v
(Z 1_(bl +k ) 2
is the normalizationfactor, then D(I|V), 4')) < 10/kE, D(, ) being the trace distance.
paper), and let
1) = E.
1
Proof. We first deal with the normalization factor:
v2
Z1:
2
Z + <E(Ib|+,E)2 +(cjI+ e)
=1+2 (b I+Ic )e +2le2.(7.8)
By Cauchy-Schwarz inequality, we have
(b1
and 21c2 < 2l1e
=
+
Icjl)
(7.9)
21,
V/le, so
V2 < 1 + 3-/21E.
(7.10)
1 - 2v/2IE < v 2.
(7.11)
Similarly,
Since
1
>
1 + 3V2,e
1 - 3v-21
> 1 - 3/2Ie,
(7.12)
<
1
1 - 2v 2lf
1 + 3v/2cl < 1 + 3/Wcl,
(7.13)
and
we have
1 - 3V ik<
-
< 1 + 3\/21E.
(7.14)
Assuming bj > 0, then
(bj - c)(1 - 3v/2ie) <
(bj - e)(1 + 3v/1e) <
ii
< (b + E)(1 + 3v/ie)
if bj-e> 0,
(7.15)
< (b + E)(1I + 3v 2e)
if bj - e < 0.
(7.16)
115
For both cases,
(bj + c)(1 + 3v/216) = bj + (1 + 3x/ilb)E + 3V/21E 2 < bj + (2 + 3v-ilb)E.
(7.17)
For b. - E > 0,
(bj - E)(1 - 3v/21E) > bj - (I + 3v 2lbj)E.
For b3
-
(7.18)
E < 0,
(bj - E)(1 + 3v/il) = bj - (1- 3Vi/bj)E - 3V\lE 2 > bj - (2 + 3v/lbj)E.
(7.19)
So if b3 > 0,
bj - b
< (2 + 3v 2lbj)E.
(7.20)
bj - b
< (2 - 3v'ilbj)E.
(7.21)
bj
< (2+ 3v'/iibl)E.
(7.22)
< (2 + 3v/1cj)E
(7.23)
Similarly, if bj < 0,
LI
-
So we get
Similarly,
vI
Ci - C
E) i &1j), where 6d = (b+ij)/v. Then
|aj -
aj|
;
bK-
-
-
=bj + ic, -
c
-
Recall that
< (4 + 3V-i(bj| + cjI))E
116
(7.24)
Therefore
1+ I( |
()
= (1-
)
1 - ( P4)
< 2 (1- ({))
2 - {@|1 ) -
@
law 2 -
|a>j|2 +
aejd*
-
<
|aj - d |2
<
(4+ 3(IbI
2 2
+Icj1))
6
,
(7.25)
where (4 + 3v'21(lbjl + Ici)) 2 = 16 + 24V 1b(I b3 + c) + 18l(bjI + icj ) 2 . With
(Ibjl + cj)
2
< 2(Ibj1 2 + c3! 2 ) and Eq. (7.9),
1 - (#kb)
12
<
E(16 + 242(lbj| + Icj) + 36l(lbjl 2 +
<
(161 + 481 + 361)e 2
Ic 312))E
2
100lE 2 .
(7.26)
Then
D(@), 4')) =
1
-
('12)
(7.27)
< 10VJE.
Conclusively, D(jO), 1 )) < 10vlc-.
E
Then we can directly argue that the probability distribution of the outcomes of a
measurement will be very close to the original one as well:
Lemma 29. Let {Pk} be a POVM, with Pk = (0|PI| '), Pk = (IPkI|).
Then
IPk - Pk| < 20v/1i
Proof. By Theorem 9.1 in [76], we directly obtain
IP
-
PkI
2D(10),| 1)) < 2OEV, where the last step came from Lemma 28.
117
k
IP
-
Pkj
<
0
7.2.2
Quantum-classical separation of communication
Now we show that for exclusion games where the allowed probability of error is
non-trivial, the quantum-classical separation of communication cost is at most exponential.
Theorem 30. Suppose that for EXCn,m,, where -y < 2-(+1) and m = Q(poly(n)),
there exists a winning quantum strategy H' such that Qcc(fV) = O(s).
can construct a classical strategy HlO+ such that Ccc(71?) = 0((m,
Then one
+ s/2)2s), whose
probability of error can be made arbitrarilysmall.
Proof. As can be shown, only one bit of classical communication is needed for m >
n,7-y= 2-(m+), so we shall only pay attention to the regime of even smaller probability of error, where the scalings of communication complexities may be interesting.
We revise Bob's local part of the protocol presented in Theorem 27 to devise this
Hl+ as follows. As for the zero error game, Alice prepares an 0 (2' log(1/iy)) classical
message that approximately registers the real and imaginary parts of all amplitudes
of the quantum message
I'P)
in H' to accuracy -, and sends it to Bob, who then
normalizes the amplitude vector.
However, instead of calculating the probability
distribution of the output as in Hc, Bob simply prepares a new quantum state P'D
according to the normalized state vector (by Lemma 28, this state remains close to
the original one when e
is small), and then feeds it into his original local quantum
computation. By Lemma 29, the probability of outputting the wrong answer
pM(x) < - + 20
28/2.
(7.28)
As long as M,(x) is not the winning output, i.e.,
pM ,(x) < 2,
(7.29)
Bob can apply amplitude amplification to suppress the error rate: he simply repeats
'It seems true from numerical results that for m = o(poly(n)), an 0(1) size of classical communication cannot guarantee an error rate of smaller than 2 -'.
For now I conjecture that for
-Y= 2-(m+1) and m = )(poly(n)), Ccc = 0(1).
118
his local protocol for t times (he can use the classical message to prepare as many
copies of 1I) as he wants), and outputs the string z that comes out for most times.
We denote the error rate after the whole procedure by -I', then by Chernoff bound,
for any r > 0, there exists a i such that as long as t > f, -' < r. That is, -y' can be
made arbitrarily small simply by increasing t. Combining Eqs. (7.28) and (7.29), we
can set
2- =
7_2-/2
(7.30)
20
in the protocol.
O(2-(m+1+s/2))
Note that the numerator is lower bounded by
2
-(m+1),
thus 2
=
= O(2-(m+s/2)) (we only care about s E w(1)). Therefore the com-
munication cost of I1'+ scales as O((m + s/2)2s). Note that the no-cloning theorem
is not violated since Bob does not need to copy quantum states, and we do not care
about the scaling of t since local computational resource is not limited.
l
This theorem enables us to resolve the following natural question: Is the same
infinite Qcc - Qic gap robust against some tiny error perturbation? The answer is
yes, as long as the allowed error is sufficiently small.
Maximum error
7.2.3
Note that the above arguments heavily depend on the lower bound of classical communication cost. Intuitively, as the allowed probability of error grows, the necessary
size of communication decreases. Then the question is: what is the maximal error
scaling that still requires a linear classical communication complexity? As mentioned
in the proof of Theorem 30, only one bit of classical communication is needed for
m ;>
Vn, y = 2 -(0+1), so this maximum error is expected to be even much smaller
than
2
-(m+1).
Indeed, the answer is given in the following theorem:
Theorem 31. Occ(EXCnm,) = Q(n), where m = 0(n0 ), 1/2 < a < 1, and y <
(n + I)~-.
Combining Theorems 30 and 31, we obtain the following
119
Corollary. The same infinite gap (O(nm-2 ) vs. Q(log n)) between quantum information and communication still holds for EXCn,n,y where m E O(na), 1/2 <
< 1,
and -y < (n + 1)-".
+
Theorem 30 also indicates that the QcC - Ccc gap is at most Q(s) vs. O((m
s/ 2 ) 2s). In the zero error case, we have ruled out the possibility of super-exponential
Qcc - Ccc gap in a certain regime. When some error is allowed, when does the same
conclusion holds? The answer is given in the following corollary:
Corollary. When m < O(2POIY(s)), the gap between Qcc and Ccc is at most exponential.
Note that the complete information of the behaviors of these gaps is encoded in
a 3-dimensional (Qic/Qcc/Cic/Cc-m-y) space, but some cross-sections (Fig. 71 being one) may be good enough to illustrate the important information in this
diagram.
120
Chapter 8
Concluding remarks
In this work, we considered the exclusion game and showed that an infinite gap exists
between its quantum communication complexity and quantum information complexity, whether we allow for zero error or a sufficiently small error. This work actually
deals with a very new regime of quantum communication: the error is extremely tiny,
and there are infinite separations. As expected, lots of interesting questions around
this topic remains open, including
1. Is the Q (log n) lower bound tight? Or in other words, is it possible to compress
the quantum message for a winning protocol?
2. The current proof of the lower bound fails for m E 0(n), i.e., a beyond exponential gap between Qcc and Ccc is not ruled out for this regime. What makes
this regime so special? Does the lower bound indeed fail for this regime?
3.
For the first question, we note that interesting connections between this game and
Bell nonlocality may be drawn, if there indeed is a gap between Qcc and Ccc [20.
Recently we also realized that a quadratic compression is possible while keeping the
error small enough, though it is still much weaker than the desired exponential compression.
The general formulation of communication tasks presented in this work may also
open up some new doors. Analogously, the task of quantum state discrimination,
121
where we essentially reduce k possibilities to 1, can be generalized to k -+
(k - m)
state exclusion task via semidefinite programming (SDP), which may provide insight
for the generalization of communication models.
quantum random access coding (QRAC)
174, 6]
For example, a duality between
and the exclusion game under certain
restrictions may be formalized in this spirit. We expect interesting new problems and
results to emerge in this direction.
122
Appendix A
QD and DD of real X states
Here I explicitly show the calculation of QD and DD for a subset of real X states
with the aid of results on the optimal measurement basis in [27].
A.1
Preparations
As explained in the main text, I shall only focus on X states with real entries:
Pu
goo
0
0
003
0
Q11
012
012
Q22
0
0
03
0
0
033
123
0
(A.1)
which has five degrees of freedom in total. Thus we can non-redundantly define the
five free parameters in terms of spin correlation functions:
G = (of) = tr(ozpij) = ooo + pii - 922
Gz = (os)
G-T = (or
G
-
(A.2)
233,
tr(ojpij) = oo - On + 922 - 933,
(A.3)
j) = tr[(o- 0 ojT)pi] = 2(012 + 003),
= tr[(ou' 0 u )pij]
=
= 2(012 -
Gzj = (ozo,) = tr[(ou D cx)pij] =oo - L1
-
(A.4)
(A.5)
003),
022
+ 033,
(A.6)
Here G = (or) (1
with all of which ranging in [-1,11.
i, j) and G
= (oc q)
x, y, z) denote the magnetization density at site 1 and two-site spin correlation
(a, #3
function of sites i, j, respectively. More importantly, pij can be naturally decomposed
as
1
Pij=
(
I1 + Gxxoy 0 ,x, + Gioi 0 oTY + Gzcxf 0 aJ + Gza
I 0 I+GzIou).
(A.7)
This decomposition is very useful in various contexts.
In quantum spin models, the matrix elements of pij can be expressed in terms of
spin correlation functions [89] as follows:
goo = 4(1 + Gz + Gz + GZZ),
(A.8)
(1 + Gf - Gz - Gf),
(A.9)
922 = '(1 - Gz + G - GZZ),
(A.10)
- Gz - Gz + G-7),
(A.11)
on =
033 = 1(1
=
(Gxx + GYY),
(A.12)
0>3 =
(Gfx - GY7).
(A.13)
012=t
003
2
124
Then the eigenvalues of p 3 can be expressed as
3=
A4 =
A.2
A = (1 + G-J + Gg - Gff),
(A.14)
A2
- Gx - GY
(A.15)
(1 + G
+ V4(Gf)2 + (G
j(+
Gff -
- Gf),
- G)2),
(A.16)
G))
(A.17)
4(Gf)2 +-G
Pairwise quantum correlation
Here we present the calculations of two-site QD and DD in Z2 -symmetric quantum
spin lattices in terms of pairwise correlation functions. As the scaling laws of these
correlation functions in a number of spin models have already been widely studied,
the results in this section can serve as powerful tools to analyse the scaling behaviors
of quantum and generic correlation measures.
A.2.1
Optimal basis
Evaluating QD, even for very simple cases, is extremely hard. Analytical results can
be established for very limited cases. Generally computing the exact value of QD has
been shown to be NP-complete [53]. For X states we have some approximate results
[4, 37], yet the accurate analytical formula is still unknown [27, 521. However, we still
have the following useful conclusions that can help us compute the one-way QD of
some regimes of X states of general interest:
Lemma 32 (optimal local measurement for real X states [27]). The local measurement on one subsystem (e.g., without loss of generality, B) that minimizes the quantum conditional entropy of postmeasurement bipartiteX states S(AfB), i.e., gives the
value of QD, is (i) ox, i.e., with respect to local projectors {I+)(+I,|-)(-t} where
{I+),I-)} form the eigenbasis of ux if
I
PooQ33
-
Ve11L221
125
IL0121 + 10031,
(A.18)
or (ii) or, i.e., with respect to local projectors {IO)(0|, |1)(1|} where
{I0), I1)} (com-
putational basis) is the eigenbasis of o7z if
(0121 + 1,031)
(A.19)
< (coo - ell)(933 - 022).
The basic idea for the proof is to parametrize the general two-qubit POVM {E'}
+ nk1Ok'B) }
as {
4
,
where
k
k
1, and
k
k
kk
0, or similarly
parametrize the von Neumann measurement (as will be shown in the next subsection),
and the value of S(Alb) (denoted as
in some literatures) turns out to be
SB(!AB)
a concave function whose minimum is located on the boundary. Note that we shall
assume IGTfl ;> IGYY (expressions in terms of matrix elements shown by Eq. (A.4)
and (A.5)) without loss of generality, since we can always switch the signs of the
involved entries via a local unitary transformation, in which case (i) and (ii) have
covered most possibilities, and in addition, even if we adopt uo
or oi as the optimal
measurement for all X states there is shown to be only a very small error for very
few cases numerically. Using o
or o-
as the optimal measurement suffices for our
analysis.
A.2.2
Explicit calculation of pairwise QD
Now I evaluate QD for the above regimes of real X states with the aid of above
results. Here I adopt the difference of quantum conditional entropy between pre- and
postmeasurement states as the mathematical form of QD and use the conclusions
of optimal measurements to explicitly express pairwise QD in terms of correlation
functions.
First from the eigenvalues given in Eq. (A.14)-(A.17),
von Neumann entropy of the joint state S(ij).
local measurement is done on
we can easily obtain the
Without loss of generality, assume
j. Taking advantage of the decomposition Eq. (A.7),
we simply trace out subsystem i and obtain the reduced density matrix
p3 =(I, + vof)
=
126
(Ij + Gio),
(A.20)
Gf)/2. Hence the von Neumann entropy of the subsystem is
with eigenvalues (1
So) =
1 + Gz
1+Gz
1-Gz
1-Gf
2
2log
2
2
2
2
2
2
2
(A.21)
So, the quantum conditional entropy of the original two-site state is given as
+Gz
1 -Gz
2G log (1 + Gz)+
2G' log (1 - Gz)-1,
41
S(ilj) = S(ij)-S(j)
=
A. log Ac+
a=1
(A.22)
with {Aa} given in Eq. (A.14)-(A.17).
Next, according to Lemma 32 and the immediate discussions, there are two classes
of postmeasurement states obtained by corresponding optimal measurement bases (i)
{I+)(+I, -)(--I} or (ii) {I0)(0I, 11)(11}. Now we compute the quantum conditional
entropies of these two classes, and hence QDs, respectively:
Class (i):
{|+)(+I,|-)(-I} as the optimal measurement basis. For pij in this class,
the local measurement operation on the original bipartite density matrix is (Ii 0
I
)(
I), with two
possible outcomes, whose corresponding postmeasurement density
matrices are given by
(Ii 0& +)(+I)pij
=Ij 91+) (+ I + Gi o
+ Gyyuiy 0
2
"
(1
+)(+|
-1
Gzorz 0 1+)(*Ij +
.
2
-1@
0
0GjIi
2
127
(.1
1 -1
+ -G.-C
1
1
-1).
(A.23)
--1
,
=
'
pZ3
I-) (-j)pij
+ Gyoy (9
2 "'
(1
G.. -1.
+
1
1:1
)
(I 0
p"
22
1i)
,]
(A.24)
.
1
respectively, with equal probabilities pi
= P2 =
1/2.
Hence the reduced density
matrices after tracing out j read
pi,+ = 2trjp _ = !(I + G oa + Gzu-),
(A.25)
pi,_ = 2trjp.:,+ = !(I-
(A.26)
G-ox
+ Gfoz),
whose spectra are the same:
(G-f)
1
(Gz) 2
2
=
A+,=
2+
(A.27)
2
The prefactors 2 come from probabilities on the denominators.
So we obtain the
quantum conditional entropy of the postmeasurement (optimal) state as
S(ilj)
= -A+,+
log A+,+
-
A+,- log A+,-,
(A.28)
which finally gives us the value of QD (for the first class we present the full expression
in terms of eigenvalues):
D(j -+ i)
=
-A+,+ log A+,+ - A+,- log A+,-
+
A, log Aa
+2G
log (I+ Gz) - 12Gi log (1 - Gf) + 1, (A.29)
Cf=1
where the A's are given earlier.
128
Class (ii): { 10) (0 1, 11) (1} as the optimal measurement basis. As the calculation procedures are similar as (i), I directly introduce some important results here.
The
postmeasurement density matrices are
0 -i
0 1
pig~ - [Is 0 |)(0|i +G of T@ +Gy~or
Pi3'O40
0
.3
0
0
(
i
+(Gzz + Gz)o 0 10)(01j + GzI 0 10)(0j,
A3114
[I
0 0
0
cj i[hI
~
io
1 0
)
11
(1I
XOX(
(A.30)
+(-Gzz + Gz)e< 0 I1)(1I| + GIi E 11)(11j,
i
0
(A.31)
and the corresponding reduced density matrices for i are
1
p,o =1 [(1 + G7)I + (Gz + Gz)o-z,
i[
P 1 2~=
Pi,1 =
(A.32)
[(1 - Gj)I + (-Gzz + Gz)o-z],
(A.33)
which are already diagonal in matrix form, but notice that the eigenvalues corresponding to the two measurement outcomes are no longer the same. We omit boring
technical steps here, and finally we will obtain
D(j
-+
i) = -
S
Ak,1logAk,1
-
S(ilj)
(A.34)
{k=0,1}
{l=+,-}
where S(ilj) is given by Eq. (A.22), and
Ao,+ =
A1,
1
Gz),
(A.35)
1
(1 - Gj -F Gzi t Gz).
2
(A.36)
( 1 + Gz t Gz
Alternatively, we can apply the following projector parametrization method to
transform the problem of optimization over measurements into optimization over
129
scalar parameters. For the two-qubit case, we can parametrize the local projectors as
{VIO)(OV,VI1)(1IV'} where
sin e--iO
cos
V=
sin Oe'q'
and 0
(A.37)
,
2
- cos
c [0, 7r], # E [0, 27), i.e., V E SU(2). Now QD can be written as an optimization
over variables 0 and
A.2.3
#,
and equivalent results are expected for real X states.
Pairwise DD
Here I consider another candidate measure of quantum correlations, DD. By symmetry, the reduced state is always diagonal in the computational basis. Therefore, it is
the Schmidt basis of real X states: o
is automatically the local measurement basis
in the definition of DD. That is,
DD(j -+ i) = -
Ak,1
log Ak,1
-
S(ilj),
(A.38)
{k=0,1}
{=+,-}
by Eq. (A.34). Note that Eq. (A.38) can be extended to all real X states that are
not in these two classes.
130
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