JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 43, NUMBER 9
SEPTEMBER 2002
The uniqueness theorem for entanglement measures
Matthew J. Donalda)
The Cavendish Laboratory, Madingley Road,
Cambridge, CB3 0HE, United Kingdom
Michał Horodeckib)
Institute of Theoretical Physics and Astrophysics, University of Gdańsk,
80-952 Gdańsk, Poland
Oliver Rudolphc)
Quantum Optics & Information Group, Dipartimento di Fisica ‘‘A. Volta,’’
Università di Pavia, Via Bassi 6, I-27100 Pavia, Italy
共Received 12 November 2001; accepted for publication 12 March 2002兲
We explore and develop the mathematics of the theory of entanglement measures.
After a careful review and analysis of definitions, of preliminary results, and of
connections between conditions on entanglement measures, we prove a sharpened
version of a uniqueness theorem which gives necessary and sufficient conditions
for an entanglement measure to coincide with the reduced von Neumann entropy on
pure states. We also prove several versions of a theorem on extreme entanglement
measures in the case of mixed states. We analyze properties of the asymptotic
regularization of entanglement measures proving, for example, convexity for the
entanglement cost and for the regularized relative entropy of entanglement. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1495917兴
I. INTRODUCTION
Quantifying entanglement1– 4 is one of the central topics of quantum information theory. Any
function that quantifies entanglement is called an entanglement measure. Entanglement is a complex property of a state and, for arbitrary states, there is no unique definitive measure. In general,
there are two ‘‘regimes’’ under which entanglement can be quantified: they may be called the
‘‘finite’’ and the ‘‘asymptotic’’ regimes. The first deals with the entanglement of a single copy of
a quantum state. In the second, one is interested in how entanglement behaves when one considers
tensor products of a large number of identical copies of a given state. It turns out that by studying
the asymptotic regime it is possible to obtain a clearer physical understanding of the nature of
entanglement. This is seen, for example, in the so-called ‘‘uniqueness theorem’’ 3– 6 which states
that, under appropriate conditions, all entanglement measures coincide on pure bipartite states and
are equal to the von Neumann entropy of the corresponding reduced density operator. However,
this theorem was never rigorously proved under unified assumptions and definitions. Rather, there
are various versions of the argument scattered through the literature.
In Ref. 4, the uniqueness theorem was put into a more general perspective. Namely, there are
two basic measures of entanglement1—entanglement of distillation (E D ) and entanglement cost
(E C )—having the following dual meanings:
共i兲
共ii兲
E D (%) is the maximal number of singlets that can be produced from the state % by means
of local quantum operations and classical communication 共LQCC operations兲.
E C (%) is the minimal number of singlets needed to produce the state % by LQCC operations.
a兲
Electronic mail: matthew.donald@phy.cam.ac.uk
Electronic mail: fizmh@univ.gda.pl
c兲
Author to whom correspondence should be addressed. Electronic mail: rudolph@fisicavolta.unipv.it
b兲
0022-2488/2002/43(9)/4252/21/$19.00
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© 2002 American Institute of Physics
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4253
关More precisely 共cf. Definitions 16 and 17兲: E D (%) (E C (%)) is the maximal 共minimal兲 number of
singlets per copy distillable from the state % 共needed to form %兲 by LQCC operations in the
asymptotic regime of n→⬁ copies.兴 It is important here that the conversion is not required to be
perfect: the transformed state needs to converge to the required state only in the limit of large n.
Now, in Ref. 4 it was shown that the two basic measures of entanglement are, respectively, a lower
and an upper bound for any entanglement measure satisfying appropriate postulates in the
asymptotic regime.7 This suggests the following clear picture: entanglement cost and entanglement
of distillation are extreme measures, and provided they coincide on pure states, all other entanglement measures coincide with them on pure states as well. However, as mentioned above, the fact
that E D and E C coincide on pure states was not proven rigorously. Moreover, it turned out that the
postulates are too strong. They include convexity, and some additivity and continuity requirements. It is not known whether any measure exists which satisfies all the requirements. E D and E C
satisfy the additivity requirement, but it is not known whether or not they are continuous in the
sense of Ref. 4. There are also indications that the entanglement of distillation is not convex.8 On
the other hand, two other important measures, the entanglement of formation 共denoted by E F 兲 and
the relative entropy of entanglement 共denoted by E R 兲 are continuous6,9 and convex, but there are
problems with additivity. The relative entropy of entanglement is certainly not additive,10 and we
do not know about the entanglement of formation.
In this situation it is desirable to prove the uniqueness theorem from first principles, and to
study to what extent we can relax the assumptions and still get uniqueness of entanglement
measures on pure states. In the present article we have solved the problem completely by providing necessary and sufficient conditions for a measure of entanglement to be equal to the von
Neumann entropy of the reduced density operator for pure states. We also show that if we relax the
postulate of asymptotic continuity, then any measure of entanglement 共not unique any longer兲 for
pure states must lie between the two analogs of E D and E C corresponding to perfect fidelity of
conversion. These are Ẽ C ( ␺ )⫽S 0 (%) and Ẽ D ( ␺ )⫽S ⬁ (%), where % is the reduced density matrix
of 兩 ␺ 典 , and S 0 , S ⬁ are Rènyi entropies. In Refs. 11 and 12, one of us has studied entanglement
measures based on cross norms and proved an alternative uniqueness theorem for entanglement
measures stemming from the Khinchin–Faddeev characterization of Shannon entropy.
The present article also contains further developments on the problem of extreme measures.
We provide two useful new versions of the theorem of Ref. 4. In one of them, we show that for
any 共suitably normalized兲 function E for which the regularization E ⬁ (%)⫽limn→⬁ E(% 丢 n )/n exists and which is (i) nonincreasing under local quantum operations and classical communication
共LQCC operations兲 and (ii) asymptotically continuous, the regularization E ⬁ must lie between E D
and E C . The theorem and its proof can easily be generalized by replacing the class of LQCC
operations by other classes of operations, or by considering conversions between any two states.
Moreover, it is valid for multipartite cases. Therefore, the result will be an important tool for
analyzing asymptotic conversion rates between different states. In particular, it follows from our
result that to establish irreversibility of conversion between two states 共see Ref. 13兲, one needs to
compare regularizations of asymptotically continuous entanglement measures for these states.
In the other new version of the extreme measures theorem, we are able to weaken the postulates of Ref. 4, so that they are at least satisfied by E C . On the other hand, we do not have a proof
that E D is asymptotically continuous for mixed states, although there is strong evidence that this
is the case. If it is, then we would finally have a form of the theorem, in which both E D and E C
could be called extreme measures, not only in the sense provided by the inequalities we prove, but
also in the sense that they belong to the set described by the postulates.
To obtain our results we perform a detailed study of possible postulates for entanglement
measures in the finite and the asymptotic regime. In particular, we examine which postulates
survive the operation of regularization. We show that if a function is convex and subadditive 关i.e.,
f (% 丢 ␴ )⭐ f (%)⫹ f ( ␴ )兴, then its regularization is convex, too. Hence, both the regularization of
the relative entropy of entanglement2 as well as of the entanglement of formation1 are convex.
It should be emphasized that our results are stated and proved in language accessible for
mathematicians or mathematical physicists who have not previously been involved in quantum
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4254
J. Math. Phys., Vol. 43, No. 9, September 2002
Donald, Horodecki, and Rudolph
information theory. This is in contrast to many papers in this field, where many implicit assumptions are obstacles for understanding the meaning of the theorems and their proofs by nonspecialists. For this reason, we devote Secs. II and III to careful statements of some essential definitions
and results. In Sec. IV we present a self-contained and straightforward proof of the difficult
implication in Nielsen’s theorem. This is a theorem which we shall use several times. Properties of
entanglement measures and relations between them are analyzed in Sec. V. The most prominent
entanglement measures—entanglement of distillation, entanglement cost, entanglement of formation and relative entropy of entanglement—are defined and studied in Sec. VI. In Sec. VII we
present our versions of the theorem on extreme measures. Finally, Sec. VIII contains our version
of the uniqueness theorem for entanglement measures, stating necessary and sufficient conditions
for a functional to coincide with the reduced von Neumann entropy on pure states.
II. PRELIMINARIES
Throughout this article, all spaces considered are assumed to be finite dimensional. The set of
trace class operators on a Hilbert space H is denoted by T共H兲 and the set of bounded operators on
H by B共H兲. A density operator 共or state兲 is a positive trace class operator with trace one. The set
of states on H is denoted by ⌺共H兲 and the set of pure states by ⌺ p (H). The trace class norm on
T共H兲 is denoted by 储 • 储 1 . For a wavefunction 兩 ␺ 典 苸H the corresponding state will be denoted by
P ␺ ⬅ 兩 ␺ 典具 ␺ 兩 . The support of a trace class operator is the subspace spanned by its eigenvectors
with nonzero eigenvalues.
In the present article we restrict ourselves mainly to the situation of a composite quantum
system consisting of two subsystems with Hilbert space H A 丢 H B where H A and H B denote the
Hilbert spaces of the subsystems. Often these systems are to be thought of as being spatially
separate and accessible to two independent observers, Alice and Bob.
Definition 1: Let H A and H B be Hilbert spaces. A density operator % on the tensor product
A
H 丢 H B is called separable or disentangled if there exist a sequence (r i ) of positive real numbers, a sequence ( ␳ Ai ) of density operators on H A and a sequence ( ␳ Bi ) of density operators on
H B such that
%⫽
兺i r i ␳ Ai 丢 ␳ Bi ,
共1兲
where the sum converges in trace class norm.
The Schmidt decomposition14 is of central importance in the characterization and quantification of entanglement associated with pure states.
Lemma 2: Let H A and H B be Hilbert spaces and let 兩 ␺ 典 苸H A 丢 H B . Then there exist a
sequence of non-negative real numbers (p i ) i summing to one and orthonormal bases ( 兩 a i 典 ) i and
( 兩 b i 典 ) i of H A and H B , respectively, such that
兩␺典⫽
兺i 冑p i兩 a i 丢 b i 典 .
By S(%) we will denote von Neumann entropy of the state % given by
S 共 % 兲 ª⫺tr% log2 %.
䊏
共2兲
The von Neumann reduced entropy for a pure state ␴ on a tensor product Hilbert space
H A 丢 H B is defined as
S vN共 ␴ 兲 ª⫺trA 共共 trB ␴ 兲 log2 共 trB ␴ 兲兲 ,
共3兲
where trA and trB denote the partial traces over H A and H B , respectively. For ␴ ⫽ P ␺ ⫽ 兩 ␺ 典具 ␺ 兩 , it
is a straightforward consequence of Lemma 2 that
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
⫺trA 共共 trB P ␺ 兲 log2 共 trB P ␺ 兲兲 ⫽⫺trB 共共 trA P ␺ 兲 log2 共 trA P ␺ 兲兲 ⫽⫺
4255
兺i p i log2 p i ,
where (p i ) i denotes the sequence of Schmidt coefficients of 兩 ␺ 典 . However, for a general mixed
state ␴, trA ((trB ␴ )log2(trB ␴ )) may not equal trB ((trA ␴ )log2(trA ␴ )).
III. CLASSES OF QUANTUM OPERATIONS
In quantum information theory it is important to distinguish between the class of quantum
operations on a composite quantum system which can be realized by separate local actions on the
subsystems 共i.e., separate actions by ‘‘Alice’’ and by ‘‘Bob’’兲 and those which cannot. The class of
local quantum operations assisted by classical communication 共LQCC兲 is of central importance in
quantum cryptography and the emerging theory of quantum entanglement.
An operation is a positive linear map ⌳:T(H1 )→T(H2 ) such that tr(⌳( ␴ ))⭐1 for all ␴
苸⌺(H1 ). Quantum operations are operations which are completely positive.15,16 We shall be
interested in the trace preserving quantum operations. By the Choi–Kraus representation,15–18
these are precisely the linear maps ⌳:T(H1 )→T(H2 ) which can be written in the form ⌳(B)
n1n2
n1n2 †
W i BW †i for B苸T(H1 ) with operators W i :H1 →H2 satisfying 兺 i⫽1
W i W i ⫽11 , where
⫽ 兺 i⫽1
n 1 ⬅dimH1 , n 2 ⬅dimH2 , and 11 is the identity operator on H1 . These can also be characterized
as precisely the linear maps which can be composed out of the following elementary operations.
共O1兲 Adding an uncorrelated ancilla:
⌳ 1 :T(H1 )→T(H1 丢 K1 ), ⌳ 1 ( ␳ )ª ␳ 丢 ␴ , where H1 and K1 denote the Hilbert spaces of the
original quantum system and of the ancilla, respectively, and where ␴ 苸⌺(K1 ).
共O2兲 Tracing out part of the system:
⌳ 2 :T(H2 丢 K2 )→T(H2 ), ⌳ 2 ( ␳ )ªtrK2 ( ␳ ), where H2 丢 K2 and K2 denote the Hilbert spaces of
the full original quantum system and of the dismissed part, respectively, and where trK2 denotes
the partial trace over K2 .
共O3兲 Unitary transformations:
⌳ 3 :T(H3 )→T(H3 ), ⌳ 3 ( ␳ )ªU ␳ U † , where U is a unitary operator on H3 .
A discussion of this material with complete proofs from first principles may be found in the
initial archived draft of this article.19
Defining a local operation as quantum operation on a individual subsystem, we now turn to
the definition of local operations assisted by classical communication. As always in this article we
consider a quantum system consisting of two 共possibly separate兲 subsystems A and B with 共initial兲
Hilbert spaces H A and H B , respectively. There are three cases: the communication between A and
B can be unidirectional 共in either direction兲 or bidirectional.
Let us first define the class of local quantum operations 共LO兲 assisted by unidirectional classical communication 共operations in this class will be called one-way LQCC operations兲 with
direction from system A 共Alice兲 to system B 共Bob兲. In this case, the operations performed by Bob
depend on Alice’s operations, but not conversely.
Definition 3: A completely positive map ⌳:T(H A1 丢 H B1 )→T(H A2 丢 H B2 ) is called a one-way
LQCC operation from A to B if it can be written in the form
K,L
⌳共 ␴ 兲⫽
兺
i, j⫽1
B
A
B†
共 1A2 丢 W Bji 兲共 V Ai 丢 1B1 兲 ␴ 共 V A†
i 丢 11 兲共 12 丢 W ji 兲
共4兲
for all ␴ 苸T(H A1 丢 H B1 ) and some sequences of operators (V Ai :H A1 →H A2 ) i and (W Bji :H B1
K
A
A
L
B† B
B
A
B
A
→H B2 ) ji with 兺 i⫽1
V A†
i V i ⫽11 and 兺 j⫽1 W ji W ji ⫽11 for each i, where 11 , 11 and 12 are the
A
B
A
unit operators acting on the Hilbert spaces H 1 , H 1 and H 2 , respectively.
Of course, by the Choi–Kraus representation any operation ⌳ of the form
⌳⫽⌳ A 丢 I B1 ,
共5兲
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4256
J. Math. Phys., Vol. 43, No. 9, September 2002
Donald, Horodecki, and Rudolph
where ⌳ A :T(H A1 )→T(H A2 ) is a completely positive trace preserving map and I B1 is the identity
operator on T(H B1 ), is a one-way LQCC operation from A to B.
Let us now define local quantum operations assisted by bidirectional classical communication
共LQCC operations兲.
Definition 4: A completely positive map ⌳:T(H A 丢 H B )→T(K A 丢 K B ) is called an LQCC
n⫹1
n⫹1
and (H Bk ) k⫽1
with
operation if there exist n⬎0 and sequences of Hilbert spaces (H Ak ) k⫽1
A(B)
A(B)
A(B)
A(B)
and H n⫹1 ⫽K
, such that ⌳ can be written in the following form,
H 1 ⫽H
K 1 ,... ,K 2n
⌳共 ␴ 兲⫽
V iAB,... ,i
兺
,... ,i ⫽1
i1
1
2n
2n
†
␴ V iAB
1 ,... ,i 2n
共6兲
for all ␴ 苸T(H A 丢 H B ) where V iAB,... ,i :H A 丢 H B →K A 丢 K B is given by
1
i
A
2n
丢 W 2n
V iAB,... ,i ª 共 1n⫹1
1
,... ,i 1
2n
2n
i
i 2n⫺2 ,... ,i 1
i ,i
i
,... ,i 1
B
A
丢 1n 兲共 1n 丢 W 2n⫺2
兲 ¯ 共 1A2 丢 W 22 1 兲共 V 11 丢 1B1 兲
i
,... ,i 1
A
n
:H Ak →H k⫹1
,
兲 k⫽1
2n⫺1
兲共 V 2n⫺1
with families of operators
2k⫺1
共 V 2k⫺1
i
共 W 2k2k
,... ,i 1
B
n
:H Bk →H k⫹1
,
兲 k⫽1
共7a兲
共7b兲
such that for k⫽0, . . . ,n⫺1 and each sequence of indices (i 2k ,... ,i 1 ),
K 2k⫹1
兺
i 2k⫹1 ⫽1
i
2k⫹1
共 V 2k⫹1
,... ,i 1 † i 2k⫹1 ,... ,i 1
A
⫽1k⫹1
,
兲 V 2k⫹1
共8a兲
and for k⫽1, . . . ,n and each sequence of indices (i 2k⫺1 ,... ,i 1 ),
K 2k
兺
i 2k ⫽1
i
共 W 2k2k
,... ,i 1 †
i
兲 W 2k2k
,... ,i 1
⫽1Bk ,
共8b兲
where for all k⬎0, 1Ak and 1Bk denote the unit operator on H Ak and H Bk , respectively.
Obviously the class of one-way LQCC operations is a subclass of the class of LQCC operations. There is another important class: separable operations. A separable operation is an operation
of the form
k
⌳:T共 H
A
丢H
B
兲 →T共 K 丢 K 兲 ,
A
B
⌳共 ␴ 兲⬅
兺 共 V i 丢 W i 兲␴共 V i 丢 W i 兲†,
i⫽1
共9兲
with 兺 i⫽1 (V i 丢 W i ) † V i 丢 W i ⫽1AB where 1AB denotes the unit operator acting on H A 丢 H B . The
class of separable operations is strictly larger than the LQCC class.20
One can also consider a small class obtained by taking the convex hull C of the set of all maps
of the form ⌳ A 丢 ⌳ B . Such operations require in general one-way classical communication, but
they do not cover the whole class of one-way LQCC operations.
All the classes above are closed under tensor multiplication, convex combinations, and composition. The results of our article apply in principle to all the classes apart from the last 共i.e., apart
from the class of all operations in the convex hull C of the set of all maps of the form ⌳ A 丢 ⌳ B 兲.
For definiteness, in the sequel we will use LQCC operations.
Finally, we conclude this section with a useful technical lemma.
Lemma 5: Let ⌳:T(H1 )→T(H2 ) be a positive trace-preserving map and suppose that
B苸T(H1 ) with B⫽B † . Then 储 ⌳(B) 储 1 ⭐ 储 B 储 1 .
n1
␤ i 兩 ␺ i 典具 ␺ i 兩 . Then
Proof: Suppose that B has eigenvalue expansion B⫽ 兺 i⫽1
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4257
n1
储 ⌳ 共 B 兲储 1 ⭐
兺 兩 ␤ i兩 储 ⌳ 共 兩 ␺ i 典具 ␺ i兩 兲储 1 ⫽ 储 B 储 1
i⫽1
n
1
as 储 B 储 1 ⫽ 兺 i⫽1
兩 ␤ i 兩 and ⌳( 兩 ␺ i 典具 ␺ i 兩 ) is a positive trace class operator with unit trace.
䊏
IV. NIELSEN’S THEOREM
A beautiful and powerful result of entanglement theory is Nielsen’s theorem.21 In one direction, the proof is straightforward, and we refer to Ref. 21. The other direction is more difficult. We
present here an entirely self-contained, simple, and direct proof. Alternative proofs have previously been given by Hardy22 and by Jensen and Schack.23
Before we state the theorem we need the following definition.
m1
m2
and (q i ) i⫽1
be two probability distributions with probabilities arDefinition 6: Let (p i ) i⫽1
ranged in decreasing order, i.e., p 1 ⭓p 2 ⭓¯⭓p m 1 and similarly for (q i ) i . Then we will say that
(q i ) i majorizes (p i ) i 关in symbols (q i ) i Ɑ(p i ) i 兴 if for all k⭐min兵m1 ,m2其 we have
k
兺
i⫽1
k
q i⭓
兺
i⫽1
pi .
共10兲
M
M
Theorem 7 „Nielsen…: Let H A and H B be Hilbert spaces and let ( 兩 ␹ m 典 ) m⫽1
and ( 兩 ␬ m 典 ) m⫽1
M
A
B
be orthonormal bases for H and H , respectively. Let 兩 ⌿ 典 ⫽ 兺 m⫽1 冑p m 兩 ␹ m ␬ m 典 and 兩 ⌽ 典
M
冑q m 兩 ␹ m ␬ m 典 be Schmidt decompositions of normalized vectors 兩 ⌿ 典 and 兩 ⌽ 典 in
⫽ 兺 m⫽1
A
H 丢 H B with p 1 ⭓p 2 ⭓¯⭓ p M and q 1 ⭓q 2 ⭓¯⭓q M . Then 兩 ⌿ 典具 ⌿ 兩 can be converted into
兩 ⌽ 典具 ⌽ 兩 by LQCC operations if and only if (q i ) majorizes (p i ).
Proof: 共One direction only.兲 Suppose that (q i ) majorizes (p i ). Set ␳ ⬅ 兩 ⌿ 典具 ⌿ 兩 and ␴
N
⬅ 兩 ⌽ 典具 ⌽ 兩 . We shall prove that there is a sequence (⌳ n ) n⫽1
with N⬍M of completely positive
A
B
maps on T(H 丢 H ) of the form
⌳ n共 ␻ 兲 ⫽ 共 C n 丢 U n 兲 ␻ 共 C n 丢 U n 兲 †⫹ 共 D n 丢 V n 兲 ␻ 共 D n 丢 V n 兲 †,
共11兲
where U n , V n 苸B(H B ) are unitary and C n , D n 苸B(H A ) satisfy C †n C n ⫹D †n D n ⫽1A such that
⌳ 1 ⴰ⌳ 2 ⴰ¯ⴰ⌳ N ( ␳ )⫽ ␴ . Note that all the ⌳ n are one-way LQCC operations from A to B and hence
their composition also is. As the Schmidt decomposition is symmetrical between A and B, we
k
k
q m ⫺ 兺 m⫽1
p m for k
could also use one-way LQCC operations from B to A. Set ␦ k ⬅ 兺 m⫽1
⫽1,2,...,M . Then ␦ M ⫽0. Let N⫽N( 兩 ⌿ 典 , 兩 ⌽ 典 ) be the number of nonzero ␦ k . We shall prove the
result by induction on N. 兩 ⌿ 典 ⫽ 兩 ⌽ 典 if and only if ␦ 1 ⫽ ␦ 2 ⫽¯⫽ ␦ M ⫺1 ⫽0. In this case
N( 兩 ⌿ 典 , 兩 ⌽ 典 )⫽0, ␳ ⫽ ␴ , and the result is certainly true.
Suppose that the result holds for all pairs ( 兩 ⌿ 典 , 兩 ⌽ 典 ) satisfying the conditions of the proposition with N( 兩 ⌿ 典 , 兩 ⌽ 典 )⫽0,...,L and that ( 兩 ⌿ 典 , 兩 ⌽ 典 ) is a pair with N( 兩 ⌿ 典 , 兩 ⌽ 典 )⫽L⫹1. Then
there exists J⭓1 such that ␦ 1 ⫽ ␦ 2 ⫽¯⫽ ␦ J⫺1 ⫽0 and ␦ J ⬎0. Setting ␦ 0 ª0, we have q j ⫺ p j
⫽ ␦ j⫺1 ⫹q j ⫺p j ⫽ ␦ j for j⫽1,...,J. This implies that p j ⫽q j for j⫽1,...,J⫺1 and that q J ⬎ p J .
Suppose that ␦ k ⬎0 for k⫽J,J⫹1,...,K⫺1 and that ␦ K ⫽0. p K ⫺q K ⫽ p K ⫺q K ⫹ ␦ K ⫽ ␦ K⫺1 and
p K ⬎q K . Moreover, if K⬍M , then q K⫹1 ⫺ p K⫹1 ⫽ ␦ K ⫹q K⫹1 ⫺ p K⫹1 ⫽ ␦ K⫹1 ⭓0. Summarizing,
we have
p J⫺1 ⫽q J⫺1 ⭓q J ⬎ p J ⭓ p K ⬎q K ⭓q K⫹1 ⭓ p K⫹1 .
M
Define (r m ) m⫽1
by r m ª p m for m⫽J,K and by r J ªp J ⫹ ␦ , r K ªp K ⫺ ␦ where ␦ ªmin兵␦k :k
⫽J,...,K⫺1其. By construction ␦ ⬎0. Now ␦ ⭐ ␦ J implies q J ⭓r J ⭓ p J , and ␦ ⭐ ␦ K⫺1 implies p K
⭓r K ⭓q K . This in turn implies that r 1 ⭓r 2 ⭓¯⭓r M . Thus for k⫽1,...,J⫺1 and for k
⫽K,...,M ,
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J. Math. Phys., Vol. 43, No. 9, September 2002
k
兺
m⫽1
Donald, Horodecki, and Rudolph
k
r m⫽
k
兺
m⫽1
p m⭐
兺
m⫽1
qm .
k
k
For k⫽J, . . . ,K⫺1, 兺 m⫽1
r m ⫽ 兺 m⫽1
p m ⫹ ␦ and so, as 0⬍ ␦ ⭐ ␦ k ,
k
兺
m⫽1
k
p m⬍
k
兺
m⫽1
r m⭐
兺
m⫽1
qm .
M
冑r m 兩 ␹ m ␬ m 典 . Then N( 兩 ⌶ 典 , 兩 ⌽ 典 )⭐L so that, by the inductive hypothesis, there
Define 兩 ⌶ 典 ª 兺 m⫽1
N
of maps of the required form with N⫽N( 兩 ⌶ 典 , 兩 ⌽ 典 ) such that
is a sequence (⌳ n ) n⫽1
⌳ 1 ⴰ⌳ 2 ⴰ¯ⴰ⌳ N 共 兩 ⌶ 典具 ⌶ 兩 兲 ⫽ ␴ .
Thus to complete the proof, we need only find a completely positive map ⌳ of the required form
such that
⌳ 共 兩 ⌿ 典具 ⌿ 兩 兲 ⫽ 兩 ⌶ 典具 ⌶ 兩 .
共12兲
To this end set Pª 兺 m⫽J,K 兩 ␹ m 典具 ␹ m 兩 . Set
Cª
冑
r J p J ⫺r K p K
r 2J ⫺r K2
冉 冑
rJ
兩 ␹ 典具 ␹ J 兩 ⫹
pJ J
P⫹
冑
rK
兩 ␹ 典具 ␹ K 兩
pK K
冊
and Uª1B . Set
Dª
冑
r J p K ⫺r K p J
r 2J ⫺r K2
冉 冑
rK
兩 ␹ 典具 ␹ J 兩 ⫹
pJ K
P⫹
冑
rJ
兩 ␹ 典具 ␹ K 兩
pK J
冊
and
Vª 兩 ␬ K 典具 ␬ J 兩 ⫹ 兩 ␬ J 典具 ␬ K 兩 ⫹
兺
m⫽J,K
兩 ␬ m 典具 ␬ m 兩 .
Note that p J ⭓p K ⬎q K ⭓0, that r J ⬎r K , that r J p J ⬎r K p K , and that r J p K ⫺r K p J ⫽(p J ⫹ ␦ )p K
⫺(p K ⫺ ␦ )p J ⫽ ␦ (p K ⫹p J )⬎0. Note also that r 2J ⫺r K2 ⫽(r J ⫺r K )(r J ⫹r K )⫽(r J ⫺r K )(p J ⫹ p K ) so
that
r J p J ⫺r K p K
r 2J ⫺r K2
⫹
r J p K ⫺r K p J
r 2J ⫺r K2
⫽1.
With these definitions and notes, the completion of the proof is straightforward.
䊏
V. ENTANGLEMENT MEASURES
Definition 8: Consider a bipartite composite quantum system with Hilbert space of the form
H A 丢 H B where H A ⬅H B ⬅Cd . Assume that isomorphisms between Cd , H A , and H B are chosen.
d
of Cd , we let
For a chosen orthonormal basis ( 兩 ␺ i 典 ) i⫽1
d
兩 ⌿ ⫹ 共 Cd 兲 典 ⬅
1
兺 兩 ␺ i 丢 ␺ i 典 苸H A 丢 H B .
i⫽1 冑d
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4259
兩 ⌿ ⫹ (Cd ) 典 is a maximally entangled wavefunction. All other maximally entangled wavefunctions in
H A 丢 H B can be obtained by applying a unitary operator of the form 1A 丢 U B to 兩 ⌿ ⫹ (Cd ) 典 where
U B is a unitary operator on H B . The pure state corresponding to 兩 ⌿ ⫹ (Cd ) 典 will be denoted by
P ⫹ (Cd )⬅ 兩 ⌿ ⫹ (Cd ) 典具 ⌿ ⫹ (Cd ) 兩 .
In an arbitrary bipartite composite system, we shall refer to any wavefunction with the same
Schmidt coefficients as 兩 ⌿ ⫹ (Cd ) 典 as a representative of 兩 ⌿ ⫹ (Cd ) 典 and to the corresponding state
as a representative of P ⫹ (Cd ).
A. Conditions on mixed states
The degree of entanglement of a density operator on the Hilbert space of a bipartite composite
quantum system can be expressed by an ‘‘entanglement measure.’’ This is a non-negative realvalued functional E defined on ⌺(H A 丢 H B ) for all finite-dimensional Hilbert spaces H A and
H B . Any of the following conditions might be imposed on E. 1– 4,24
共E0兲 If ␴ is separable, then E( ␴ )⫽0.
d
d
is any representative of P ⫹ (Cd ), then E( P ⫹
)⫽log2 d for d
共E1兲 共Normalization兲 If P ⫹
⫽1,2, . . . .
A weaker condition is the following.
(E1 ⬘ ) E( P ⫹ (C2 ))⫽1.
共E2兲 共LQCC Monotonicity兲 Entanglement cannot increase under procedures consisting of local
operations on the two quantum systems and classical communication. If ⌳ is an LQCC operation,
then
E 共 ⌳ 共 ␴ 兲兲 ⭐E 共 ␴ 兲
共13兲
for all ␴ 苸⌺(H A 丢 H B ).
A condition which, as we shall confirm below 共Lemma 9兲, is weaker than 共E2兲, is the following.
(E2 ⬘ ) E(⌳( ␴ ))⫽E( ␴ ) whenever ␴ 苸⌺(H A 丢 H B ) and ⌳ is a strictly local operation which
is either unitary or which adds extraneous dimensions. On Alice’s side, these local operations take
the form of either ⌳ 1 (%)⫽(U A 丢 I B )%(U A 丢 I B ) † where U A :H A →H A is unitary or ⌳ 2 (%)
⫽(W A 丢 I B )%(W A 丢 I B ) † where H A 傺K A and W A :H A →K A is the inclusion map. There are
equivalent local operations on Bob’s side.
(E2 ⬙ ) E(⌳( ␴ ))⫽E( ␴ ) whenever ␴ 苸⌺(H A 丢 H B ) and ⌳ is a strictly local unitary operation.
Without further remark, we shall always assume that all our measures satisfy (E2 ⬙ ).
共E3兲 共Continuity兲 Let (H An ) n苸N and (H Bn ) n苸N be sequences of Hilbert spaces and let Hn
⬅H An 丢 H Bn for all n. For all sequences (% n ) n苸N and ( ␴ n ) n苸N of states with % n , ␴ n 苸
⌺(H An 丢 H Bn ), such that 储 % n ⫺ ␴ n 储 1 →0, we require that
E 共 % n 兲 ⫺E 共 ␴ n 兲
→0.
1⫹log2 dimHn
A weaker condition deals only with approximations to pure states:
(E3 ⬘ ) Same as 共E3兲 but with % n 苸⌺ p (H An 丢 H Bn ) for all n.
Sometimes we are interested in entanglement measures which satisfy an additivity property:
共E4兲 共Additivity兲 For all n⭓1 and all %苸⌺(H A 丢 H B )
E共 % 丢 n兲
⫽E 共 % 兲 .
n
Here % 丢 n denotes the n-fold tensor product of % by itself which acts on the tensor product
(H A ) 丢 n 丢 (H B ) 丢 n .
An apparently weaker property, which as we shall see in Lemma 10 is actually equivalent to
共E4兲, is the following.
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4260
J. Math. Phys., Vol. 43, No. 9, September 2002
Donald, Horodecki, and Rudolph
(E4 ⬘ ) 共Asymptotic Additivity兲 Given ⑀ ⬎0 and %苸⌺(H A 丢 H B ), there exists an integer
N⬎0 such that n⭓N implies
E共 % 丢 n兲
E共 % 丢 n兲
⫺ ⑀ ⭐E 共 % 兲 ⭐
⫹⑀.
n
n
共E5兲 共Subadditivity兲 For all %, ␴ 苸⌺(H A 丢 H B ),
E 共 % 丢 ␴ 兲 ⭐E 共 % 兲 ⫹E 共 ␴ 兲 .
(E5 ⬘ ) For all %苸⌺(H A 丢 H B ) and m,n⭓1,
E 共 % 丢 (m⫹n) 兲 ⭐E 共 % 丢 m 兲 ⫹E 共 % 丢 n 兲 .
(E5 ⬙ ) 共Existence of a regularization兲 For all %苸⌺(H A 丢 H B ), the limit
E共 % 丢 n兲
n
n→⬁
E ⬁ 共 % 兲 ⬅ lim
exists.
In Lemma 12 we shall prove the well-known result that (E5 ⬘ ) is a sufficient condition for (E5 ⬙ ).
When (E5 ⬙ ) holds, we shall refer to E ⬁ as the regularization of E. We shall discuss some general
properties of E ⬁ in Proposition 13.
共E6兲 共Convexity兲 Mixing of states does not increase entanglement:
E 共 ␭%⫹ 共 1⫺␭ 兲 ␴ 兲 ⭐␭E 共 % 兲 ⫹ 共 1⫺␭ 兲 E 共 ␴ 兲
for all 0⭐␭⭐1 and all %, ␴ 苸⌺(H A 丢 H B ).
共E6兲 might seem to be essential for a measure of entanglement. Nevertheless, there is some
evidence that an important entanglement measure 共the entanglement of distillation兲 which describes asymptotic properties of multiple copies of identical states may not be convex.8 A weaker
condition is to require convexity only on decompositions into pure states. We shall prove below
that this property is satisfied by the entanglement of distillation.
(E6 ⬘ ) For any state %苸⌺(H A 丢 H B ) and any decomposition %⫽ 兺 i p i 兩 ␺ i 典具 ␺ i 兩 with 兩 ␺ i 典
苸H A 丢 H B , p i ⭓0 for all i and 兺 i p i ⫽1, we require
E共 % 兲⭐
兺i p i E 共 P ␺ 兲 .
i
B. Conditions on pure states
The conditions imposed on an entanglement measure can be weakened by requiring that they
only apply for pure states. Indeed, it might not even be required that the measure is defined except
on pure states. Recall that ⌺ p (H A 丢 H B ) denotes the set of pure states on the composite space.
共P0兲 If ␴ 苸⌺ p (H A 丢 H B ) is separable, then E( ␴ )⫽0.
d
d
is any representative of P ⫹ (Cd ), then E( P ⫹
)⫽log2 d for
(P1)⫽(E1) 共Normalization兲 If P ⫹
d⫽1,2, . . . .
(P1 ⬘ )⫽(E1 ⬘ ) E( P ⫹ (C2 ))⫽1.
共P2兲 Let ⌳ be an operation which can be realized by means of local operations and classical
communications. If ␴ 苸⌺ p (H A 丢 H B ) is such that ⌳( ␴ ) is also pure, then
E 共 ⌳ 共 ␴ 兲兲 ⭐E 共 ␴ 兲 .
(P2 ⬘ ) For ␴ 苸⌺ p (H A 丢 H B ), E( ␴ ) depends only on the nonzero coefficients of a Schmidt
decomposition of ␴.
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4261
By Nielsen’s theorem and the proof of Lemma 9 below, 共P2兲 is equivalent to assuming (P2 ⬘ ) and
that if the Schmidt coefficients of % majorize those of ␴, then E(%)⭐E( ␴ ). Our proof of the
theorem shows that, given (P2 ⬘ ), only local operations and operations of the specific form of Eq.
共11兲 need be considered for 共P2兲 共cf. Ref. 25兲.
Below we will in particular be interested in entanglement measures satisfying the following
additional conditions:
共P3兲 Let (H An ) n苸N and (H Bn ) n苸N be sequences of Hilbert spaces and let Hn ⬅H An 丢 H Bn for all
n. For all sequences (% n ) n苸N and ( ␴ n ) n苸N of states with % n , ␴ n 苸⌺ p (H An 丢 H Bn ), such that
储 % n ⫺ ␴ n 储 1 →0, we require that
E 共 % n 兲 ⫺E 共 ␴ n 兲
→0.
1⫹log2 dimHn
共P4兲 For all n⭓1 and all %苸⌺ p (H A 丢 H B ),
E共 % 丢 n兲
⫽E 共 % 兲 .
n
Of course, when % is pure, so is % 丢 n .
(P4 ⬘ ) Given ⑀ ⬎0 and %苸⌺ p (H A 丢 H B ), there exists an integer N⬎0 such that n⭓N implies
E共 % 丢 n兲
E共 % 丢 n兲
⫺ ⑀ ⭐E 共 % 兲 ⭐
⫹⑀.
n
n
(P5 ⬙ ) 共Existence of a regularization on pure states兲 For all %苸⌺ p (H A 丢 H B ), the limit
E共 % 丢 n兲
n
n→⬁
E ⬁ 共 % 兲 ⬅ lim
exists.
C. Some connections between the conditions
Lemma 9: (E2⬘ ) is implied by (E2).
Proof: By Eq. 共5兲, the operations considered in (E2 ⬘ ) are LQCC. To see this for ⌳ 2 , note that
W A† W A ⫽1 H A . Thus 共E2兲 implies E(⌳ i ( ␴ ))⭐E( ␴ ) for i⫽1,2. Unitary maps are invertible and so
E(⌳ 1 ( ␴ ))⭓E( ␴ ). On the other hand, if H A 傺K A and P A is the projection onto H A , then, for any
␶ A 苸⌺ p (H A ), the map ⌳ A3 :⌺(K A )→⌺(H A ) defined by ⌳ A3 (%)ª P A % P A† ⫹tr(%(1⫺ P A )) ␶ A is
completely positive and trace preserving, so by Eq. 共5兲, the map on ⌺(K A 丢 H A ) defined by ⌳ 3
⫽⌳ A3 丢 I B is LQCC.
䊏
⌳ 3 (⌳ 2 ( ␴ ))⫽ ␴ and hence 共E2兲 implies E( ␴ )⭐E(⌳ 2 ( ␴ )).
Lemma 10: (E4⬘ ) is equivalent to (E4) and ( P4⬘ ) is equivalent to ( P4).
Proof: That 共E4兲 implies (E4 ⬘ ) is immediate. Suppose (E4 ⬘ ) and choose m, %, and ⑀.
By (E4 ⬘ ), there exists N such that n⭓N implies 兩 E(%)⫺E(% 丢 n )/n 兩 ⭐ ⑀ and 兩 E(% 丢 m )
⫺(E(% 丢 m ) 丢 n )/n 兩 ⭐ ⑀ . But, by definition, (% 丢 m ) 丢 n ⫽% 丢 mn where the equality relates equivalent
density matrices on products of isomorphic local spaces. Thus n⭓N implies
冏
E共 % 兲⫺
冏冏
冏冏
冏
E共 % 丢 m兲
E 共 % 丢 mn 兲
E 共 % 丢 mn 兲 E 共 % 丢 m 兲
⭐ E共 % 兲⫺
⫹
⭐2 ⑀ .
⫺
m
mn
mn
m
共E4兲 follows. The same proof shows the equivalence of (P4 ⬘ ) and 共P4兲.
䊏
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4262
J. Math. Phys., Vol. 43, No. 9, September 2002
Donald, Horodecki, and Rudolph
Lemma 11: Let E be an entanglement measure which satisfies ( P1⬘ ), ( P2), and ( P4). Then
E satisfies ( P0) and ( P1). Moreover, if E is defined on mixed states and satisfies either (E2) or
(E6⬘ ), then (E0) is satisfied.
Proof: First we deal with separable states.
Choose ⑀ ⬎0. Any pair of separable pure states is interconvertible by local unitary operators.
If ␴ is such a state, then so is ␴ 丢 n , and so, by 共P2兲, E( ␴ )⫽E( ␴ 丢 n ). But 共P4兲 implies that
E( ␴ )⫽E( ␴ 丢 n )/n and hence E( ␴ )⫽0. This gives 共P0兲 and the d⫽1 case of 共P1兲.
Now let %苸⌺(H A 丢 H B ) be a mixed separable state. Expanding the states % Ai and % Bi of Eq.
共1兲 into pure components shows that ␴ is a convex combination of pure separable states: ␴
⫽ 兺 i p i␴ i .
Thus (E6 ⬘ ) is sufficient to go from 共P0兲 to 共E0兲. But 共E2兲 is also sufficient, because if
⌳ i :T(H A 丢 H B )→T(H A 丢 H B ) is a local operation such that ⌳ i ( ␴ 1 )⫽ ␴ i , then ⌳ª 兺 i p i ⌳ i is a
LQCC operation such that ⌳( ␴ 1 )⫽ ␴ and so 共E2兲 and 共P0兲 yield E( ␴ )⭐E( ␴ 1 )⫽0.
d
)⫽log2 d follows from (P1 ⬘ ), 共P2兲, and 共P4兲.
Now we turn to showing that, for d⭓2, E( P ⫹
d
By (P2 ⬘ ), E( P ⫹ ) is independent of the representative of P ⫹ (Cd ) considered.
n
).
Choose ⑀ ⬎0 and d⭓2. Choose N⬎1/⑀ . Set w(n)⬅E( P ⫹
By Nielsen’s theorem, 共P2兲 implies that w(d 1 )⭐w(d 2 ) whenever d 1 ⭐d 2 .
d 丢n
dn
) ⫽ P⫹
, so that, by 共P4兲, w(d)⫽w(d n )/n for all n and, by
Up to local isomorphisms, ( P ⫹
n
(P1 ⬘ ), w(2)⫽w(2 )/n⫽1.
Choose n 1 ,n 2 ⬎N such that 2 n 2 ⫹1 ⭓d n 1 ⭓2 n 2 . Then log2 d⭓n2 /n1 , 兩 n 2 /n 1 ⫺log2 d兩⭐1/n 1
⬍ ⑀ , and, using 共P4兲,
兩 w 共 d 兲 ⫺log2 d 兩 ⭐ 兩 w 共 d n 1 兲 ⫺n 2 兩 /n 1 ⫹ 兩 n 2 /n 1 ⫺log2 d 兩 ⭐ 兩 w 共 d n 1 兲 ⫺n 2 w 共 2 兲 兩 /n 1 ⫹ ⑀
and
兩 w 共 d n 1 兲 ⫺n 2 w 共 2 兲 兩 /n 1 ⫽ 兩 w 共 d n 1 兲 ⫺w 共 2 n 2 兲 兩 /n 1 ⭐ 兩 w 共 2 n 2 ⫹1 兲 ⫺w 共 2 n 2 兲 兩 /n 1 ⫽1/n 1 ⭐ ⑀ .
It follows that w(d) is arbitrarily close to log2 d.
Lemma 12: (E5 ⬘ ) implies (E5 ⬙ ). Indeed, (E5 ⬘ ) implies that
䊏
E 共 % 丢 m 兲 /m →inf兵 E 共 % 丢 m 兲 /m :m⭓1 其 .
Proof: 共See Ref. 26, Theorem 4.9.兲 Fix k⬎0. Every m⭓1 can be written m⫽nk⫹r with 0
⭐r⬍k. Then for all m⬎0 set f (m)ªE(% 丢 m ). (E5 ⬘ ) implies that
f 共 m 兲 n f 共 k 兲⫹ f 共 r 兲 n f 共 k 兲 f 共 r 兲 f 共 k 兲 f 共 r 兲
⭐
⭐
⫹
⫽
⫹
.
m
nk⫹r
nk
nk
k
nk
As m→⬁ then n→⬁ so lim supm→⬁ f (m)/m ⭐ f (k)/k and thus lim supm→⬁ f (m)/m
⭐infk⭓1 f (k)/k. Now infk⭓1 f (k)/k ⭐lim infm→⬁ f (m)/m shows that limm→⬁ f (m)/m exists and
䊏
equals infm⭓1 f (m)/m .
Proposition 13: Let E be an entanglement measure which satisfies (E5⬙ ). Then we have the
following.
共1兲
共2兲
共3兲
共4兲
共5兲
共6兲
E ⬁ satisfies (E4)
If E satisfies (E0), then so does E ⬁ .
If E satisfies (E1), then so does E ⬁ .
If E satisfies (E2), then so does E ⬁ .
If E satisfies (E5), then so does E ⬁ .
If E satisfies (E5) and (E6), then so does E ⬁ .
Proof:
共1兲 For all m and %,
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4263
E ⬁共 % 丢 m 兲
E 共 % 丢 nm 兲
⫽ lim
⫽E ⬁ 共 % 兲 .
m
nm
n→⬁
共2兲
共3兲
共4兲
共5兲
If ␴ is separable, then so is ␴ 丢 n for all n.
n
d
d 丢n
is a representative of P ⫹ (Cd ), then ( P ⫹
) is a representative of P ⫹ (Cd ).
If P ⫹
If ⌳ is LQCC, then so is ⌳ 丢 n and ⌳( ␴ ) 丢 n ⫽⌳ 丢 n ( ␴ 丢 n ).
For all %,␴ and k⭓1, 共E5兲 implies that
E 共共 % 丢 ␴ 兲 丢 k 兲 E 共 % 丢 k 兲 E 共 ␴ 丢 k 兲
⭐
⫹
.
k
k
k
共6兲 Suppose that E satisfies 共E5兲 and 共E6兲. Let %, ␴ 苸⌺(H A 丢 H B ) and choose x 1 ,x 2
苸 关 0,1 兴 with x 1 ⫹x 2 ⫽1. Let ␻ ⫽x 1 %⫹x 2 ␴ . Expanding ␻ 丢 n as a sum of products, using convexity of E, and then using local isomorphisms to reorder the terms in each product gives
n
E共 ␻ 丢 n兲⭐
兺
k⫽0
冉冊
n
冉冊
n k n⫺k
n k n⫺k
x 1 x 2 E 共 % 丢 k 丢 ␴ 丢 (n⫺k) 兲 ⭐
x 1 x 2 共 E 共 % 丢 k 兲 ⫹E 共 ␴ 丢 (n⫺k) 兲兲 ,
k
k⫽0 k
兺
where the second inequality is a consequence of 共E5兲. To complete the proof, we need the
following lemma:
n
n
( nk )x k1 x n⫺k
E(% 丢 k )→x 1 E ⬁ (%) and (1/n) 兺 k⫽0
( nk )x k1 x n⫺k
Lemma 14: As n→⬁, (1/n) 兺 k⫽0
2
2
⫻E( ␴ 丢 (n⫺k) )→x 2 E ⬁ ( ␴ ).
Proof: It is sufficient to prove the first limit. Set g(m)⫽E(% 丢 m )/m and L⫽E ⬁ (%). Choose
⑀ ⬎0. By Lemma 12, there exists K such that k⭓K implies 兩 g(k)⫺L 兩 ⬍ ⑀ /2 and there is a constant
C⬎0 such that 兩 g(k)⫺L 兩 ⬍C for all k. N⬎K implies that
K
冉冊
N
冉冊
K
1
N
N k N⫺k K
kx k1 x N⫺k
⭐
x x
⫽ .
2
N k⫽0 k
N k⫽0 k 1 2
N
兺
兺
n
n
( nk )x k y n⫺k . xh ⬘ (x)⫽nx(x⫹y) n⫺1 ⫽ 兺 k⫽0
( nk )kx k y n⫺k . Thus x 1
Set h(x)⫽(x⫹y) n ⫽ 兺 k⫽0
n
n
k n⫺k
⫹x 2 ⫽1 implies that 兺 k⫽0 ( k )kx 1 x 2 ⫽nx 1 .
Choose N 0 ⬎K such that KC/N 0 ⬍ ⑀ /2. Then N⬎N 0 implies
冏 兺冉 冊
N
冏 冏 兺冉 冊
冏 兺冉 冊
冏
冏
N
1
1
N k N⫺k
N
x x
E 共 % 丢 k 兲 ⫺x 1 E ⬁ 共 % 兲 ⫽
kx k1 x N⫺k
g 共 k 兲 ⫺x 1 L
2
N k⫽0 k 1 2
N k⫽0 k
N
1
N
⫽
kx k1 x N⫺k
共 g 共 k 兲 ⫺L 兲
2
N k⫽0 k
K
⭐
冉冊
1
N
kx k1 x N⫺k
C
2
N k⫽0 k
兺
冉冊
N
⫹
1
N
kx k1 x N⫺k
共 g 共 k 兲 ⫺L 兲
2
N k⫽K⫹1 k
兺
N
兺
k⫽K⫹1
⭐KC/N⫹ ⑀ /2
冉冊
N k N⫺k
x x
k 1 2
⭐⑀.
䊏
Continuity 共E3兲 is not mentioned in Proposition 13, although we could use Lemma 12 to
deduce upper-semicontinuity from 共E3兲 and (E5 ⬘ ), as the infimum of a family of real continuous
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4264
J. Math. Phys., Vol. 43, No. 9, September 2002
Donald, Horodecki, and Rudolph
functions is upper-semicontinuous. For an example which may be relevant, consider the sequence
of functions on 关0,1兴 defined by f n (x)⫽nx n . Clearly f m⫹n (x)⭐ f m (x)⫹ f n (x). g n (x)⫽x n converges 共pointwise兲 as n→⬁ to a discontinuous but upper-semicontinuous function.
VI. EXAMPLES OF IMPORTANT ENTANGLEMENT MEASURES
In this section we will present some important entanglement measures and check which of the
postulates from Sec. V they satisfy.
A. Operational measures
Here we shall describe two entanglement measures, entanglement of distillation and entanglement cost1 共see also Refs. 27 and 28兲, which are defined in terms of specific state conversions.
Lemma 15: Let %苸⌺(H A 丢 H B ) with H A ⬅H B ⬅H and dimH⫽d. Let 兩 ␾ 典 ⫽ 兩 ␾ A 典 丢 兩 ␾ B 典
d
苸H A 丢 H B be a separable wavefunction and P ⫹
be a representative of P ⫹ (Cd ) on H A 丢 H B .
d
)⫽%.
Then there exist LQCC operations ⌳ 1 and ⌳ 2 such that ⌳ 1 (%)⫽ 兩 ␾ 典具 ␾ 兩 and ⌳ 2 ( P ⫹
A d
B d
A
B
Proof: Let ( ␺ i ) i⫽1 关resp. ( ␺ i ) i⫽1 兴 be an orthonormal basis for H 关resp. H ] and define ⌳ 1
by
d
⌳ 1共 ␴ 兲 ⬅
兺 共1
j⫽1
冉兺
冊
d
A
丢 兩␾
B
典具 ␺ Bj 兩 兲
i⫽1
共 兩 ␾ A 典具 ␺ Ai 兩 丢 1 B 兲 ␴ 共 兩 ␺ Ai 典具 ␾ A 兩 丢 1 B 兲 共 1 A 丢 兩 ␺ Bj 典具 ␾ B 兩 兲
d
⫽
兺
i, j⫽1
兩 ␾ A 丢 ␾ B 典具 ␺ Ai 丢 ␺ Bj 兩 ␴ 兩 ␺ Ai 丢 ␺ Bj 典具 ␾ A 丢 ␾ B 兩 ⫽ 兩 ␾ 典 tr共 ␴ 兲 具 ␾ 兩 ⫽ 兩 ␾ 典具 ␾ 兩
for all ␴ 苸⌺(H A 丢 H B ).
For ⌳ 2 , we note that if 兩 ⌿ 典具 ⌿ 兩 苸⌺(H A 丢 H B ) is any pure state, then, by Nielsen’s theorem,
d
d
to 兩 ⌿ 典具 ⌿ 兩 because the distribution (1/d) i⫽1
is
there exists an LQCC operation mapping P ⫹
majorized by any probability distribution on 兵 1, . . . ,d 其 . Now, as in the proof of Lemma 11, we
d
to pure components of %.䊏
can construct ⌳ 2 as a convex combination of operations mapping P ⫹
A
B
Given a state % on H 丢 H , consider a sequence of LQCC operations (⌳ n ) with
⌳ n :T((H A ) 丢 n 丢 (H B ) 丢 n )→T((H A ) 丢 n 丢 (H B ) 丢 n ). Suppose that ␴ n ⬅⌳ n (% 丢 n ) satisfies
d
储 P ⫹n ⫺ ␴ n 储 1 →0
d
for some representative P ⫹n of P ⫹ (Cd n ) on (H A ) 丢 n 丢 (H B ) 丢 n . We call such a sequence (⌳ n ) an
LQCC distillation protocol. The asymptotic ratio attainable via this protocol is then defined by
E D 共共 ⌳ n 兲 ,% 兲 ⬅lim sup
n→⬁
log2 d n
.
n
共14兲
Lemma 15 shows that, for any state, a distillation protocol always exists with d n ⬅1.
Definition 16: The distillable entanglement or entanglement of distillation E D is defined as the
supremum of Eq. (14) over all possible LQCC distillation protocols:
E D 共 % 兲 ⬅ sup E D 共共 ⌳ n 兲 ,% 兲 .
共15兲
(⌳ n )
By construction E D satisfies the properties 共E2兲 and 共E4兲 of entanglement measures. The proof
is analogous to the proof of Lemma 1 in Ref. 28. It is not known whether E D satisfies 共E3兲 or 共E6兲.
共Indeed, as already mentioned, there is evidence that 共E6兲 may not be satisfied.8兲 We shall confirm
in Lemma 24 that 共E0兲 and 共E1兲 are satisfied.
The so-called entanglement cost E C is defined in a complementary way. Given a state %,
consider a sequence of LQCC operations ⌳ n :T(Cd n 丢 Cd n )→T((H A ) 丢 n 丢 (H B ) 丢 n ) transforming a
representative of P ⫹ (Cd n ) into a state ␴ n such that
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4265
储 ␴ n ⫺% 丢 n 储 1 →0.
The asymptotic ratio attainable via this formation-protocol is then given by
E C 共共 ⌳ n 兲 ,% 兲 ⬅lim inf
n→⬁
log2 d n
.
n
共16兲
Once again Lemma 15 shows that, for any state, a formation protocol always exists with d n
⬅d n where d⫽max兵dimH A ,dimH B 其 .
Definition 17: The entanglement cost E C is defined as the infimum of Eq. (16) over all possible
LQCC formation protocols:
E C 共 % 兲 ⬅ inf E C 共共 ⌳ n 兲 ,% 兲 .
兵⌳n其
共17兲
By construction E C satisfies property 共E2兲. As we shall discuss in the next section, by Ref. 28
and Proposition 13, it also satisfies 共E0兲, 共E1兲, 共E2兲, 共E4兲, 共E5兲, and 共E6兲. It is not known whether
it satisfies 共E3兲. We shall also prove below that for pure states both E D and E C are equal to the
reduced von Neumann entropy given by Eq. 共3兲. 共This was first realized in Ref. 29 and a rigorous
proof was sketched in Ref. 21.兲
B. Abstract measures
The entanglement measures discussed in this subsection quantify entanglement mathematically, but their definitions do not admit a direct operational interpretation in terms of entanglement
manipulations. The first one is the so-called entanglement of formation1 which is defined as
follows:
Definition 18: Let H A and H B be finite dimensional Hilbert spaces and let 兩 ␺ 典 苸H A 丢 H B .
Then the entanglement of formation is defined for pure states as
E F 共 P ␺ 兲 ªS vN共 P ␺ 兲 ,
共18a兲
where S vN( P ␺ ) 关defined in Eq. (3)兴 is the von Neumann entropy of either of the reduced density
matrices of 兩 ␺ 典 . For mixed states %苸⌺(H A 丢 H B ) we define
兺i p i E F共 P ␺ 兲 ,
E F 共 % 兲 ªinf
i
共18b兲
where the infimum is taken over all possible decompositions of % of the form %⫽ 兺 i p i 兩 ␺ i 典具 ␺ i 兩 with
p i ⭓0 for all i and 兺 i p i ⫽1.
The entanglement of formation satisfies 共E0兲–共E3兲, 共E5兲, and 共E6兲. In particular, 共E2兲 was
shown in Ref. 1, 共E3兲 in Ref. 6, and 共E0兲, 共E1兲, 共E5兲, and 共E6兲 follow directly from the definition
of E F .
The entanglement of formation E F is believed but not known to be equal to the entanglement
cost E C . However, it is known that the regularized entanglement of formation E F⬁ 关which exists by
(E5 ⬘ )兴 is equal to the entanglement cost.28 This allows us to apply Proposition 13 to E C .
Let us now present another important measure, namely, the relative entropy of
entanglement.24,2 It is defined as follows:
E R 共 % 兲 ⬅inf S rel共 % 兩 ␴ 兲 ,
␴
共19兲
where S rel(% 兩 ␴ )⬅tr% log2 %⫺tr% log2 ␴ is the quantum relative entropy, and where the infimum is
taken over all separable states ␴. One can consider variations of the above measure, by changing
the set of states over which the infimum is taken 共this set should be closed under LQCC
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4266
J. Math. Phys., Vol. 43, No. 9, September 2002
Donald, Horodecki, and Rudolph
operations though兲. Like the entanglement of formation, E R satisfies 共E0兲–共E3兲, 共E5兲, and 共E6兲. In
particular, 共E1兲 and 共E2兲 were shown in Ref. 24, 共E3兲 in Ref. 9, 共E0兲 follows immediately and 共E5兲
almost immediately from the definition of E R , and 共E6兲 follows from the convexity of the quantum relative entropy S rel .
The properties of E R and Proposition 13 show that the regularized relative entropy of entanglement E R⬁ exists and satisfies 共E0兲, 共E1兲, 共E2兲, 共E4兲, 共E5兲, and 共E6兲. It is shown in Ref. 10 that
E R does not satisfy 共E4兲. This implies, of course, that E R and E R⬁ are not always equal 共cf. Ref.
30兲.
Finally, let us note that for pure states both the entanglement of formation 共by definition兲 and
the relative entropy of entanglement 共as shown in Refs. 2 and 31兲 are equal to the reduced von
Neumann entropy S vN 关defined in Eq. 共3兲兴. An immediate consequence of the additivity of S vN is
that E F⬁ ⫽E C and E R⬁ are also equal to S vN on pure states 共see also Theorem 23兲.
VII. ENTANGLEMENT OF DISTILLATION AND ENTANGLEMENT COST AS EXTREME
MEASURES
In this section we improve the theorem of Ref. 4 by giving precise conditions under which E D
and E C are lower and upper bounds for entanglement measures. We propose three versions of the
theorem.
Proposition 19: Suppose that E is an entanglement measure defined on mixed states which
satisfies (E1) – (E4). Then for all states %苸⌺(H A 丢 H B )
E D 共 % 兲 ⭐E 共 % 兲 ⭐E C 共 % 兲 .
共20兲
Proof: Choose ⑀ ⬎0. We shall prove the result in three steps:
共I兲 First we prove that, having if necessary passed to a subsequence, there exists an integer
N 1 ⬎0 such that n⭓N 1 implies
E共 % 丢 n兲
⭓E D 共 % 兲 ⫺ ⑀ .
n
共21兲
Consider a near-optimal LQCC protocol (⌳ n ) n . By the definition of distillable entanglement, there
exists a LQCC protocol (⌳ n ) n such that, after possibly passing to a subsequence,
d
储 P ⫹n ⫺⌳ n 共 % 丢 n 兲储 1 →0
共22a兲
冏
共22b兲
and
E D共 % 兲 ⫺
冏
log2 d n ⑀
⭐
n
2
for all n⭓N 1⬘ . 共E3兲 implies that
冏
d
冏
E 共 ⌳ n 共 % 丢 n 兲兲 ⫺E 共 P ⫹n 兲
1⫹n log2 d
→0
共23兲
as n→⬁ where d⫽dimH A 丢 H B . It follows that we can choose N 1⬙ ⬎0 such that n⭓N 1⬙ implies
冏
d
冏
E 共 ⌳ n 共 % 丢 n 兲兲 E 共 P ⫹n 兲 ⑀
⭐ ,
⫺
n
n
2
共24兲
and, so, using 共E2兲, for n⭓N 1 ⫽max兵N1⬘ ,N⬙1其,
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4267
d
E 共 % 丢 n 兲 E 共 ⌳ n 共 % 丢 n 兲兲 E 共 P ⫹n 兲 ⑀ log2 d n ⑀
⭓
⭓
⫺ ⫽
⫺ ⭓E D 共 % 兲 ⫺ ⑀ .
n
n
n
2
n
2
共25兲
共II兲 As a second step, we prove that, having if necessary passed to another 共perhaps disjoint兲
subsequence, there exists an integer N 2 ⭓N 1 such that n⭓N 2 implies
E共 % 丢 n兲
⭐E C 共 % 兲 ⫹ ⑀ .
n
共26兲
This is similar to the first step. Consider a near-optimal protocol (⌳ n ) n for %. We have 关after
possibly passing to a suitable subsequence of (⌳ n ) n 兴, for all sufficiently large n,
d
d
E 共 % 丢 n 兲 E 共 ⌳ n 共 P ⫹n 兲兲 ⑀ E 共 P ⫹n 兲 ⑀ log2 d n ⑀
⭐
⫹ ⭐
⫹ ⫽
⫹ ⭐E C 共 % 兲 ⫹ ⑀ .
n
n
2
n
2
n
2
共27兲
共III兲 The final step is to invoke 共E4兲 to give
E D 共 % 兲 ⫺ ⑀ ⭐E 共 % 兲 ⫽
E共 % 丢 n兲
⭐E C 共 % 兲 ⫹ ⑀ .
n
共28兲
䊏
Unfortunately, as we do not at present know of any function for which we can prove that
postulates 共E1兲–共E4兲 hold for all states, it is possible that Proposition 19 may be empty. Nevertheless, by modifying the final step of the proof, we can obtain the following:
Proposition 20: Let E be an entanglement measure defined on mixed states and satisfying
(E1), (E2), (E3), and (E5⬙ ). Then for all states %苸⌺(H A 丢 H B ),
E D 共 % 兲 ⭐E ⬁ 共 % 兲 ⭐E C 共 % 兲 .
共29兲
Proof: Without using condition 共E4兲 or any properties of E ⬁ except its existence, we can
䊏
maintain the structure of the previous proof, simply by replacing E(%) in 共28兲 by E ⬁ (%).
Proposition 20 is certainly nonempty. Indeed, as mentioned in the previous section, both the
entanglement of formation and the relative entropy of entanglement satisfy all assumptions of the
proposition. We obtain the following corollary.
Corollary 21: The entanglement of distillation E D is less than or equal to the entanglement
cost E C for all states.
Although, in physical terms, Corollary 21 seems almost necessary, a rigorous proof requires
some control both over changes in state and over changes in dimension.
Let us now consider yet another version, where we weaken the assumptions in the theorem on
extreme measures of Ref. 4. We impose the condition (E3 ⬘ ), which is stronger than 共P3兲 but
weaker than 共E3兲.
One mechanism for deriving condition (E3 ⬘ ) for a given function E might be to establish the
inequalities
f 共 % 兲 ⭐E 共 % 兲 ⭐g 共 % 兲 ,
共30兲
where f , g are functions satisfying (E3 ⬘ ) which coincide on pure states. We will take f (%)
⬅S(% A )⫺S(%) and g(%)⬅S(% A ) 共where % A ªtrHB %兲. Both of these functions f and g do satisfy
(E3 ⬘ ). This follows immediately from two facts:
共i兲 Fannes inequality32,33
兩 S 共 ␴ 兲 ⫺S 共 % 兲 兩 ⭐ 储 ␴ ⫺% 储 1 log2 dimH⫹ ␩ 共 储 ␴ ⫺% 储 1 兲 ,
共31兲
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4268
J. Math. Phys., Vol. 43, No. 9, September 2002
Donald, Horodecki, and Rudolph
which holds for any two states ␴ and % acting on the Hilbert space H and satisfying 储 ␴ ⫺% 储 1
⭐ 1/3; here ␩ (s)⬅⫺s log s and S denotes the standard von Neumann entropy as above.
共ii兲 储 ␴ A ⫺% A 储 1 ⭐ 储 ␴ ⫺% 储 1 , where ␴ A and % A are the reduced density operators of % and ␴,
respectively.
With the above choices for f and g one can show that E F and E R satisfy the inequalities in
共30兲 共see Refs. 1, 2, 31, and 34兲. Then, E R⬁ and E F⬁ also satisfy inequalities 共30兲, because the
additivity of the von Neumann entropy implies that both f and g satisfy 共E4兲. E D also satisfies the
inequality E D ⭐g, but we do not know whether or not it satisfies the second inequality. However,
a stronger inequality 共the so-called hashing inequality兲, which would have many interesting implications, was conjectured in Ref. 35. Strong evidence for this conjecture was collected there.
We shall also use the weak form of convexity (E6 ⬘ ).
Proposition 22: Let E be an entanglement measure defined on mixed states and satisfying
(E1), (E2), (E3⬘ ), and (E6⬘ ). Then for all states %苸⌺(H A 丢 H B ) we have
E D 共 % 兲 ⭐E 共 % 兲 ⭐E C 共 % 兲
共32兲
E D 共 % 兲 ⭐E ⬁ 共 % 兲 ⭐E C 共 % 兲
共33兲
if (E4) holds and
if (E5) holds.
Proof: Step I of the proof of Proposition 19 goes through with (E3 ⬘ ) replacing 共E3兲 in
inequality 共24兲.
To replace step II, we use the estimate E C ⭓E F⬁ . This follows from Proposition 20 共but also,
of course, from Ref. 28, where it was shown that E C ⫽E F⬁ 兲. For any state % consider its finite
decompositions into pure states
%⫽
兺i p i兩 ␺ i 典具 ␺ i兩
for which
E F共 % 兲 ⫽
兺i p i S vN共 P ␺ 兲 .
i
In Ref. 36 it was shown that such a decomposition exists.
As (E1)⫽(P1)⇒(P1 ⬘ ), (E2)⇒(P2), and (E3 ⬘ )⇒(P3), we can apply Theorem 23 共to follow兲 to show that E( P ␺ i )⫽S vN( P ␺ i ) if E satisfies 共E4兲 and E ⬁ ( P ␺ i )⫽S vN( P ␺ i ) if E satisfies 共E5兲.
Now (E6 ⬘ ) implies, in the first case, that E(%)⭐E F (%) 共cf. Ref. 36兲 and hence
E共 % 兲⫽
E 共 % 丢 n 兲 E F共 % 丢 n 兲
⭐
,
n
n
which yields the required upper bound when n→⬁. For the second case, we can use the proof of
part 共6兲 of Proposition 13 to show that (E6 ⬘ ) holds for E ⬁ . This yields E ⬁ (%)⭐E F (%) and
E ⬁共 % 兲 ⫽
E ⬁共 % 丢 n 兲 E F共 % 丢 n 兲
⭐
.
n
n
Again the required bound follows on taking n→⬁.
䊏
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4269
VIII. THE UNIQUENESS THEOREM FOR ENTANGLEMENT MEASURES
Theorem 23: Let E be a functional on pure states. Then the following are equivalent:
共1兲 E satisfies ( P1⬘ ), ( P2), ( P3), and ( P4⬘ ).
共2兲 E satisfies ( P0), ( P1), ( P2), ( P3), and ( P4).
共3兲 E coincides with the reduced von Neumann entropy E⫽S vN .
On the other hand, if E satisfies ( P0), ( P1), ( P2), and ( P3), then E satisfies ( P5⬙ ) and, on pure
states, E ⬁ ⫽S vN .
Proof: The equivalence of 共1兲 and 共2兲 is proved in Lemmas 10 and 11.
It is clear that the reduced von Neumann entropy satisfies 共P0兲, 共P1兲, and 共P4兲. 共P3兲 follows
from the facts (i) and (ii) of the previous section. Finally 共P2兲 is a consequence of Nielsen’s
theorem and the fact that the von Neumann entropy is a Schur-concave function.37 Indeed, with
the inductive decomposition of LQCC operations introduced in our proof of Nielsen’s theorem, we
can prove 共P2兲 just by showing, in the notation of Eq. 共12兲, that S vN(⌳( 兩 ⌿ 典具 ⌿ 兩 ))⭐S vN( 兩 ⌿ 典
⫻具 ⌿ 兩 ). This amounts to proving that, for p J ⭓ p K and suitable ␦,
⫺ 共 p J ⫹ ␦ 兲 log2 共 p J ⫹ ␦ 兲 ⫺ 共 p K ⫺ ␦ 兲 log2 共 p K ⫺ ␦ 兲 ⭐⫺ p J log2 p J ⫺ p K log2 p K ,
and this is easily confirmed by differentiating with respect to ␦.
Now suppose that E satisfies 共P0兲, 共P1兲, 共P2兲, and 共P3兲. Using (P2 ⬘ ), we may assume that
H A ⬅H B ⬅H. Suppose that dimH⫽d and let 兩 ␺ 典 苸H 丢 H. Write S⬅S vN( 兩 ␺ 典具 ␺ 兩 ) for the von
Neumann entropy of the reduced density matrix of 兩 ␺ 典 . Consider n copies of the wavefunction
兩 ␺ 典 : 兩 ␺ 丢 n 典 苸Htot⬅H 丢 n 丢 H 丢 n . Let 兵 q j : j⫽1, . . . ,d 其 be the set of eigenvalues of the reduced
density matrix of 兩 ␺ 典 and 兵 p i :i⫽1, . . . ,d 2n 其 be the set of eigenvalues of the reduced density
matrix of 兩 ␺ 丢 n 典 . Again using (P2 ⬘ ), we may adjust d so that q j ⬎0 for j⫽1, . . . ,d. In view of
共P0兲, we may also assume that S⬎0. Considered as a probability distribution, 兵 p i 其 is the distribution for n independent trials each with distribution 兵 q j 其 . Choose bases (e i )傺H 丢 n and
( f i )傺H 丢 n such that
兩 ␺ 丢 n典 ⫽
兺i 冑p i兩 e i 典 丢 兩 f i 典 .
Choose ⑀ ⬎0. By the asymptotic equipartition theorem 共Ref. 38, Theorem 3.1.2兲, there exists an
integer N⬅N( ⑀ ) such that, for all n⭓N, one can find a subset TYP⬅TYP(n, ⑀ ) of the set of
d 2n
with the following properties:
indices 兵 i 其 i⫽1
2 ⫺n(S⫹ ⑀ ) ⭐ p i ⭐2 ⫺n(S⫺ ⑀ ) ,
p⬅
兺
i苸TYP
for i苸TYP,
p i ⭓1⫺ ⑀ ,
#TYP⭐2 n(S⫹ ⑀ ) .
共34a兲
共34b兲
共34c兲
Here #TYP denotes the number of elements in TYP.
Introduce another wavefunction 兩 ␾ n 典 苸Htot given by
兩 ␾ n典 ⬅
1
兺
冑p i苸TYP
冑p i 兩 e i 典 丢 兩 f i 典 .
This wavefunction satisfies
兩 具 ␺ 丢 n 兩 ␾ n 典 兩 2 ⫽ p⭓1⫺ ⑀
共35兲
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4270
J. Math. Phys., Vol. 43, No. 9, September 2002
Donald, Horodecki, and Rudolph
and so
储 兩 ␺ 丢 n 典具 ␺ 丢 n 兩 ⫺ 兩 ␾ n 典具 ␾ n 兩 储 1 ⫽2 冑共 1⫺ 兩 具 ␺ 丢 n 兩 ␾ n 典 兩 2 兲 ⭐2 冑⑀ .
共36兲
Now, the crucial observation 共cf. Ref. 21兲 is that for ⑀ ⬍min兵 1/2S, 1/2 其 and n sufficiently
large, there exist completely positive maps ⌳ n and ⌳ ⬘n such that
a
⌳ n 共 兩 ␾ n 典具 ␾ n 兩 兲 ⫽ P ⫹
共37a兲
a
a representative of P ⫹ (Ca ) in Htot with 兩 (log2 a)/n ⫺S兩 ⬍ ⑀ ⫹ 2/n and
for P ⫹
b
⌳ ⬘n 共 P ⫹
兲 ⫽ 兩 ␾ n 典具 ␾ n 兩
共37b兲
b
a representative of P ⫹ (Cb ) in Htot with 兩 (log2 b)/n ⫺S兩 ⬍ ⑀ ⫹ 1/n. Indeed, to see Eq. 共37a兲,
for P ⫹
set a⬅ b p2 n(S⫺ ⑀ ) c , i.e., a is the largest integer smaller than or equal to p2 n(S⫺ ⑀ ) . Then a
a
, hence
⭐ p2 n(S⫺ ⑀ ) ⭐p/p i and we see that the distribution (p i /p) i苸TYP is majorized by (1/a) i⫽1
Eq. 共37a兲 follows from Nielsen’s theorem. Equation 共37b兲 follows by a similar argument when we
take b⬅ d p2 n(S⫹ ⑀ ) e , i.e., b is the smallest integer larger than or equal to p2 n(S⫹ ⑀ ) . The conditions
on ⑀ and n are sufficient to go from a⬅ b p2 n(S⫺ ⑀ ) c to 兩 (log2 a)/n ⫺S兩 ⬍ ⑀ ⫹ 2/n and from b
⬅ d p2 n(S⫹ ⑀ ) e to 兩 (log2 b)/n ⫺S兩 ⬍ ⑀ ⫹ 1/n, ensuring, for example, that a⫽0.
Now choose a sequence ( ⑀ j ) j苸N of positive numbers such that ⑀ j →0 for j→⬁. Suppose that
(n k ) k苸N is a sequence of integers such that n k →⬁ and E( 兩 ␺ 丢 n k 典具 ␺ 丢 n k 兩 )/n k →L for some L.
For each j, choose n k j ⭓max兵N(⑀ j),1/⑀ j 其 . We can apply the postulates 共P0兲–共P3兲 to obtain the
following estimates:
E 共 兩 ␺ 丢 n k j 典具 ␺ 丢 n k j 兩 兲 E 共 兩 ␺
⫽
nkj
⭓
⫽
⫽
丢 nk
j
典具 ␺ 丢 n k j 兩 兲 ⫺E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
j
nkj
E 共 兩 ␺ 丢 n k j 典具 ␺ 丢 n k j 兩 兲 ⫺E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
j
nkj
E 共 兩 ␺ 丢 n k j 典具 ␺ 丢 n k j 兩 兲 ⫺E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
j
nkj
E 共 兩 ␺ 丢 n k j 典具 ␺ 丢 n k j 兩 兲 ⫺E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
j
nkj
⫹
⫹
⫹
⫹
E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
j
nkj
E 共 ⌳ n k 共 兩 ␾ n k 典具 ␾ n k 兩 兲兲
j
j
j
nkj
E共 P
an
⫹
kj
兲
nkj
log2 a n k
j
nkj
.
As j→⬁, the first term vanishes due to 共P3兲 and the second approaches S vN( P ␺ ) 共cf. Ref. 6兲. This
implies that L⭓S vN( 兩 ␺ 典具 ␺ 兩 ). The proof of the inequality L⭐S vN( 兩 ␺ 典具 ␺ 兩 ) is similar:
E 共 兩 ␺ 丢 n k j 典具 ␺ 丢 n k j 兩 兲 E 共 兩 ␺
⫽
nkj
⭐
⫽
丢 nk
j
典具 ␺ 丢 n k j 兩 兲 ⫺E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
j
nkj
E 共 兩 ␺ 丢 n k j 典具 ␺ 丢 n k j 兩 兲 ⫺E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
j
nkj
E 共 兩 ␺ 丢 n k j 典具 ␺ 丢 n k j 兩 兲 ⫺E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
nkj
j
⫹
⫹
⫹
E 共 兩 ␾ n k 典具 ␾ n k 兩 兲
j
j
nkj
E共 P
bn
⫹
kj
兲
nkj
log2 b n k
nkj
j
.
We have now shown that every limit point of the sequence E( 兩 ␺ 丢 n 典具 ␺ 丢 n 兩 )/n has the value
L⫽S vN( 兩 ␺ 典具 ␺ 兩 ). But, by 共P1兲, 共P2兲, and Lemma 15, this sequence is bounded, and so (P5 ⬙ ) holds
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J. Math. Phys., Vol. 43, No. 9, September 2002
Uniqueness theorem—entanglement measures
4271
with E ⬁ ( 兩 ␺ 典具 ␺ 兩 )⫽L⫽S vN( 兩 ␺ 典具 ␺ 兩 ). This proves the final statement of the theorem. On the other
hand, if 共P4兲 holds, then L⫽E( 兩 ␺ 典具 ␺ 兩 ), and so we have proved that 共2兲 implies 共3兲. This completes the proof of Theorem 23.
䊏
It is natural to wonder whether the conditions in Theorem 23 can be weakened, and, in
particular, whether 共P3兲 is necessary. That it is has been noted by Vidal.3 Consider the entanglement measures defined on pure states by S ⬁ ( ␴ )⫽⫺log2 p1(␴) where p 1 ( ␴ ) is the largest coefficient in a Schmidt decomposition of ␴ and by S 0 ( ␴ )⫽log d(␴), where d is the number of nonzero
coefficients. S 0 and S ⬁ both satisfy 共P0兲, 共P1兲, 共P2兲 共by Nielsen’s theorem兲, and 共P4兲. S ⬁ is even
trace norm continuous on Hilbert spaces of fixed dimension. 共P3兲, however, does not hold for
either. This is, of course, a consequence of Theorem 23. An explicit example of the failure of 共P3兲
for S ⬁ is provided by the states ␴ n ⬅ 兩 ⌿ n 典具 ⌿ n 兩 , % n ⬅ 兩 ⌽ n 典具 ⌽ n 兩 with Schmidt decompositions
4 n ⫺2 n ⫹1
4n
兩 ⌿ n 典 ⬅ 冑1/2n 兩 ␺ 1 ␺ 1 典 ⫹ 兺 i⫽2
(1/2n ) 兩 ␺ i ␺ i 典 and 兩 ⌽ n 典 ⬅ 兺 i⫽1
(1/2n ) 兩 ␺ i ␺ i 典 for some orthonormal family ( 兩 ␺ i 典 ) of wavefunctions. In fact, any entanglement measure E defined on pure states
and satisfying 共P0兲, 共P1兲, 共P2兲, and 共P4兲 will satisfy S ⬁ ( ␴ )⭐E( ␴ )⭐S 0 ( ␴ ) for all pure ␴. The
upper bound here is a consequence of Lemma 15 while, for the lower bound, we modify the proof
of Theorem 23 using the fact that 兩 ␺ 丢 n 典具 ␺ 丢 n 兩 can always be converted without approximation
c
where c is the largest integer smaller than or equal to 1/p 1 .
into P ⫹
An example of a measure on pure states satisfying 共P0兲, 共P1兲, 共P2兲, and 共P3兲, but not 共P4兲, is
given by E( ␴ )⫽2(1⫺p 1 ( ␴ ))S vN( ␴ ) for p 1 ( ␴ )⭓ 1/2 , E( ␴ )⫽S vN( ␴ ) for p 1 ( ␴ )⭐ 1/2.
Finally, let us consider entanglement of distillation and entanglement cost in the above context. Using the maps constructed in Theorem 23, we show that they are equal to S vN . We have
already noted that for E C this also follows from Ref. 28.
Lemma 24: The entanglement of distillation E D and the entanglement cost E C both coincide
on pure states with the von Neumann reduced entropy E D ( P ␺ )⫽E C ( P ␺ )⫽S vN( P ␺ ) for all 兩 ␺ 典
苸H 丢 H.
Proof: From Sec. VII we know that E D ⭐E C . It suffices to show that on pure states E D
⭓S vN and E C ⭐S vN . We will continue to use the notation from the proof of Theorem 23.
That E C ( P ␺ )⭐S vN( P ␺ ) follows directly from the definition of E C , using the operations
defined by the ⌳ n⬘ which satisfy Eq. 共37b兲 and estimate 共36兲.
j
To show that E D ( P ␺ )⭓S vN( P ␺ ), let us apply the map ⌳ n j from Eq. 共37a兲 to the state
a
兩 ␺ 丢 n j 典具 ␺ 丢 n j 兩 . We only need check that the resulting state ⌳ n j ( 兩 ␺ 丢 n j 典具 ␺ 丢 n j 兩 ) approaches P ⫹
as
j→⬁. But, by Lemma 5,
a
储 ⌳ n j 共 兩 ␺ 丢 n j 典具 ␺ 丢 n j 兩 兲 ⫺ P ⫹
储 1 ⫽ 储 ⌳ n j 共 兩 ␺ 丢 n j 典具 ␺ 丢 n j 兩 兲 ⫺⌳ n j 共 兩 ␾ n j 典具 ␾ n j 兩 兲储 1
⭐ 储 兩 ␺ 丢 n j 典具 ␺ 丢 n j 兩 ⫺ 兩 ␾ n j 典具 ␾ n j 兩 储 1
and once again estimate 共36兲 is sufficient.
䊏
With the results obtained in this article, we can now prove that E D is convex on pure decompositions, i.e., we have the following Lemma.
Lemma 25:
ED
冉兺
i
冊
p i 兩 ␺ i 典具 ␺ i 兩 ⭐
兺i p i E D共 兩 ␺ i 典具 ␺ i兩 兲 ,
共38兲
where p i ⭓0 for all i and 兺 i p i ⫽1.
Proof: We have seen that E C is convex and satisfies E D ⭐E C . Using Lemma 24 gives
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4272
J. Math. Phys., Vol. 43, No. 9, September 2002
ED
冉兺
i
Donald, Horodecki, and Rudolph
冊 冉兺
p i 兩 ␺ i 典具 ␺ i 兩 ⭐E C
i
p i 兩 ␺ i 典具 ␺ i 兩
冊
⭐
兺i p i E C共 兩 ␺ i 典具 ␺ i兩 兲
⫽
兺i p i S vN共 P ␺ 兲
⫽
兺i p i E D共 兩 ␺ i 典具 ␺ i兩 兲 .
i
共39兲
䊏
C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 54, 3824 共1996兲; quant-ph/9604024.
V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 共1998兲; quant-ph/9707035.
3
G. Vidal, J. Mod. Opt. 47, 355 共2000兲; quant-ph/9807077.
4
M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 84, 2014 共2000兲; quant-ph/9908065.
5
S. Popescu and D. Rohrlich, Phys. Rev. A 56, R3319 共1997兲; quant-ph/9610044.
6
M. A. Nielsen, Phys. Rev. A 61, 064301 共2000兲; quant-ph/9908086.
7
An analogous result for the finite regime was obtained in Ref. 3.
8
P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 86, 2681 共2001兲; quant-ph/0010054.
9
M. J. Donald and M. Horodecki, Phys. Lett. A 264, 257 共1999兲; quant-ph/9910002.
10
K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64, 062307 共2001兲; quant-ph/0010095.
11
O. Rudolph, J. Math. Phys. 42, 2507 共2001兲; quant-ph/0105104.
12
O. Rudolph, J. Math. Phys. 42, 5306 共2001兲; math-ph/0005011.
13
N. Linden, S. Popescu, B. Schumacher, and M. W. Westmoreland, ‘‘Reversibility of local transformations of multiparticle
entanglement,’’ quant-ph/9912039.
14
E. Schmidt, Math. Ann. 63, 433 共1907兲.
15
K. Kraus, Ann. Phys. 共N.Y.兲 64, 311 共1971兲.
16
E. B. Davies, Quantum Theory of Open Systems 共Academic, London, 1976兲.
17
K. Kraus, States, Effects and Operations 共Springer-Verlag, Berlin, 1983兲.
18
M.-D. Choi, Linear Algebra Appl. 10, 285 共1975兲.
19
M. J. Donald, M. Horodecki, and O. Rudolph, ‘‘The uniqueness theorem for entanglement measures,’’ quant-ph/0105017
v1.
20
C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Phys. Rev.
A 59, 1070 共1999兲; quant-ph/9804053.
21
M. A. Nielsen, Phys. Rev. Lett. 83, 436 共1999兲; quant-ph/9811053.
22
L. Hardy, Phys. Rev. A 60, 1912 共1999兲; quant-ph/9903001.
23
J. G. Jensen and R. Schack, Phys. Rev. A 63, 062303 共2001兲; quant-ph/0006049.
24
V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, 2275 共1997兲; quant-ph/9702027.
25
D. Jonathan and M. Plenio, Phys. Rev. Lett. 83, 1455 共1999兲; quant-ph/9903054.
26
P. Walters, An Introduction to Ergodic Theory 共Springer-Verlag, New York, 1982兲.
27
E. M. Rains, Phys. Rev. A 60, 179 共1999兲; quant-ph/9809082.
28
P. Hayden, M. Horodecki, and B. Terhal, J. Phys. A 34, 6891 共2001兲; quant-ph/0008134.
29
C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 共1996兲; quant-ph/9511030.
30
K. Audenaert, J. Eisert, E. Jané, M. B. Plenio, S. Virmani, and B. De Moor, Phys. Rev. Lett. 87, 217902 共2001兲;
quant-ph/0103096.
31
M. B. Plenio, S. Virmani, and P. Papadopoulos, J. Phys. A 33, L193 共2000兲; quant-ph/0002075.
32
M. Fannes, Commun. Math. Phys. 31, 291 共1973兲.
33
M. Ohya and D. Petz, Quantum Entropy and Its Use 共Springer-Verlag, New York, 1993兲, p. 22.
34
P. Horodecki, M. Horodecki, and R. Horodecki, Acta Phys. Slov. 48, 141 共1998兲; quant-ph/9805072.
35
M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 85, 433 共2000兲; quant-ph/0003040.
36
A. Uhlmann, Open Syst. Inf. Dyn. 5, 209 共1998兲; quant-ph/9704017.
37
M. A. Nielsen, Phys. Rev. A 63, 022114 共2001兲; quant-ph/0008073.
38
T. M. Cover and J. A. Thomas, Elements of Information Theory 共Wiley, New York, 1991兲.
1
2
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