Computable measure of entanglement
G. Vidal
Institut fu
¨
r Theoretische Physik, Universita
¨
t Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria
R. F. Werner
Institut fu
¨
r Mathematische Physik, TU Braunschweig, Mendelssohnstrasse 3, 38304 Braunschweig, Germany
共Received 28 April 2001; published 22 February 2002兲
We present a measure of entanglement that can be computed effectively for any mixed state of an arbitrary
bipartite system. We show that it does not increase under local manipulations of the system, and use it to obtain
a bound on the teleportation capacity and on the distillable entanglement of mixed states.
DOI: 10.1103/PhysRevA.65.032314 PACS number共s兲: 03.67.⫺a, 03.65.⫺w
I. INTRODUCTION
In recent years it has been realized that quantum mechan-
ics offers unexpected possibilities in information transmis-
sion and processing, and that quantum entanglement of com-
posite systems plays a major role in many of them. Since
then, a remarkable theoretical effort has been devoted both to
classifying and quantifying entanglement.
Pure-state entanglement of a bipartite system is presently
well understood, in that the relevant parameters for its opti-
mal manipulation under local operations and classical
communication 共LOCC兲 have been identified, in some
asymptotic sense 关1兴 as well as for the single-copy case 关2兴.
Given an arbitrary bipartite pure state
兩
AB
典
, the entropy of
entanglement E(
AB
) 关1兴, namely, the von-Neumann en-
tropy of the reduced density matrix
A
⬅Tr
B
兩
AB
典具
AB
兩
,
tells us exhaustively about the possibilities of transforming,
using LOCC,
兩
AB
典
into other pure states, in an asymptotic
sense. When manipulating a single copy of
兩
AB
典
, this infor-
mation is provided by the n entanglement monotones E
l
⫽ 兺
i⫽l
n
i
(l⫽ 1,...,n) 关2兴, where
i
are the eigenvalues of
A
in decreasing order.
Many efforts have also been devoted to the study of the
mixed-state entanglement. In this case several measures have
been proposed. The entanglement of formation E
F
(
) 关3兴
—or, more precisely, its renormalized version, the entangle-
ment cost E
C
(
) 关4兴— and the distillable entanglement
E
D
(
) 关3兴 quantify, respectively, the asymptotic pure-state
entanglement required to create
, and that which can be
extracted from
, by means of LOCC. The relative entropy
of entanglement 关5兴 appears as a third, related measure 关6兴
that interpolates between E
C
and E
D
关7兴.
However, in practice, it is not known how to effectively
compute these measures, nor any other, for a generic mixed
state, because they involve variational expressions. To our
knowledge, the only exceptions are Wootter’s closed expres-
sion for the entanglement of formation E
F
(
) 关and concur-
rence C(
)兴 of two-qubit states 关8兴, and its single-copy ana-
log E
2
(
) also for two qubits 关9兴.
Multipartite pure-state entanglement represents the next
order of complexity in the study of entanglement, and is of
interest, because one hopes to gain a better understanding of
the correlations between different registers of a quantum
computer. Consider a tripartite state
兩
ABC
典
. Some of its en-
tanglement properties depend on those of the two-party re-
duced density matrices, which are in a mixed state. For in-
stance, the relative entropy of
AB
⬅ tr
C
兩
ABC
典具
ABC
兩
has
been used to prove that bipartite and tripartite pure-state en-
tanglements are asymptotically inequivalent 关10兴. Thus, the
lack of an entanglement measure that can be easily computed
for bipartite mixed states is not only a serious drawback in
the study of mixed-state entanglement, but also a limitation
for understanding multipartite pure-state entanglement.
The aim of this paper is to introduce a computable mea-
sure of entanglement 关11兴, and thereby fill an important gap
in the study of entanglement. It is based on the trace norm of
the partial transpose
T
A
of the bipartite mixed state
,a
quantity whose evaluation is completely straightforward us-
ing standard linear algebra packages. It essentially measures
the degree to which
T
A
fails to be positive, and therefore it
can be regarded as a quantitative version of Peres’ criterion
for separability 关12兴. From the trace norm of
T
A
, denoted by
兩兩
T
A
兩兩
1
, we will actually construct two useful quantities. The
first one is the negativity
N
共
兲
⬅
储
T
A
储
1
⫺ 1
2
, 共1兲
which corresponds to the absolute value of the sum of nega-
tive eigenvalues of
T
A
关13兴, and which vanishes for unen-
tangled states. As we will prove here, N(
) does not increase
under LOCC, i.e., it is an entanglement monotone 关14兴, and
as such it can be used to quantify the degree of the entangle-
ment in composite systems. We will also consider the loga-
rithmic negativity
E
N
共
兲
⬅log
2
兩兩
T
A
兩兩
1
, 共2兲
which again exhibits some form of monotonicity under
LOCC 共it does not increase during deterministic distillation
protocols兲 and is, remarkably, an additive quantity.
The importance of N and E
N
is boosted, however, be-
yond their practical computability by two results that link
these measures with relevant parameters characterizing en-
tangled mixed states. The negativity will be shown to bound
the extent to which a single copy of the state
can be used,
together with LOCC, to perform quantum teleportation 关15兴.
PHYSICAL REVIEW A, VOLUME 65, 032314
1050-2947/2002/65共3兲/032314共11兲/$20.00 ©2002 The American Physical Society65 032314-1
In turn, the logarithmic negativity bounds the distillable en-
tanglement E
D
⑀
contained in
, that is, the amount of ‘‘almost
pure’’-state entanglement that can be asymptotically distilled
from
丢 N
, where ‘‘almost’’ means that some small degree
⑀
of imperfection is allowed in the output of the distillation
process.
Remarkably, this last result has already found an applica-
tion in the context of asymptotic transformations of bipartite
entanglement 关16兴, as a means to prove that 关positive partial
transposition 共PPT兲兴 bound entangled states 关17兴 cannot be
distilled into entangled pure states even if loaned 共i.e., sub-
sequently recovered for replacement兲 pure-state entangle-
ment is used to assist the distillation process. In this way, the
bound on distillability implied by E
N
has contributed to
prove that, in a bipartite setting, asymptotic local manipula-
tion of the mixed-state entanglement is sometimes, in con-
trast to its pure-state counterpart, an inherently irreversible
process.
We have divided this paper into seven sections. In Sec. II
some properties of the negativity N, such as its monotonicity
under LOCC, and of the logarithmic negativity E
N
are
proved. We also discuss a more general construction leading
to several other 共nonincreasing under LOCC兲 negativities. In
Secs. III and IV we derive, respectively, the bounds on tele-
portation capacity and on asymptotic distillability. Then in
Sec. V we calculate the explicit expression of N and E
N
for
pure states and for some highly symmetric mixed states, also
for Gaussian states of light field. In Sec. VI extensions of
these quantities to multipartite systems are briefly consid-
ered, and Sec. VII contains some discussion and conclusions.
II. MONOTONICITY OF N„
… UNDER LOCC
In this section we show that the negativity N(
)isan
entanglement monotone. We first give a rather detailed proof
of this result. Then we sketch an argument extending this
observation to several other similarly constructed
negativities—e.g., the robustness of entanglement 关18兴.
A. Definition and basic properties
From now on we will denote by
a generic state of a
bipartite system with finite-dimensional Hilbert space H
A
丢 H
B
⬅C
d
A
丢 C
d
B
shared by two parties, Alice and Bob.
T
A
denotes the partial transpose of
with respect to Alice’s
subsystem, that is the Hermitian, trace-normalized operator
defined to have matrix elements
具
i
A
,j
B
兩
T
A
兩
k
A
,l
B
典
⬅
具
k
兩
A
,j
B
兩
i
A
,l
B
典
共3兲
for a fixed but otherwise arbitrary orthonormal product basis
兩
i
A
,j
B
典
⬅
兩
i
典
A
丢
兩
j
典
B
苸 H
A
丢 H
B
. The trace norm of any Her-
mitian operator A is
储
A
储
1
⬅tr
冑
A
†
A 共关19兴 Sec.VI6兲, which is
equal to the sum of the absolute values of the eigenvalues
of A, when A is Hermitian 关20兴. For density matrices, all
eigenvalues are positive and thus
储
储
1
⫽ tr
⫽ 1. The partial
transpose
T
A
also satisfies tr
关
T
A
兴
⫽ 1, but since it may have
negative eigenvalues
i
⬍ 0, its trace norm reads in general
储
T
A
储
1
⫽ 1⫹ 2
冏
兺
i
i
冏
⬅1⫹ 2N
共
兲
. 共4兲
Therefore, the negativity N(
)—the sum
兩
兺
i
i
兩
of the nega-
tive eigenvalues
i
of
T
A
—measures by how much
T
A
fails
to be positive definite. Notice that for any separable or un-
entangled state
s
关21兴,
s
⫽
兺
k
p
k
兩
e
k
,f
k
典具
e
k
,f
k
兩
; p
k
⭓0,
兺
k
p
k
⫽ 1, 共5兲
its partial transposition is also a separable state 关12兴
s
T
A
⫽
兺
k
p
k
兩
e
k
*
,f
k
典具
e
k
*
,f
k
兩
⭓0, 共6兲
and therefore
储
s
T
A
储
1
⫽ 1 and N(
s
)⫽ 0.
The practical computation of N(
) is straightforward, us-
ing standard linear algebra packages for eigenvalue compu-
tation of Hermitian matrices. On the other hand, this repre-
sentation is not necessarily the best for proving estimates and
general properties of N(
). To begin with a simple example,
consider the property that N(
) does not increase under mix-
ing
Proposition 1. N is a convex function, i.e.,
N
冉
兺
i
p
i
i
冊
⭐
兺
i
p
i
N
共
i
兲
, 共7兲
whenever the
i
are Hermitian, and p
i
⭓0 with 兺
i
p
i
⫽ 1.
There is nothing to prove here, when we write N(
)
⫽ (
储
T
A
储
1
⫺ 1)/2, and observe that
储
•
储
1
, as any norm, satis-
fies the triangle inequality and is homogeneous of degree 1
for positive factors, hence convex.
However, the fact that
储
储
1
is indeed a norm is not so
obvious, when it is defined in terms of the eigenvalues. This
is shown best by rewriting it as a variational expression. Our
reason for recalling this standard observation from the theory
of the trace norm is that the same variational expression will
be crucial for showing monotonicity under LOCC opera-
tions. The variational expression is simply the representation
of a general Hermitian matrix A as a difference of positive
operators: Since we are in finite dimension we can always
write
A⫽ a
⫹
⫹
⫺ a
⫺
⫺
, 共8兲
where
⫾
⭓0 are density matrices (tr
关
⫾
兴
⫽ 1) and a
⫾
⭓0
are positive numbers. Note that by taking the trace of this
equation we simply have tr
关
A
兴
⫽ a
⫹
⫺ a
⫺
.
Lemma 2. For any Hermitian matrix A there is a decom-
position of the form 共8兲 for which a
⫹
⫹ a
⫺
is minimal. For
this decomposition,
储
A
储
1
⫽ a
⫹
⫹ a
⫺
, and a
⫺
is the absolute
sum of the negative eigenvalues of A.
Proof. Let P
⫺
be the projector onto the negative eigen-
valued subspace of A, and N⫽⫺tr
关
AP
⫺
兴
the absolute sum
of the negative eigenvalues. We can reverse the decomposi-
tion 共8兲 to obtain that A⫹ a
⫺
⫺
is positive semidefinite. This
implies that
G. VIDAL AND R. F. WERNER PHYSICAL REVIEW A 65 032314
032314-2
0⭐tr
关共
A⫹ a
⫺
⫺
兲
P
⫺
兴
⫽⫺N⫹ a
⫺
tr
关
⫺
P
⫺
兴
. 共9兲
But tr
关
⫺
P
⫺
兴
⭐1, that is a
⫺
⭓N. This bound can be satu-
rated with the choice a
⫺
⫺
⬅⫺ P
⫺
AP
⫺
共corresponding to
the Jordan decomposition of A, where
⫺
and
⫹
have dis-
joint support兲, which ends the proof. 䊐
For the negativity we, therefore, get the formula
N
共
A
兲
⫽ inf
兵
a
⫺
兩
A
T
A
⫽ a
⫹
⫹
⫺ a
⫺
⫺
其
, 共10兲
where the infimum is over all density matrices
⫾
and a
⫾
⭓0.
Another remarkable property of N(
) is the easy way in
which N(
1
丢
2
) relates to the negativity of
1
and that of
2
. This relationship is an important, but notoriously difficult
issue for discussing asymptotic properties of entanglement
measures 共see, e.g., 关22兴 for a discussion and a counterexam-
ple to the conjectured additivity of the relative entropy of
entanglement兲.
For the entanglement measure proposed in this paper we
get additivity for free. We start from the identity
储
1
丢
2
储
1
⫽
储
1
储
1
储
2
储
1
, which is best shown by using the definition of
the trace norm via eigenvalues, and we observe that partial
transposition commutes with taking tensor products. After
taking logarithms, we find for the logarithmic negativity
E
N
共
1
丢
2
兲
⫽ E
N
共
1
兲
⫹ E
N
共
2
兲
. 共11兲
It might seem from this that E
N
is a candidate for the much
sought for canonical measure of entanglement. However, it
has other drawbacks. For instance, it is not convex, as is
already suggested by the combination of a convex functional
共the trace norm兲 with the concave log function, which im-
plies that it increases under some LOCC. And although it has
an interesting, monotonic behavior during asymptotic distil-
lation 共as shown in Sec. IV兲, it does not correspond to the
entropy of entanglement for pure states 共see Sec. V兲.
B. Negativity as a mixed-state entanglement monotone
By definition, a LOCC operation 共possibly for many par-
ties兲 consists of a sequence of steps, in each of which one of
the parties performs a local measurement and broadcasts the
result to all other parties. In each round the local measure-
ment chosen is allowed to depend on the results of all prior
measurements. If at the end of a LOCC operation with initial
state
the classical information available is ‘‘i,’’ which oc-
curs with probability p
i
, and final state conditional on this
occurrence is
i
⬘
, we require of an entanglement monotone
关14兴 E that
E
共
兲
⭓
兺
i
p
i
E
共
i
⬘
兲
. 共12兲
It is clear by iteration that this may be proved by looking at
just one round of a LOCC protocol, consisting of a single
local operation. In the present case, since N makes no dis-
tinction between Alice and Bob, it suffices to consider just
one local measurement by Bob.
Now the most general local measurement is described by
a family M
i
of completely positive linear maps such that, in
the notation used in the previous paragraph, M
i
(
)⫽ p
i
i
⬘
.
These maps satisfy the normalization condition
兺
i
tr
关
M
i
(
)
兴
⫽ tr(
). This can be further simplified 关14兴
when some M
i
can be decomposed further into completely
positive maps, e.g., M
i
⫽ M
i
⬘
⫹ M
i
⬙
. Then we may simply
consider the finer decomposition as a finer measurement,
with the result i replaced by two others, i
⬘
and i
⬙
. Using the
convexity already established it is clear that it suffices to
prove Eq. 共12兲 for the finer measurement. That is, we can
assume that there are no proper decompositions of the M
i
,
or that M
i
is ‘‘pure.’’ This is equivalent to M
i
taking pure
states to pure states, or to the property 关23兴 that it can be
written with a single Kraus summand. Taking into account
that this describes a local measurement by Bob, we can write
M
i
共
兲
⫽
共
I
A
丢 M
i
兲
共
I
A
丢 M
i
†
兲
, 共13兲
where the Kraus operators M
i
must satisfy the normalization
condition
兺
i
M
i
†
M
i
⭐I
B
. For computing the right-hand side
of Eq. 共12兲 we need that
M
i
共
兲
T
A
⫽ M
i
共
T
A
兲
, 共14兲
which immediately follows from Eq. 共13兲 by expanding
as
a sum of 共not necessarily positive兲 tensor products. A similar
formula holds for Alice’s local operations, but with a modi-
fied operation M
i
on the 共rhs兲 right-hand side, in which the
Kraus operators have been replaced by their complex conju-
gates. Consider the decomposition
T
A
⫽
共
1⫹ N
兲
⫹
⫺ N
⫺
共15兲
with density operators
⫾
and N⫽ N(
). Then we can also
decompose the partially transposed output states
p
i
共
i
⬘
兲
T
A
⫽ M
i
共
兲
T
A
⫽ M
i
共
T
A
兲
⫽
共
1⫹ N
兲
M
i
共
⫹
兲
⫺ NM
i
共
⫺
兲
.
共16兲
Dividing by p
i
we get a decomposition of precisely the sort,
Eq. 共10兲, defining N(
i
⬘
). The coefficient a
⫺
⫽ N/p
i
must be
larger than the infimum, i.e., N(
i
⬘
)⭐N/p
i
. Multiplying by
p
i
and summing, we find the following inequality.
Proposition 3.
兺
i
p
i
N
共
i
⬘
兲
⭐N
共
兲
, 共17兲
i.e., N(
) is indeed an entanglement monotone.
C. Other negativities
Both the proofs, of convexity and of monotonicity, are
based on the variational representation of the trace norm in
lemma 2. The abstract version of this lemma is the definition
of the so-called base norm
储
•
储
S
associated with a compact
set S in a real vector space 关24兴. The negativity introduced
COMPUTABLE MEASURE OF ENTANGLEMENT PHYSICAL REVIEW A 65 032314
032314-3
above then corresponds to a special choice of S, and we can
easily find the property of S required for proving LOCC
monotonicity in the abstract setting. Other choices of S then
lead to other entanglement monotones, some of which have
been proposed in the literature.
For our purposes, we can take S as an arbitrary compact
convex subset of the Hermitian operators with unit trace,
whose real linear hull equals all Hermitian operators. Then,
in analogy to lemma 2, we define the associated base norm
and ‘‘S negativity’’ as
储
A
储
S
⫽ inf
兵
a
⫹
⫹ a
⫺
兩
A⫽ a
⫹
⫹
⫺ a
⫺
⫺
,a
⫾
⭓0,
⫾
苸 S
其
,
共18兲
N
S
共
A
兲
⫽ inf
兵
a
⫺
兩
A⫽ a
⫹
⫹
⫺ a
⫺
⫺
,a
⫾
⭓0,
⫾
苸 S
其
.
共19兲
Note that once again, if A has trace 1 we have that
储
A
储
S
⫽ 1⫹ 2N
S
(A). Then norm and convexity properties of N
S
and
储
•
储
S
follow exactly as before.
Taking S as the set of all density matrices, we get
储
A
储
S
⫽
储
A
储
1
, for all Hermitian A, and a totally uninteresting en-
tanglement quantity, as N
S
(
) vanishes for all density matri-
ces. The negativity of the preceding section corresponds to
the choice of S equal to the set of all matrices A such that
A⫽ A
†
,trA⫽ 1, and A
T
A
⭓0 关additionally, we have replaced
A
T
A
with A in the lhs of Eq. 共10兲 A, so that we can write
N(
) instead of N(
T
A
)兴.
We could have also taken S as the subset of density ma-
trices with positive partial transpose,
⫾
⭓0 and
⫾ T
A
⭓0. In
this case S corresponds to all states such that its partial trans-
pose is also a state. The resulting quantity we will denote by
N
PPT
. Even more restrictively, if we take for S the set of
separable density operators, i.e., we take
⫾
共and therefore
also
⫾ T
A
) in Eqs. 共18兲 and 共19兲 to be separable, the corre-
sponding quantity N
SS
amounts to the robustness of the en-
tanglement, originally introduced in 关18兴共see also 关25兴兲 as
the minimal amount of separable noise needed to destroy the
entanglement of
. From the inclusions between the respec-
tive sets S we immediately get the inequalities
N
SS
共
兲
⭓N
PPT
共
兲
⭓N
共
兲
⭓0. 共20兲
In general, all these inequalities are strict. For example,
N
SS
(
) vanishes only on separable states 共SS兲, whereas
N
PPT
(
) and N(
) vanish for all PPT states.
We claim that also N
SS
and N
PPT
are entanglement mono-
tones. The proof is quite simple. An analysis of the argu-
ments given in the preceding section shows that we really
used only one property of S, namely, for all operations M
i
appearing in a LOCC protocol, we have M
i
(
)苸 S
, when-
ever
苸 S
, where S
notes the cone generated by S
共equivalently the set of
with ⭓0,
苸 S). But this is ob-
vious for both separable states and PPT states.
III. UPPER BOUND TO TELEPORTATION CAPACITY
Sections III and IV are devoted to discuss applications of
the previous results. More specifically, we derive bounds to
some properties characterizing the entanglement both of a
single copy of a mixed state
共this section兲 and of asymp-
totically many copies of it 共following section兲.
For a single copy of a bipartite state
acting on C
d
1
丢 C
d
2
, where we set d
1
⫽ d
2
⬅m for simplicity, an important
question in quantum-information theory is to what extent this
state can be used to implement some given tasks requiring
entanglement, such as teleportation. The best approximation
P
opt
(
) to a maximally entangled state
兩
⌽
⫹
典
⬅
1
冑
m
兺
␣
⫽ 1
m
兩
␣
A
丢
␣
B
典
共21兲
that can be obtained from
by means of LOCC is then
interesting, because it determines, for instance, how useful
the state
is to approximately teleport log
2
m qubits of infor-
mation. In this section we will show that the negativity N(
)
provides us with an explicit lower bound on how close
can
be taken, by means of LOCC, to the state ⌽
⫹
. From here a
lower bound on the teleportation distance 共i.e., an upper
bound on how good teleportation results from
) will also
follow.
A. Singlet distance
In order to characterize the optimal state P
opt
(
) achiev-
able from
by means of LOCC, we need to quantify its
closeness to the maximally entangled state P
⫹
⬅
兩
⌽
⫹
典具
⌽
⫹
兩
.
Let
1
and
2
be two density matrices. The trace norm of
1
⫺
2
, 共or absolute distance 关26兴兲, is a measure of the de-
gree of distinguishability of
1
and
2
, and it is, therefore,
reasonable to use it to measure how much P(
)—the state
resulting from applying a local protocol P to state
—resembles P
⫹
. In what follows we will prove that the
negativity is a lower bound to the singlet distance of
,
⌬
共
P
⫹
,
兲
⬅inf
P
兩兩
P
⫹
⫺ P
共
兲
兩兩
1
, 共22兲
where the infimum is taken over local protocols P.
We start by recalling that the absolute distance
D(
1
,
2
)⬅
兩兩
1
⫺
2
兩兩
1
is a convex function 关26兴
兺
i
p
i
D
共
,
i
兲
⭓D
冉
,
兺
i
p
i
i
冊
, 共23兲
which confirms, as already assumed, that the optimal ap-
proximation P(
)toP
⫹
can always be chosen to be a single
state—as opposed to a distribution of states
兵
p
i
,
i
其
corre-
sponding to the output of a probabilistic transformation.
Therefore, in Eq. 共22兲 we need only consider deterministic
protocols P based on LOCC.
A second feature of the absolute distance that we need is
that
D
共
W
1
W
†
,W
2
W
†
兲
⫽ D
共
1
,
2
兲
, 共24兲
for any unitary transformation W. Properties 共23兲 and 共24兲
together imply that the best approximation to the maximally
entangled state P
⫹
can always be ‘‘twirled’’ without losing
optimality. Consider the state
G. VIDAL AND R. F. WERNER PHYSICAL REVIEW A 65 032314
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冕
dU U丢 U
*
P
opt
共
兲
U
†
丢 U
†
*
, 共25兲
which the parties can locally obtain from P
opt
(
) by Alice
applying an arbitrary unitary U, by Bob applying U
*
, and
then by deleting the classical information concerning which
unitary has been applied. It follows from the invariance of
P
⫹
under U 丢 U
*
and from property 共24兲 that D„U
丢 U
*
P
opt
(
)U
†
丢 U
†
*
,P
⫹
…⫽ D„P
opt
(
),P
⫹
… for any U.
Then property 共23兲 implies that the mixture in Eq. 共25兲 is not
further away from P
⫹
than P
opt
(
). But P
opt
(
) was al-
ready minimizing Eq. 共22兲, and therefore state 共25兲 must also
be optimal.
We can then assume that P
opt
(
) has already undergone a
twirling operation. This means that it is a noisy singlet 关27兴
p
⫽ pP
⫹
⫹
共
1⫺ p
兲
I丢 I
m
2
, 共26兲
from which the absolute distance to P
⫹
can be easily com-
puted, D(P
⫹
,
p
)⫽ 2(1⫺ p)(m
2
⫺ 1)/m
2
. Similarly, the
trace norm of
p
T
A
reads
兩兩
p
T
A
兩兩
1
⫽ mp⫹(1⫺ p)/m, and there-
fore
D
共
P
⫹
,
p
兲
⫽ 2
冉
1⫺
兩兩
p
T
A
兩兩
1
m
冊
. 共27兲
The lower bound to the singlet distance 共22兲 follows now
straightforwardly from the monotonicity of
兩兩
T
A
兩兩
1
关or
N(
)兴 under LOCC, that is,
兩兩
T
A
兩兩
1
⭓
兩兩
P
opt
(
)
T
A
兩兩
1
, and
reads
⌬
共
P
⫹
,
兲
⭓2
冉
1⫺
兩兩
T
A
兩兩
1
m
冊
. 共28兲
Therefore, we have proved the following bound for the sin-
glet distance.
Proposition 4.
⌬
共
P
⫹
,
兲
⭓2
冉
1⫺
1⫹ 2N
共
兲
m
冊
. 共29兲
B. Teleportation distance
A quantum state
shared by Alice and Bob can be used
as a teleportation channel ⌳关15兴. That is, given the shared
state
and a classical channel between the parties, Alice can
transmit an arbitrary 共unknown兲 state
苸 C
m
to Bob with
some degree of approximation. Let ⌳
T,
(
) be the state that
Bob obtains when Alice sends
using
and some protocol
T involving LOCC only. The teleportation distance
d
共
⌳
兲
⬅
冕
d
D„
,⌳
共
兲
…, 共30兲
where D„
,⌳(
)…⬅
兩兩兩
典具
兩
⫺ ⌳(
)
兩兩
1
, can be used to
quantify the degree of performance of the channel. The mea-
sure d
is consistent with the Haar measure dU in SU(m),
and thus d(⌳) is invariant under the twirling of the channel,
that is the application of an arbitrary unitary U to
previous
to the teleportation, followed by the application of U
†
after
the teleportation scheme. Indeed,
d
共
⌳
兲
⫽
冕
dW D„WP
0
W
†
,⌳
共
WP
0
W
†
兲
…, 共31兲
for some reference state P
0
⬅
兩
0
典具
0
兩
, and using property
共24兲 of the trace norm, Eq. 共31兲 is also equal to
冕
dW d„WP
0
W
†
,U
†
⌳
共
UWP
0
W
†
U
†
兲
U…. 共32兲
We can now average over U to obtain
d
共
⌳
兲
⫽
冕
dU
冕
d
D„
,U
†
⌳
共
U
兲
U…, 共33兲
where the right side of the equation corresponds to the tele-
portation distance of the twirled channel.
We next adapt a reasoning of the Horodecki 关27兴 to our
present situation. It uses an isomorphism between states
⌳
and channels ⌳ due to Jamiołkowski 关28兴 and first exploited
by Bennett et al. 关3兴. Let us ascribe the channel ⌳ to the state
⌳
⫽ (I丢 ⌳)P
⫹
. The state
⌳
can be produced by sending
Bob’s part of the bipartite system in state P
⫹
down the chan-
nel ⌳. Conversely, the standard teleportation protocol 关15兴
共or a slight and obvious modification of it兲 applied to state
⌳
reproduces the channel ⌳ with probability 1/m
2
. How-
ever, if the state
⌳
is a noisy singlet
p
, then the corre-
sponding channel is the depolarizing channel
⌳
p
dep
共
%
兲
⫽ p% ⫹
共
1⫺ p
兲
I
m
, 共34兲
which the standard teleportation scheme reproduces with
certainty using state
p
. For this case d(⌳
p
dep
)⫽ 2(1
⫺ p)(m⫺ 1)/m. Therefore, there is a complete physical
equivalence between noisy singlets and depolarizing telepor-
tation channels. In addition,
d
共
⌳
p
dep
兲
⫽
m
m⫹ 1
D
共
P
⫹
,
p
兲
. 共35兲
Now, since both quantities d and D are invariant under twirl-
ing, and any channel 共state兲 can be taken into the depolariz-
ing 共noisy singlet兲 form, this equality holds for any channel
⌳ and state
⌳
.
Lemma 5. 共adapted from 关27兴兲. The minimal distance
d
min
(
) that can be achieved when using the bipartite state
to construct an arbitrary teleportation channel is given by
d
min
共
兲
⫽
m
m⫹ 1
⌬
共
P
⫹
,
兲
. 共36兲
Proof. d
min
(
)⭐m⌬(P
⫹
,
)/(m⫹ 1), because a possible
way to use
as a teleportation channel is by using a twirled
version of an optimal state P(
) and the standard teleporta-
tion scheme, which produces a depolarizing teleportation
channel with d⫽ mD„P
⫹
,P(
)…/(m⫹ 1) 关recall Eq. 共35兲兴.
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