PHYSICAL REVIEW A 67, 012303 共2003兲
Efficient distillation beyond qubits
Karl Gerd H. Vollbrecht* and Michael M. Wolf†
Institute for Mathematical Physics, TU Braunschweig, Braunschweig, Germany
共Received 4 September 2002; published 7 January 2003兲
We provide generalizations of known two-qubit entanglement distillation protocols for arbitrary Hilbert
space dimensions. The protocols, which are analogs of the hashing and breeding procedures, are adapted to
bipartite quantum states that are diagonal in a basis of maximally entangled states. We show that the obtained
rates are optimal, and thus equal to the distillable entanglement, for a (d⫺1)-parameter family of rank
deficient states. Methods to improve the rates for other states are discussed. In particular, for isotropic states it
is shown that the rate can be improved such that it approaches the relative entropy of entanglement in the limit
of large dimensions.
DOI: 10.1103/PhysRevA.67.012303
PACS number共s兲: 03.67.⫺a, 03.65.Ta
I. INTRODUCTION
The possibility of entanglement distillation plays a crucial
role in quantum communication and quantum information
processing 共cf. 关1兴兲. Together with quantum error correction
it enables all the fascinating applications provided by quantum information theory in the presence of a noisy and interacting environment. It provides a method to overcome decoherence without requiring a complete isolation from the
environment.
The usual abstract way of thinking about entanglement
distillation is the following. Two parties, Alice and Bob, situated at distant locations share n copies of a mixed entangled
quantum state ␳ , which they may have obtained by sending
one part of a pure maximally entangled state through a noisy
quantum channel. Assume that both parties are able to perform any collective quantum operation, which merely acts
locally on their part of the n copies. Moreover, Alice and Bob
are connected via a classical channel, such that they can
perform arbitrary many rounds of local quantum operations,
where each round may depend on the measurement outcomes of all the preceding operations on both sides. The set
of operations accessible in this way is called LOCC 共local
operations and classical communication兲.
It was shown in one of the seminal works from the early
years of quantum information theory 关2兴 that under the above
conditions Alice and Bob can for certain two-qubit states
distill a smaller number m of states ␳ ⬘ , which are closer to
the maximally entangled state than the initial ones, from a
larger number n⭓m of more weakly entangled states ␳ . This
can be done in such a way that in the limit n→⬁ the output
states ␳ ⬘ become maximally entangled. The asymptotic ratio
m/n is then called the rate of the distillation protocol and the
maximally accessible rate under all LOCC protocols is an
important measure of entanglement, the distillable entanglement D( ␳ ). The latter quantifies in some sense the amount of
useful entanglement contained in the state ␳ .
Up to now the hashing/breeding distillation protocol presented in 关2,3兴, which is adapted to Bell diagonal states of
*Electronic address: k.vollbrecht@tu-bs.de
†
Electronic address: mm.wolf@tu-bs.de
1050-2947/2003/67共1兲/012303共9兲/$20.00
two qubits, is essentially the only protocol leading to a nonzero rate in the asymptotic limit. The present paper is devoted to generalizing these protocols to higher dimensions.
We will therefore closely follow the ideas of Refs. 关2,3兴. The
protocols are adapted to states that are diagonal in a basis of
maximally entangled states or mapped onto such states by a
LOCC twirl operation. The main step is to translate the apparent quantum task into a classical problem in such a way
that all the operations involved are LOCC. The following
results are obtained.
共1兲 We provide an entanglement assisted distillation protocol 共breeding兲 that works for any finite dimension d. It is in
this case assumed that Alice and Bob share maximally entangled states initially, which they may have obtained with a
different protocol 共e.g., hashing in a prime-dimensional subspace兲 and with a smaller rate. The attained distillation rate is
given by
log2 d⫺S„T共 ␳ 兲 …,
共1兲
where S(T( ␳ )) is the von Neumann entropy of the twirled
state ␳ .
共2兲 We show that a hashing protocol exists for prime dimensions d, which does not require any predistilled states.
The rate obtained is the same as in Eq. 共1兲. The method can
easily be generalized to dimensions that are powers of
primes, if one uses a different basis of maximally entangled
states.
共3兲 Both protocols, breeding and hashing, are shown to be
optimal for a (d⫺1)-parameter family of low rank states.
This generalizes the observation of Rains 关4兴 for the case d
⫽2.
共4兲 We discuss methods for improving the rates for isotropic states. In particular, the projection onto local subspaces will turn out to yield the optimal rate in the limit of
large dimensions.
Despite the quite considerable effort in the theory of entanglement distillation and its practical relevance, many of
the basic questions are yet unanswered. So we know neither
a decidable necessary and sufficient criterion for distillability
in Hilbert spaces Cd 丢 Cd with d⬎2, nor D( ␳ ), even for
otherwise simple quantum states ␳ . What makes the investigation of distillability so difficult is, on the one hand, the
67 012303-1
©2003 The American Physical Society
PHYSICAL REVIEW A 67, 012303 共2003兲
KARL GERD H.VOLLBRECHT AND MICHAEL M. WOLF
asymptotic limit (n→⬁) and, on the other hand, the mathematically intractable set of LOCC operations.
However, many partial results have been obtained in various directions, and before we go into the details of the
present article we want at least briefly to recall some of them.
(a) Distillability. For the case of two qubits it was shown
in 关6兴 that every entangled state is distillable. A general necessary condition for the distillability of a state ␳ is the fact
that its partial transpose ␳ T A , defined with respect to a given
product basis by 具 i j 兩 ␳ T A 兩 kl 典 ⫽ 具 k j 兩 ␳ 兩 il 典 , has a negative eigenvalue 关5兴. Except for special cases like states on C2 丢 Cn
关6,7兴 and Gaussian states 关8兴, it is, however, unclear whether
this condition is sufficient as well. There is some evidence
presented in 关7,9兴 that this may not be the case and that there
are indeed undistillable states, whose partial transpose is not
positive 共NPPT兲. However, it was shown in 关10兴 共see also
关11兴兲 that every NPPT state becomes distillable when adding
a certain bound entangled state with positive partial transpose 共PPT兲. Moreover, this activation process was shown to
require only an infinitesimal amount of entanglement contained in the additional PPT state 关10兴.
(b) Distillation protocols. Up to now, essentially the only
mixed state distillation protocol leading to a nonzero rate in
the asymptotic limit is the breeding/hashing protocol presented in 关2,3兴, which is adapted to Bell-diagonal states of
two qubits. There are, however, several purification schemes
that increase the purity and entanglement of a state without
yielding a nonzero rate on their own. The best known examples are the recurrence protocols presented in 关2,3兴 共see
关12兴 for the discussion of imperfect operations兲 and 关13兴.
These were investigated also for higher dimensions in 关14兴
and 关15兴. In 关16兴 it was shown that these methods can be
improved by using more than one source state per target. A
systematic generalization of the local unitaries applied in the
recurrence method for two-qubit systems was recently provided in 关17兴. A protocol which is analogous to the hashing
procedure but uses PPT preserving rather than LOCC operations is described in 关18兴.
(c) Distillable entanglement. For pure states the distillable
entanglement is equal to the von Neumann entropy of the
reduced state 关19兴. For mixed states only bounds are known,
which can be calculated for some very special cases 共cf.
关20兴兲. The best known upper bound for the distillable entanglement can be found in 关21兴. A closely related bound,
which we will describe in detail later, is given by the relative
entropy of entanglement 关4,22兴.
(d) Relations to other quantum information tasks. It was
already noticed in 关3兴 that a distillation protocol involving
only one-way classical communication 共like the hashing protocol兲 corresponds to a quantum error correcting code, and
vice versa. Moreover, upper bounds for the distillable entanglement allow one to obtain upper bounds for quantum
channel capacities 共cf. 关23兴兲. In particular, the inequality
D 1 共 ␳ 兲 ⭓S 共 trB 关 ␳ 兴 兲 ⫺S 共 ␳ 兲 ,
共2兲
where D 1 ( ␳ ) is the distillable entanglement for one-way
communication protocols, implies Shannon-like formulas for
quantum capacities, providing the quantum noisy coding
theorem 关23兴. The inequality 共2兲 is consistent with the results
we obtain in Sec. V.
An important application of distillation and purification
schemes is also the possibility of factoring out an eavesdropper for secret quantum communication 关13,24兴. This is
known as quantum privacy amplification.
II. PRELIMINARIES
Let us first introduce the preliminaries we need for the
distillation protocols described in Secs. III and IV. We begin
by characterizing the states for which the protocols are
adapted and we will subsequently specify the set of required
local operations.
A. Basis of maximally entangled states
Bases of maximally entangled states 兵 ⌿ kl 其 , k,l
苸 兵 0, . . . ,d⫺1 其 , exist for any Hilbert space H⫽Cd 丢 Cd and
correspond to an orthonormal operator basis of d 2 unitary
d⫻d matrices 兵 U kl 其 via
兩 ⌿ kl 典 ⬅ 共 1 丢 U kl 兲 兩 ⍀ 典 ,
共3兲
with 兩 ⍀ 典 being a maximally entangled state. The latter is in
the following chosen to be 兩 ⍀ 典 ⫽ 兩 ⌿ 00典 ⫽1/冑d 兺 d⫺1
j⫽0 兩 j j 典 . A
general construction procedure 关25兴 for unitary operator
bases involves Latin squares and complex Hadamard matrices and the best known example constructed in this way is of
the form
d⫺1
U kl ⫽
兺
r⫽0
␩ rl 兩 k 丣 r 典具 r 兩 , ␩ ⫽e 2 ␲ i/d ,
共4兲
where 丣 means addition modulo d. The set of unitaries in
Eq. 共4兲 is orthonormal with respect to the Hilbert-Schmidt
scalar product, i.e., tr关 U *
i j U kl 兴 ⫽d ␦ ik ␦ jl , and forms a discrete Weyl system since U i j U kl ⫽ ␩ jk U i 丣 k, j 丣 l .
In the following we will solely use the maximally entangled basis from Eqs. 共3兲 and 共4兲 or tensor products thereof
and denote the first index as the shift and the second as the
phase index.
B. Symmetric states
Symmetric states commuting with a group of local unitaries play an important and paradigmatic role in quantum information theory and in particular in the context of entanglement distillation. The best known examples are Werner
states, isotropic states, and Bell-diagonal states. The latter
are convex combinations of the four maximally entangled
Bell states of two qubits and contain in this case the isotropic
states, which for two-qubit systems are in turn equivalent to
Werner states.
In analogy to the two-qubit case, we will call a state on
Cd 丢 Cd Bell diagonal if it can be written as a convex combination of maximally entangled states P i j ⫽ 兩 ⌿ i j 典具 ⌿ i j 兩 . The
corresponding symmetry group is given by
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PHYSICAL REVIEW A 67, 012303 共2003兲
EFFICIENT DISTILLATION BEYOND QUBITS
G⫽ 兵 U i, j 丢 U i,⫺ j 其 ,
共5兲
which is an Abelian group with the property that its commutant, which is again spanned by G, contains d 2 onedimensional projectors 兵 P i j 其 . Hence, for 兵 ␭ i j 其 being convex
weights,
␳⫽
兺i j ␭ i j P i j
⇔ ᭙ g苸G: 关 ␳ ,g 兴 ⫽0.
共6兲
Moreover, every nonsymmetric state ␳ can be mapped onto a
Bell-diagonal state T( ␳ ) by means of a discrete twirl operation
T共 ␳ 兲 ⫽
1
d2
兺
g苸G
g * ␳ g,
共7兲
which can be implemented by means of local operations and
classical communication.
Isotropic states, which are completely characterized by
their fidelity f ª 具 ⍀ 兩 ␳ 兩 ⍀ 典 , are a special instance of Belldiagonal states with ␭ 00⫽ f and ␭ i j ⫽(1⫺ f )/(d 2 ⫺1) for
(i, j)⫽(0,0). An isotropic state is entangled and distillable if
and only if f ⬎1/d 关14兴.
C. Local operations and measurements
An essential ingredient for any distillation protocol acting
on two-qubit systems is the controlled-NOT 共CNOT兲 operation.
One possibility of generalizing this operation to higher dimensions is to use a controlled shift 共CS兲operation 关14兴,
which acts as
C兩i典 丢 兩 j典⫽兩i典 丢 兩 j 丣 i典,
共8兲
where the first tensor factor is the source and the second the
target. It is readily verified that a bilateral controlled shift
共BCS兲 operation, where both parties in a bipartite system
apply CS operations locally on a tensor product of two maximally entangled states, acts as
共 C 丢 C 兲 兩 ⌿ i j 典 丢 兩 ⌿ kl 典 ⫽ 兩 ⌿ i, j両l 典 丢 兩 ⌿ k 丣 i,l 典 ,
共9兲
where the first tensor product on the left-hand side in Eq. 共9兲
corresponds to the Alice-Bob split, whereas the others correspond to the source-target split. Note that the target state
picks up the shift index of the source, i.e., k哫k 丣 i, while the
source remains unchanged if and only if l⫽0.
If we measure a target state of the form 兩 ⌿ kl 典 in a computational basis, then k is the difference of the measurement
outcomes. In this way we can obtain information about the
shift index of an unknown source state, if we know the target
state: We first apply a BCS operation and then measure the
target state in the computational basis.
An analogous transformation can be defined for the phase
indices, since we can simply interchange the two indices by
the following local unitary operation:
共 V 丢 V̄ 兲 P i j 共 V 丢 V̄ 兲 * ⫽ P j,⫺i ,
V⫽
1
冑d
␩ kl 兩 k 典具 l 兩 .
兺
k,l
共10兲
So we can define a modified bilateral controlled shift
共MBCS兲 operation by first applying a V 丢 V̄ transformation
to the source state, followed by a BCS operation on source
and target and then undoing the local operation on the source
state by a (V 丢 V̄) * rotation. The MBCS operation then acts
as
P i j 丢 P kl 哫 P i 丣 l, j 丢 P k 丣 j,l ,
共11兲
where the second tensor factor corresponds to the target
again.
Assume now that we have one target state 兩 ⌿ 00典 and n
source states with shift indices k 1 , . . . ,k n and phase indices
l 1 , . . . ,l n . We now apply s a BCS operations and p a MBCS
operations to the ath source and the target. If we then measure the target state again in the computational basis, the
difference of the measurement outcomes will be
n
n
a⫽1
a⫽1
共 丣 k a s a 兲 丣 共 丣 l a p a 兲 ⫽: 具 kជ ,sជ 典 丣 具 ជl ,pជ 典 .
共12兲
This enables us to obtain some information about the distribution of the shift and phase indices of the source states
without disturbing them.
In the following we will organize the shift and phase indices of a sequence of maximally entangled states in a single
vector Sជ ª(k 1 , . . . ,k n ,l 1 , . . . ,l n ) and accordingly the mulជ
tiplicities of the BCS and MBCS operations in a vector M
ជ then belong to
ª(s 1 , . . . ,s n ,p 1 , . . . ,p n ). Both Sជ and M
兵 0, . . . ,d⫺1 其 2n , i.e., they are elements of the group Z2n
d
with addition modulo d.
ជ folThe sequence of local operations characterized by M
lowed by a measurement of an additionally required target
state P 00 thus yields the information
ជ ,Sជ 典 ª
具M
2n
丣
M i S i ⫽ 具 kជ ,sជ 典 丣 具 ជl ,pជ 典
共13兲
i⫽1
about a sequence Sជ of maximally entangled states, without
disturbing the latter.
To complete the list of basic local operations required
later on we have still to introduce a generalization of a ␲ /2
rotation given by
u共 g 兲ª
兺k e i ␲ /dk g兩 k 典具 k 兩 ,
2
共14兲
where g苸Z is an arbitrary number. We will use this within
bilateral operations of the form u(g) 丢 ū(g) and v (g)
丢 v̄ (g), where v (g)ªV * u(g)V. These unitary operations
act on a maximally entangled state P kl as
012303-3
关 u 共 g 兲 丢 ū 共 g 兲兴 P kl 关 u 共 g 兲 丢 ū 共 g 兲兴 * ⫽ P k,l両gk ,
共15兲
PHYSICAL REVIEW A 67, 012303 共2003兲
KARL GERD H.VOLLBRECHT AND MICHAEL M. WOLF
关v共 g 兲 丢 v̄共 g 兲兴 P kl 关v共 g 兲 丢 v̄共 g 兲兴 * ⫽ P k 丣 gl,l .
D. Why primes are special
Measurements based on the Z2n
d scalar product in Eq. 共13兲
play a crucial role in the entanglement distillation protocols
discussed in the following. One of the main tasks will
ជ such that we
thereby be to choose the measurement vector M
obtain as much information about the sequence Sជ as possible.
In general this can be a highly nontrivial problem. However,
ជ randomly
if the dimension d is a prime, then choosing M
turns out to be a pretty good choice. The reason why this
works well only for prime dimensions is that in this case Zd
is a field, which means that it is an Abelian group not only
with respect to the addition modulo d, but also with respect
to the multiplication if we exclude the zero element. That is,
for a,b,x苸Zd , a⫽0, every equation of the form
共17兲
b⫽ax mod d
has a unique solution for x if and only if d is prime. This
leads us to the following lemma.
Lemma 1. Let Sជ ⫽Sជ ⬘ be elements of Zm
d with d prime.
m
Given a scalar product 具 xជ ,yជ 典 ª 丣 i⫽1
x i y i and a uniformly
ជ 苸Zmd , the probability for
distributed random vector M
ជ ,Sជ 典 ⫽ 具 M
ជ ,Sជ ⬘ 典 is equal to 1/d.
具M
ជ 典 ⫽0.
Proof. We ask for the probability that 具 Sជ ⫺Sជ ⬘ ,M
Since Sជ ⫽Sជ ⬘ , there exists a component x where S x ⫽S x⬘ and
we can write
ជ 典 ⫽M x 共 S x ⫺S x⬘ 兲 ⫹
0⬅ 具 Sជ ⫺Sជ ⬘ ,M
兺 M i共 S i ⫺S ⬘i 兲 .
i⫽x
共18兲
Assume now that all M i ,i⫽x, are already randomly chosen.
Then Eq. 共18兲 has the form of Eq. 共17兲 with x⫽M x , a
⫽(S x ⫺S x⬘ ), and b⫽ 兺 i⫽x M i (S ⬘i ⫺S i ). Since this equation
has a unique solution for x and M x is a uniformly distributed
random variable, the probability that M x matches the solution is indeed 1/d.
䊏
III. THE BREEDING PROTOCOL
The breeding protocol is the preliminary stage of the
hashing protocol discussed in the next section. Both are
adapted to the distillation of Bell-diagonal states and the
main idea is to transform this quantum problem into the classical problem of identifying a word given the probability
distribution of the corresponding alphabet and a restricted set
of measurements.
Assume that Alice and Bob share n copies of a Belldiagonal state ␳ , such that
␳ 丢 n⫽
兺
k 1 , . . . ,k n ,l 1 , . . . ,l n
P k 1 l 1 丢 ••• 丢 P k n l n ª P Sជ
共16兲
␭ k 1 l 1 •••␭ k n l n P k 1 l 1 丢 ••• 丢 P k n l n .
共19兲
共20兲
with probability ␭ k 1 l 1 •••␭ k n l n . Note that, if they knew the
sequence Sជ ⫽(k 1 , . . . ,k n ,l 1 , . . . ,l n ), they could apply appropriate local unitary operations in order to obtain the stan丢n
and thus gain nlog2d
dard maximally entangled state P 00
ebits of entanglement.
Let us further assume that they have already a sufficiently
large set of predistilled maximally entangled states P 00 and
that they are able to perform all the local operations such as
BCS and MBCS operations described in the previous section. In this case they can utilize the predistilled states for
ជ ,Sជ 典
performing local measurements leading to the result 具 M
as discussed in Eq. 共13兲 in order to obtain some information
about the sequence Sជ . However, every single measurement
ជ will destroy one of the predistilled maximally entangled
M
states, and the task is therefore to identify the sequence Sជ
using as few measurements as possible.
At this point the quantum task of distillation is transformed into an entirely classical problem: Given an unknown
vector Sជ and the possibility of measuring functions of the
ជ ,Sជ 典 , how many measurements r do we need to idenform 具 M
ជ
tify S ? Knowing Sជ , we gained n maximally entangled pairs,
but destroyed r, such that the overall gain would be (1
⫺r/n)log2d ebits per copy.
We would need r⫽2n measurements to identify a completely random Sជ 苸Z2n
d with independently and uniformly
distributed components. So we would destroy more entangled pairs than we gain.
Fortunately, however, the set of possible vectors Sជ is not
completely random but has a distribution depending on the
spectrum of the Bell-diagonal state ␳ . In the limit of many
copies there exists thus a set of likely sequences containing
only 2 nS( ␳ ) ⭐d 2n different vectors Sជ with the property that
the probability that a vector Sជ is contained within this set of
likely sequences approaches 1 in the limit n→⬁ 关26兴. Here
S( ␳ ) is the von Neumann entropy of the density matrix,
which is equal to the Shannon entropy of the spectrum of ␳ ,
i.e.,
S 共 ␳ 兲 ⫽⫺
␭ kl log2 ␭ kl .
兺
k,l
共21兲
ជ ranNow let Alice and Bob choose each measurement M
domly. Then Lemma 1 tells us that if d is prime they can
reduce their list of possible sequences by a factor of 1/d after
every measurement by deleting every sequence that is not
consistent with the obtained result. Since their list initially
contains 2 nS( ␳ ) possible sequences, r⫽nS( ␳ )/log2d measurements are required in order to identify Sជ in the limit of
large n. The rate obtained in this way in the asymptotic limit
is a lower bound to the 共entanglement assisted兲 distillable
entanglement, which is thus
An appropriate interpretation of Eq. 共19兲 is to say that Alice
and Bob share the state
012303-4
D 共 ␳ 兲 ⭓log2 d⫺S 共 ␳ 兲 .
共22兲
PHYSICAL REVIEW A 67, 012303 共2003兲
EFFICIENT DISTILLATION BEYOND QUBITS
Let us now extend the obtained result, expressed in Eq. 共22兲,
from prime dimensions to arbitrary ones. As we have already
mentioned, Lemma 1 holds only for primes and cannot be
extended. A way out of this dilemma is to use controlled shift
operations between Hilbert spaces of different dimensions.
The definition looks similar to Eq. 共23兲:
surement outcomes, i.e., the final shift index of the target,
will be
C ⬘兩 i 典 丢 兩 j 典 ª 兩 i 典 丢 兩 j 丣 i 典 ,
ជ ,Rជ 典 in Eq. 共25兲 is the familiar outcome if
The first term 具 M
n⫺1
M i M i⫹n⫺1 S 2n
the target state is P 00 . The terms ⫺ 兺 i⫽1
⫹S n correspond to the other possible targets and their back
action onto the source states, and the last term gS 2n is due to
the initial v (g) 丢 v̄ (g) rotation.
Note that in general the outcome also depends on the
order in which the different BCS and MBCS operations are
carried out. For Eq. 共25兲 first all BCS and then all MBCS
operations were applied.
With the possibility of measuring functions of the form
共25兲 we will now follow the line of argumentation of the
previous section and show that the gain of information is
共asymptotically兲 at least log2d bits per measurement, if the
dimension d is prime. Therefore we ask again for the probability that two different sequences Sជ ⫽Sជ ⬘ lead to the same
ជ and g
measurement outcome under the assumption that M
are uniformly distributed random variables. To this end we
have to distinguish between three different cases.
共1兲 The two sequences differ only in S n ⫽S n⬘ . This is the
trivial case where Sជ and Sជ ⬘ are distinguished with unit probability.
共2兲 Sជ and Sជ ⬘ differ in the phase index of their target, i.e.,
ជ is already cho⬘ . If we assume in this case that M
S 2n ⫽S 2n
ជ ;g 兩 Sជ 典 ⫽ 具 M
ជ ;g 兩 Sជ ⬘ 典 is of the form
sen, then the equation 具 M
b⫽ax with g playing the role of x. Hence, the probability
that a randomly chosen g matches the right solution is 1/d, if
d is prime.
共3兲 Sជ and Sជ ⬘ have a target state with the same phase
⬘ and Rជ ⫽Rជ ⬘ . Then Lemma 1 tells us that
index, i.e., S 2n ⫽S 2n
ជ distinguishes the two sequences Rជ
a randomly chosen M
ជ ⬘ with probability 1/d, if d is prime.
⫽R
The hashing protocol consists now of several rounds of
such measurements, where each round destroys the entanglement of one of the n pairs characterized by Sជ . The relevant
part of the system after r rounds is thus described by a vector
and the measurements applied to Sជ (r) are in
Sជ (r)苸Z2(n⫺r)
d
ជ (r)苸Z2(n⫺r⫺2)
and g(r)苸Zd . Note
turn characterized by M
d
that, given the sequence of measurements, the vector Sជ (r)
deterministically depends on Sជ ⫽Sជ (0).
The above discussion implies now that the probability
共23兲
with the only difference that i runs from 0 to d⫺1 whereas
j runs from 0 to d ⬘ ⫺1, where d ⬘ is now supposed to be a
prime with d ⬘ ⭓d. Moreover, the addition in the second ket
共target兲 is modulo d ⬘ .
The penalty of the BCS and MBCS operations based on
such a controlled shift operation is that the source states after
the operation will in general no longer be of the form P kl
given by Eqs. 共3兲 and 共4兲. However, they still are, if the
target state, acting on Cd ⬘ 丢 Cd ⬘ , has a phase index equal to
zero. Then,
⬘ 典 ⫽ 兩 ⌿ i, j 典 丢 兩 ⌿ ⬘k 丣 i,0典 .
共 C ⬘ 丢 C ⬘ 兲 兩 ⌿ i j 典 丢 兩 ⌿ k0
共24兲
In this way we can straightforward generalize the result of
Eq. 共22兲 to Bell-diagonal states of arbitrary dimension, just
by reshuffling the degrees of freedom of the predistilled
maximally entangled states. Hence, the crucial point is that
the target states, which are in the case of the breeding protocol the predistilled resources, have prime dimensions.
IV. THE HASHING PROTOCOL
So far we have assumed that Alice and Bob initially share
a set of predistilled maximally entangled states. We took into
account this resource when calculating the rate, but, since the
additivity of the distillable entanglement is not yet decided
for bipartite systems, we might expect a smaller rate for the
共non entanglement assisted兲 distillation rate. The hashing
protocol, however, achieves the same asymptotic rate as the
previously discussed breeding protocol without requiring additional predistilled states. The main idea, though, is still the
same: Alice and Bob identify the sequence Sជ by means of
LOCC.
The protocol, however, is now a bit more complicated
since if we use one of the n states characterized by Sជ 苸Z2n
d as
target for the BCS and MBCS operations then we have to
take into account the back action from the unknown target to
the remaining source states.
We will in the following choose the nth state characterized by S n and S 2n as the target and consider the remainជ
ing n⫺1 states characterized by a vector R
⫽(S 1 , . . . ,S n⫺1 ,S n⫹1 , . . . ,S 2n⫺1 ) as source states.
In the first step of the protocol a bilateral v (g) 丢 v̄ (g)
rotation with a randomly chosen number g苸 兵 0, . . . ,d⫺1 其
is applied to the target state. This shifts some information
about the phase index of the target state into its shift index.
ជ 苸Z2n⫺2
is chosen and the two
Then a random vector M
d
parties perform the respective BCS and MBCS operations.
After measuring the target state, the difference of the mea-
n⫺1
ជ ;g 兩 Sជ 典 ª 具 M
ជ ,Rជ 典 ⫺
具M
兺
i⫽1
M i M i⫹n⫺1 S 2n ⫹S n ⫹gS 2n .
共25兲
r⫺1
ជ 共 r 兲 ;g 共 r 兲 兩 Sជ 共 r 兲 ⫺Sជ 共 r 兲 ⬘ 典 ⫽0 兴
P 关 Sជ 共 r 兲 ⫽Sជ 共 r 兲 ⬘ ⵩᭙ k⫽0
:具M
共26兲
that Sជ (r) and Sជ (r) ⬘ agree on all r measurement results, and
thereby remain distinct, is at most d ⫺r , if d is prime. Since
we have initially again about 2 nS( ␳ ) likely sequences, we
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PHYSICAL REVIEW A 67, 012303 共2003兲
KARL GERD H.VOLLBRECHT AND MICHAEL M. WOLF
need 共in the limit of large n) r⫽nS( ␳ )/log2d rounds in order
to identify the remaining sequence Sជ (r). This leads us again
to the rate in Eq. 共22兲.
What if d is not prime? Unfortunately, we cannot extend
the result to other dimensions in the way we did in the case
of the breeding protocol, since we do not have any target
states of prime dimension with zero phase indices. However,
if the dimension is a power of a prime d⫽d ⬘ p , then we can
easily extend the result for states that are diagonal with respect to a different basis of maximally entangled states,
given by
P kជ ជl ª
p
丢
i⫽1
P kili,
共27兲
1
␳ ␮⫽
d
丢 C . The 共LOCC兲 twirl operation, which maps an arbitrary
state onto a state diagonal in this basis, is given by
␳ 哫T 丢 p ( ␳ ).
The hashing protocol can then be applied in the described
manner to pn tensor factors of prime dimension d ⬘ and we
obtain again in this way the rate in Eq. 共22兲 for states in d
⫽d ⬘ p dimensions. Moreover, note that isotropic states are
diagonal in both bases given by Eqs. 共3兲 and 共4兲 and Eq.
共27兲.
V. OPTIMALITY FOR LOW RANK STATES
It is well known from the case of two-qubit systems that
the hashing/breeding protocol is not optimal in general.
However, for a certain (d⫺1)-parameter family of rank deficient states, we will show that the rate obtained is equal to
the relative entropy of entanglement. Since the latter is an
upper bound to the distillable entanglement, this implies that
the protocols are optimal in this case. For d⫽2 this was first
noticed by Rains in 关4兴.
Consider mixed states of rank smaller than or equal to d:
兺 ␮ l兩 ␾ l 典具 ␾ l兩 ,
l⫽1
共28兲
which are diagonal with respect to
兩 ␾ l 典 ª 兩 ⌿ 0,l 典 ⫽
共30兲
冋
1
⫽
兺 exp
d 2 l,r,s
⫽
1
d
册
2␲i
l 共 r⫺s 兲 兩 r,r 典具 s,s 兩
d
共31兲
d⫺1
兺
r⫽0
兩 r 典具 r 兩 丢 兩 r 典具 r 兩
共32兲
E R 共 ␳ 兲 ⫽inf␴ 共 tr关 ␳ 共 log2 ␳ ⫺log2 ␴ 兲兴 兲 ,
共33兲
where the infimum is taken over all separable states ␴ .
Hence, if we choose ␴ ⫽ ␳ sep , the distillable entanglement is
bounded by
D 共 ␳ ␮ 兲 ⭐E R 共 ␳ ␮ 兲 ⭐⫺tr关 ␳ log2 ␳ sep兴 ⫺S 共 ␳ ␮ 兲 .
共34兲
However, ⫺tr关 ␳ log2␳sep兴 ⫽log2d and the rate achieved by the
breeding/hashing protocol is thus optimal for every ␳ ␮ .
Strictly speaking, the hashing protocol leads to distillable
entanglement and the breeding rate is equal to the distillable
entanglement D ⬘ assisted by a maximally entangled resource
␻ . The latter can easily be seen by noting that
D ⬘ 共 ␳ 兲 ⫽D 共 ␳ 丢 ␻ 兲 ⫺E R 共 ␻ 兲
⭐E R 共 ␳ 丢 ␻ 兲 ⫺E R 共 ␻ 兲 ⭐E R 共 ␳ 兲 .
共35兲
共36兲
That is, the relative entropy of entanglement is an upper
bound for the entanglement assisted distillable entanglement
as well.
VI. IMPROVING THE RATES FOR ISOTROPIC STATES
d
␳ ␮⫽
兺 兩 ␾ l 典具 ␾ l兩
l⫽1
is evidently a separable state, which is said to be maximally
correlated. We will denote this state by ␳ sep in the following.
An upper bound to the distillable entanglement is given
by the relative entropy of entanglement 关4,22兴, which is defined by
where P k i l i is a maximally entangled state acting on Cd ⬘
d⬘
d
1
冑d
d⫺1
兺
r⫽0
e (2 ␲ i/d)rl 兩 rr 典 .
共29兲
The set of these states forms a simplex in Rd⫺1 and has the
property that every state ␳ ␮ except for the barycenter, for
which ␮ l ⫽1/d, is entangled. To see this, note first that
maxl兵␮l其 is the fully entangled fraction. That is, if not all ␮ l
are equal, then the fully entangled fraction is larger than 1/d
and the corresponding state is thus entangled. On the other
hand, if ␮ l ⫽1/d, then
For a general Bell-diagonal state the protocols introduced
are not optimal. In particular, for states with a large entropy
the rate is poor or even zero. In order to improve this, one
can make use of hybrid protocols, where the first step decreases the entropy while conserving most of the entanglement, and in a second step the hashing/breeding protocol is
applied. There are many ways of performing such an entropy
decreasing preprocessing. We will in the following discuss
two of them for the case of isotropic states: the recurrence
method which is well known for qubits 关2,3,13兴 and has already been investigated in 关14,15兴 for d⬎2, and the projection onto local subspaces. For the latter we will show that in
the limit of large dimensions the rate approaches the relative
entropy of entanglement. Hence, the achieved rate is optimal
in that limit.
Isotropic states, for which we will discuss both methods
in more detail, have the form
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PHYSICAL REVIEW A 67, 012303 共2003兲
EFFICIENT DISTILLATION BEYOND QUBITS
␳ ⫽ f 兩 ⍀ 典具 ⍀ 兩 ⫹
1⫺ f
d 2 ⫺1
共 1⫺ 兩 ⍀ 典具 ⍀ 兩 兲 .
共37兲
That is, they are depolarized maximally entangled states depending on a single fidelity parameter f 苸 关 0,1兴 . The twirl
operation mapping every state onto an isotropic state has the
form ␳ 哫 兰 dU(U 丢 Ū) ␳ (U 丢 Ū) * , i.e., it is an averaging
over the unitary group U(d) with respect to the Haar measure dU. The states in Eq. 共37兲 are entangled 共and distillable兲 if and only if f ⬎1/d 关14兴.
A. The recurrence method
The recurrence method for higher dimensions and variations thereof has already been discussed in detail in 关14兴 and
关15兴. For completeness, however, we will recall the main
steps. The idea is to apply a BCS operation on two copies of
the state and then to measure the target state in the computational basis. The source states are kept whenever the measurement outcomes coincide, otherwise they are discarded.
The remaining states may then be twirled onto isotropic
states again and we can proceed by either iterating the recurrence method or applying the hashing/breeding procedure if
the entropy of the remaining states is already sufficiently
small, i.e., S( ␳ )⬍log2d. We note that, as already mentioned
in 关13兴 for d⫽2, the isotropic twirling after each step is not
really necessary and in general increases the entropy and
therefore decreases the rate of a subsequently applied
breeding/hashing protocol. However, it is sufficient to look at
the fidelity parameter f in order to see that a recurrence preprocessing improves the rates such that every entangled isotropic state becomes distillable.
Straightforward calculation shows that the fidelity f ⬘ after
applying the recurrence method once is given by
f ⬘⫽
1⫹ f 关 d f 共 d 2 ⫹d⫺1 兲 ⫺2 兴
d 3 f 2 ⫺2d f ⫹d 2 ⫹d⫺1
,
共38兲
and the probability for equal measurement outcomes is
p rec⫽
d 3 f 2 ⫺2d f ⫹d 2 ⫹d⫺1
共 d⫹1 兲 d 共 d⫺1 兲
2
.
FIG. 1. Normalized rates for isotropic states of dimension d
⫽7. The upper bound 共thick兲 is given by the relative entropy of
entanglement. The hashing rate 共solid兲 can be improved in the region S( ␳ )⬃log2d if one first applies one round of the recurrence
method 共dotted兲 or locally projects the state onto subspaces of dimension 3 and 4 共dashed兲.
In Fig. 1 we have applied the method as described above
to an isotropic state of dimension d⫽7. As expected this
leads to an improvement of the rate in the region where
S( ␳ )⬃log2d.
B. Projecting onto local subspaces
A surprisingly efficient method for improving the distillation rate is to project the initial state locally onto blocks of
smaller dimensions and to apply the hashing/breeding protocol to these subspaces.
Let 兵 Q i ⫽ 兺 k 兩 k 典具 k 兩 其 be a set of b projectors of dimension
q i corresponding to a local projective observable with b outb
b
q i ⫽d and 兺 i⫽1
Q i ⫽1d . Assume further
comes, i.e., 兺 i⫽1
that Alice and Bob have the same observable 兵 Q Ai 其 ⫽ 兵 Q Bi 其
and that they apply the corresponding measurement to an
isotropic state ␳ . Again the state is kept if the results of the
measurements coincide and discarded otherwise. If both parties obtain the same measurement outcome i, which happens
with a probability
p i⫽
共39兲
Note that the recurrence method alone does not lead to a
nonzero rate since in every round we destroy or discard at
least half of the resources 共all the target states兲 and maximally entangled states are obtained only in the limit of infinitely many rounds. However, it holds that f ⬘ ⬎ f and therefore S( ␳ ⬘ )⬍S( ␳ ) for every entangled isotropic ␳ . This
means that every entangled isotropic state can be distilled
with a nonzero rate by applying sufficiently many rounds of
the recurrence protocol followed by hashing/breeding.
There are many directions in which the recurrence method
can be modified or improved. One could use more than one
source state 关16兴, a different bilateral operation 关15兴, neglect
the isotropic twirling 关13兴, apply it to other kinds of states
关15兴, or use target and source states with different fidelities.
冉
冊
qi
qi
共 1⫺ f 兲
f⫹ 2
q 2i ⫺
,
d
d
共 d ⫺1 兲
共40兲
the state after the measurement ␳ i ⫽(Q Ai 丢 Q Bi ) ␳ (Q Ai
B
丢 Q i ) * /p i is again an isotropic state on Cq i 丢 Cq i with fidelity
冋
f i⫽ f
冉 冊册
qi
qi
共 1⫺ f 兲
1⫺
⫹ 2
d 共 d ⫺1 兲
d
/p i .
共41兲
If we now apply the hashing/breeding protocol to the b
blocks, the rate obtained,
b
兺
i⫽1
p i max关 0,log2 q i ⫺H 共 ␳ i 兲兴 ,
共42兲
exceeds log2d⫺H(␳) for some values of f depending on the
dimensions 兵 q i 其 of the subspaces. If the dimensions of the
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PHYSICAL REVIEW A 67, 012303 共2003兲
KARL GERD H.VOLLBRECHT AND MICHAEL M. WOLF
Assume that d⫽2 m , i.e., a single state ␳ consists of m
pairs of qubits, and let the local measurements have m outcomes corresponding to subspaces of equal dimensions q i
⫽2 m /m. We can then estimate the probability of success and
the achieved fidelity 关for m⭓(1⫺ f ) ⫺1 ] by
FIG. 2. Normalized rates for the case where ␳ consists of ten
pairs of qubit systems (d⫽2 10) and the overall state is isotropic.
The upper bound 共thick兲 is given by the relative entropy of entanglement. The hashing rate 共solid兲 can be significantly improved
when projecting onto subspaces of dimension 2 m , m⫽1, . . . ,9.
The dashed line is the envelope of the rates obtained in this way.
The dotted curve is the rate one would achieve when considering
␳ as a set of ten pairs of qubits, twirling onto isotropic states on
C2 丢 C2 , and applying the hashing protocol then on the two-qubit
level.
blocks are about the same, then an increasing number of
blocks leads to a larger 共smaller兲 rate for small 共large兲 f.
Figures 1 and 2 show the rates obtained for d⫽7 and for the
case where the two parties share 10 pairs of qubits and the
overall state is isotropic. For the latter case the rate is already
not too far below the relative entropy of entanglement. In
fact, the content of the following subsection is to sketch that
in the limit of large dimensions d→⬁ the relative distance
between these quantities vanishes for all f.
C. The limit of large dimensions
As we have already mentioned, an upper bound for the
distillable entanglement is the relative entropy of entanglement, i.e., the minimal distance between ␳ and a separable
state ␴ measured in terms of the relative entropy
S 共 ␳ , ␴ 兲 ⫽tr关 ␳ log2 ␳ 兴 ⫺tr关 ␳ log2 ␴ 兴 .
共43兲
For entangled isotropic states the nearest separable state is
the isotropic state with f ⫽1/d 关21,28,29兴 and the relative
entropy of entanglement is thus
E R 共 ␳ 兲 ⫽log2 d⫺ 共 1⫺ f 兲 log2 共 d⫺1 兲 ⫺S 共 f ,1⫺ f 兲
共44兲
for f ⬎1/d and zero otherwise, where S is again the Shannon
entropy. Note that in the limit of large dimensions this becomes after normalization
E R共 ␳ 兲
⫽f.
d→⬁ log2 d
lim
共45兲
The same limit, however, appears for the distillation rate
obtained with the block projection method described in the
previous subsection.
p i ⭓ p̃ª
f ⫺4 ⫺m
,
m
共46兲
f i ⭓ f̃ ª
fm
.
f m⫹1
共47兲
In the limit of large m, the Eqs. 共46兲 and 共47兲 tell us that we
gain m 共almost兲 maximally entangled states of dimension q
⫽2 m /m with probability f /m. Looking at the normalized
entanglement as in Eq. 共45兲 leads then indeed to the same
limit as for the relative entropy of entanglement. Hence, we
共relatively兲 approach the distillable entanglement in the limit
of large dimensions. However, the main reason for this is that
the relative entanglement difference of two maximally entangled states in dimensions 2 m /m and 2 m vanishes for m
→⬁. Hence, the result is not as deep as it might seem at first
glance.
Nevertheless, projecting onto blocks of lower dimensions
can significantly improve the rates already for finite dimensions as shown in Figs. 1 and 2.
VII. CONCLUSION
Despite considerable efforts in the theory of entanglement
distillation, this field of investigation provides many open
questions even on a very basic level. Due to the complexity
of the underlying variational problem it is hard to find good
upper and lower bounds to the distillable entanglement, not
to mention an explicit calculation of this measure of ‘‘useful
entanglement.’’ So far, the most important distillation protocol leading to a nonzero rate, and thus to a nontrivial lower
bound to the distillable entanglement, has been adapted to
Bell-diagonal states of two qubits. Of course, this protocol
can also be applied to higher dimensional states either by
projecting down to qubits or by interpreting the state as a
tensor product of qubits if possible 共compare Fig. 2兲. However, both methods will in general discard most of the entanglement.
The present article shows how to generalize the breeding
and hashing protocols to higher dimensions. The rates obtained are optimal only for special cases; however, they provide an improved lower bound to the distillable entanglement in general. Both protocols consist of two steps: first the
quantum task is translated to a classical problem and then the
latter is solved. The classical part of the protocol is already
essentially optimal; however, some information and hence
entanglement is lost during the ‘‘translation process.’’
We think that the presented results admit further generali
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PHYSICAL REVIEW A 67, 012303 共2003兲
EFFICIENT DISTILLATION BEYOND QUBITS
ACKNOWLEDGMENTS
zations toward other classes of states and maybe also with
respect to the restriction of the hashing protocols to dimensions that are 共powers of兲 primes.
We hope that our work initiates further investigations concerning the distillation of entanglement, including explora
tions of the implications from recent work in the field of
quantum error correcting codes.
The authors would like to thank R. F. Werner, H. Barnum,
and D. Schlingemann for valuable discussions. Funding by
the European Union project EQUIP 共Contract No. IST-199911053兲 and financial support from the DFG 共Bonn兲 are gratefully acknowledged.
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