PHYSICAL REVIEW A 75, 012305 共2007兲
Class of positive-partial-transpose bound entangled states associated with almost any set of pure
entangled states
Marco Piani* and Caterina E. Mora†
Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80–952 Gdańsk, Poland
Institut für Quantenoptik und Quanteninformation, Österreichischen Akademie der Wissenschaften, Innsbruck, Austria
共Received 25 August 2006; revised manuscript received 20 October 2006; published 4 January 2007兲
We analyze a class of entangled states for bipartite d 丢 d systems, with d nonprime. The entanglement of
such states is revealed by the construction of canonically associated entanglement witnesses. The structure of
the states is very simple and similar to the one of isotropic states: they are a mixture of a separable and a pure
entangled state whose supports are orthogonal. Despite such a simple structure, in an opportune interval of the
mixing parameter their entanglement is not revealed by partial transposition. Moreover, for a restricted set of
such states, we prove that there exists an interval of the mixing parameter such that both partial transposition
and realignment 共i.e., all permutational criteria in the bipartite setting兲 fail to detect them as entangled. In the
range in which the states are positive under partial transposition 共PPT兲, they are not distillable; on the other
hand, the states in the considered class are provably distillable as soon as they are nonpositive under partial
transposition. The states are associated to any set of more than two pure states. The analysis is extended to the
multipartite setting. By an opportune selection of the set of multipartite pure states, it is possible to construct
mixed states which are PPT with respect to any choice of bipartite cuts and nevertheless exhibit genuine
multipartite entanglement. Finally, we show that every k-positive but not completely positive map is associated
to a family of nondecomposable maps.
DOI: 10.1103/PhysRevA.75.012305
PACS number共s兲: 03.67.Mn, 03.65.Ud, 02.10.Ud
I. INTRODUCTION
Entanglement is a resource required in many tasks typical
of the fields of quantum information and quantum computation 关1,2兴, like quantum teleportation 关3兴 and superdense
coding 关4兴. Although there is a clear definition of what an
entangled state is 关5兴, it is, in general, difficult to determine
whether a given state is entangled or not. Correspondingly,
the structure of the space of states, as classified with respect
to the entanglement property, is still a central issue of investigation. Moreover, in the multipartite case the picture is
even more complicated, since it appears that there are qualitatively different kinds of entanglement 关6兴.
We remark that the study of very specific and simply parametrized classes of states, typically satisfying some symmetry 共such as Werner states 关5兴 or isotropic states 关7兴兲, has
always turned out to be very useful to improve our understanding of the entanglement phenomenon and of the geometry and properties of the set of states. In this paper, we
provide examples of states that, despite a simple structure,
exhibit interesting properties both in the bipartite and multipartite setting.
One of the means to investigate the entanglement of states
is based on the use of linear maps 关8,9兴, which are positive
共P兲 but not completely positive 共CP兲: we shall refer to them
as PnCP maps. A map is P if it transforms any state into
another positive operator. It is moreover CP if also its partial
action on a subsystem of any larger system gives rise to a P
map. In the case of a bipartite system, a state is entangled if
and only if there exists a PnCP map such that the operator
*Electronic address: piani@ts.infn.it
†
Electronic address: caterina.mora@uibk.ac.at
1050-2947/2007/75共1兲/012305共11兲
obtained acting with the map on only one of the two subsystems is not positive any more. The simplest example of
PnCP map is the operation of transposition T 共with respect to
a given basis兲. The action of transposition on one of the
subsystems is called partial transposition 共PT兲 and is also
known as the Peres-Horodecki criterion 关8,9兴. In the bipartite
共2 丢 2兲- and 共2 丢 3兲-dimensional cases, PT can “detect” all
entangled states: only states that develop negative eigenvalues under PT 关i.e., nonpositive under partial transposition
共NPT兲 states兴 are entangled. In higher dimensions, there are
states that are positive under partial transposition 共PPT兲 even
if entangled. The latter states have the interesting property
that their entanglement cannot be distilled 共see 关2兴, for a
review兲; thus, it is considered to be “bound.”
Beside partial transposition, there is another easily computable entanglement criterion, realignment 关10,11兴. Both PT
and realignment are part of the larger family of permutational criteria 关12,13兴 and constitute the only two independent criteria of such type in the bipartite scenario. It must be
remarked that realignment: 共i兲 is not related to a positive
linear map and 共ii兲 can detect some PPT bound entangled
states.
If we want to use linear PnCP maps to detect PPT bound
entangled states, it is necessary to use PnCP maps that are
not decomposable, i.e., that cannot be written as the sum of a
CP map and a CP map composed with transposition. Indeed,
the study of P maps is strictly related to the study of entanglement, the link being provided by the ChoiJamiołkowsky isomorphism 关14,15兴.
It was proved that every entangled state is useful for tasks
that it would be impossible to perform classically 关16兴; in
this sense bound entanglement can be “activated” 关17兴. Quite
interestingly, it was found that PPT bound entangled states
provide probabilistic interconvertibility among multipartite
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©2007 The American Physical Society
PHYSICAL REVIEW A 75, 012305 共2007兲
MARCO PIANI AND CATERINA E. MORA
pure states, which are not interconvertible by local operations and classical communication 共LOCC兲 alone 关18,19兴.
The first explicit examples of PPT entangled states were
given in 关20兴, and, since then, many other examples have
been found 关21–25兴. It has been shown 关26兴 that most of
these states are part of a same family of PPT—but not a
priori entangled—states. A first systematic method to construct PPT bound entangled states was proposed in 关27兴 and
is based on the concept of unextendible product basis.
In the present work, we first consider a class of bipartite
d 丢 d states, with d a nonprime dimension, which are described by a given set of pure states and a mixing parameter
共a probability兲. Given two states ␺共1兲 and ␺共2兲, in a d1 丢 d1
and d2 丢 d2 Hilbert space, respectively, we consider the set of
mixed states parametrized by the mixing parameter p
␳ p共␺共1兲, ␺共2兲兲 =
1−p
共d21
− 1兲共d22 − 1兲
共1 − P␺共1兲兲 丢 共1 − P␺共2兲兲
+ pP␺共1兲 丢 P␺共2兲 ,
of the class of states relating it to the choice of a set of M
ⱖ 2 pure states. Section VII is devoted to some considerations regarding the construction of the canonical witnesses.
In Sec. VII, the multipartite setting is studied. In Sec. IX,
starting from considering tensorlike witnesses, we provide a
general theorem relating the properties of k-positivity and
nondecomposability.
II. DEFINITIONS AND BASIC NOTIONS
A d-dimensional system is associated to the Hilbert space
Cd, and operators on such system are described by the algebra of d ⫻ d matrices with complex entries M d. A state ␳
corresponds to a normalized 关Tr共␳兲 = 1兴 positive semidefinite
共␳ ⱖ 0兲 matrix. We will denote 共normalized兲 vectors in the
Hilbert space by 兩␺典 or ␺, and the projector onto the pure
state ␺ by P␺ = 兩␺典具␺兩.
共1兲
where P␺共i兲 is the projector onto state ␺共i兲. We will see that,
for almost any choice of ␺共1兲 and ␺共2兲 entangled, there exists
a p⌫ ⬎ 0 such that ␳ p is PPT entangled for all choices of p
⬍ p⌫. The structure of these states can be considered very
simple in comparison to the PPT bound entangled states already known in literature. The class is a generalization of
states that already appeared in 关18,19兴, where it was proved
that some states in the class, even though PPT, are entangled
because they allow operations that are impossible by LOCC.
In our case, we prove that they are entangled by constructing
canonically associated entanglement witnesses. Moreover,
we extend our analysis to the realignment criterion, showing
that it is inequivalent to partial tranposition 共in the sense that
there are states of the form 共1兲 detected by one of the two
criteria but not by the other兲, and that there are states in the
class not revealed by either criteria. This class of states is
naturally rich. Furthermore, it can be verified by direct inspection that it is not contained in the class described in 关26兴,
thus it contributes effectively to the variety of the known
PPT 共entangled兲 states.
We also study the multipartite setting, to which the family
of states can be naturally extended 关18,19兴. It is possible to
show 关18兴 that these states can be PPT entangled with respect
to every bipartite cut. As in the bipartite case, this is proved
by associating to each state a canonical witness. Furthermore, we find conditions for which the states contain genuine multipartite entanglement and show that it is possible to
have a genuinely multipartite entangled state, which is PPT
with respect to any bipartite cut. Finally, we relate the properties of k-positivity and nondecomposability of linear maps
共to be defined in the following兲, and show that even a decomposable map can become useful to detect PPT entangled
states just by considering its trivial extensions.
The paper is organized as follows. In Sec. II, we provide
definitions and basic notions. In Sec. III, we introduce the
basic set of states of interest, involving the choice of two
pure states, and in Sec. IV, we associate to them canonical
witnesses. In Sec. V, we discuss partial transposition and
realignment, and in Sec. VI, we generalize the construction
A. Entanglement and separable states
A bipartite system AB is associated to a tensor-product
Hilbert space HAB = HA 丢 HB. A pure bipartite state ␺AB is
entangled if it is not factorized, i.e., not of the form ␺AB
= ␺A 丢 ␺B. A bipartite mixed state ␳AB is separable if it can be
written as a convex combination of factorized states
␳AB = 兺 pi P␺ i
i
A
丢
P␺ i ,
B
pi ⱖ 0,
兺i pi = 1,
共2兲
otherwise it is entangled.
More in general, one can consider N-partite systems,
which are associated to tensor-product Hilbert spaces of the
N
Hi, where Hi is the Hilbert space associated to
form 丢 i=1
system i. In this case, it is possible to study the separability
issue with respect to different groupings of the parties. A
pure N-partite state ␺N is k-separable if it can be written as a
k
␺Si, with Pk
tensor product of k states, i.e., as ␺N = 丢 i=1
k
= 兵Si其i=1 a partition of the parties in k subsets. In particular,
␺N is biseparable if ␺N = ␺S1 丢 ␺S2. A pure state is k-partite
entangled if it cannot be written as the tensor product of
states, each of which pertains to less than k parties. Similarly,
a mixed state is k-separable if it can be written as a convex
combination of k-separable pure states. The k-partition need
not be the same for all the k-separable pure states entering in
the convex combination; if all the pure states can be chosen
to be k-separable with respect to the same partition Pk, we
say that the state is k-separable with respect to the partition
Pk. In particular, we say that a state is biseparable if it is
2-separable, and that it is separable along a cut S1 : S2 if it is
2-separable with respect to the partition 兵S1 , S2其. A mixed
state is k-partite entangled if every possible convex decomposition of the state contains at least a k-partite entangled
pure state. Note that a N-partite state is biseparable if and
only if it is not N-partite entangled. Any result valid in the
bipartite setting can be applied to the multipartite case when
considering a given cut.
In the bipartite case, any pure state ␺ can be written in its
standard Schmidt decomposition
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CLASS OF POSITIVE-PARTIAL-TRANSPOSE BOUND…
r
兩␺典 = 兺 ␮i兩iA 丢 iB典,
i=1
r
␮2i = 1, are the Schmidt coefficients, r
where ␮i ⬎ 0, 兺i=1
ⱕ min共dA , dB兲 is the Schmidt rank 共or number兲 and 兩iA共B兲典 are
orthogonal states 共i.e., they can be extended to an orthonormal basis兲. We say that a bipartite density matrix ␴ has
Schmidt number k if 共i兲 for any decomposition 兵pi ⱖ 0 , ␾i其 of
␴, i.e., ␴ = 兺i pi 兩 ␾i典具␾i兩, at least one of the vectors ␾i has at
least Schmidt rank k and 共ii兲 there exists a decomposition of
␴ with all vectors 兵␾i其 of Schmidt rank at most k 关28兴.
B. Partial transposition and realignment
We recall now the two separability criteria that we will
use in the following and that are based on the reordering of
the entries of the density matrix: partial transposition 关8,9兴
and realignment 关10,11兴. Given a bipartite density matrix ␳
= 兺ijkl␳ij,kl 兩 ij典具kl兩 the linear operations of partial transposition
and realignment are defined as follows. Partial transposition
共with respect to the first system兲 corresponds to the reordering 共兩ij典具kl 兩 兲⌫A = 兩kj典具il兩, and realignment to R共兩ij典具kl 兩 兲
= 兩ik典具jl兩. It is immediate to see that, if a state is separable,
⌫A
储1 ⱕ 1 and 储R共␳AB兲储1 ⱕ 1 must hold, with 储X储1
then both 储␳AB
⌫A
†
= Tr冑X X the trace norm of X. The condition 储␳AB
储1 ⱕ 1 is
equivalent to requiring that ␳AB stays positive under partial
⌫A
transposition, i.e., ␳AB
ⱖ 0.
With regard to partial transposition, we note that for any
bipartite state 兩␺典 = 兺 j␮ j 兩 ii典 共here written in its Schmidt decomposition兲, we have
detect PPT entangled states a witness must be nondecomposable. Indeed, Tr共W␳兲 ⱖ 0 for all PPT state ␳ and all decomposable witnesses W.
In 关28,31兴, the concept of Schmidt-number witness was
introduced. A 共nontrivial兲 Schmidt-number k witness W is an
observable, which is positive semidefinite with respect to
共mixed兲 states of Schmidt number k − 1, but such that there
exists a Schmidt-number k state ␳ such that Tr共W␳兲 ⬍ 0.
Moreover, witnesses are able to distinguish between different kinds of multipartite entanglement 关32兴. Indeed, there
always exists an observable whose expectation value is able
to discriminate between states in a convex subset and a state
outside it. Therefore, for any state ␳ that is 共k + 1兲-partite
entangled there exists a witness W such that Tr共W␳兲 ⬍ 0,
whereas Tr共W␴兲 ⱖ 0 for all states ␴ that are at most k-partite
entangled. Similarly, there is always a witness that distinguishes a state that is not k-separable from states that are. In
particular, for an N-partite state that is N-partite entangled,
there exists a witness that tells it from biseparable states.
D. Maps and entanglement
A linear map ⌳ : M d → M d⬘ is positive if ⌳关X兴 ⱖ 0 for all
X ⱖ 0; k-positive if idk 丢 ⌳ is positive, with idk the identity
map on M k; completely positive if it is k-positive for all k
ⱖ 1. It is remarkable that ⌳ : M d → M d⬘ is completely positive
if and only if it is d-positive 关14兴.
Operators W in M dd⬘ ⬵ M d 丢 M d⬘ are isomorphic to linear
maps ⌳ : M d → M d⬘, through the Choi-Jamiołkowski isomorphism 关14,15兴
共兩␺典具␺兩兲⌫ = 兺 ␮2j 兩jj典具jj兩 + 兺 ␮i␮ j共兩␺+ij典具␺+ij兩 − 兩␺−ij典具␺−ij兩兲,
j
j⬎i
共3兲
with 兩␺±ij典 = 共兩ij典 ± 兩ji典兲 / 冑2, and where partial transposition
was operated in the Schmidt basis. The eigenvalues of 共兩␺典
⫻具␺ 兩 兲⌫ are ␭i0 = ␮2i , for i = 1 , . . . , d, and ␭±ij = ± ␮i␮ j, for j ⬎ i,
corresponding to Schmidt-rank-one eigenstates 兩ii典 and
Schmidt-rank-two eigenstates 兩␺±ij典, respectively. Thus, either
␺ is factorized, i.e., there is only one nonvanishing Schmidt
coefficient 共=1兲, or all the eigenvalues of 共兩␺典具␺ 兩 兲⌫ have
modulus strictly ⬍1. With regard to realignment, we have
R共兩␺典具␺ 兩 兲 = 兺ij␮i␮ j 兩 i典具i 兩 丢 兩j典具j兩. For any pure state ␺, thus,
储共兩␺典具␺ 兩 兲⌫A储1 = 储R共兩␺典具␺ 兩 兲储1 = 共兺i␮i兲2. Therefore, both partial
transposition and realignment detect all pure entangled 共bipartite兲 states.
C. Entanglement witnesses
It is well known that any bipartite entangled state ␳AB can
be detected by means of a suitable entanglement witness
关9,29兴: for every entangled state ␳AB there exists an observable W = WAB such that Tr共W␳AB兲 ⬍ 0, while Tr共W␴sep兲 ⱖ 0
for all separable states ␴sep. It is clear that a nontrivial entanglement witness, i.e., an observable able to detect at least
some entangled state, is not positive semidefinite.
A witness is decomposable 关30兴 if it can be written as
W = P + Q⌫, with P , Q ⱖ 0 positive semidefinite operators. To
W = W⌳ = d共idd 丢 ⌳兲关P+d 兴
共4兲
⌳关X兴 = ⌳W关X兴 = Tr1关共XT 丢 1兲W兴,
共5兲
where the trace in 共5兲 is on the first subsystem only, and
P+d ⬅ P⌿+,
d
兩⌿+d 典 =
1
冑d 兺i 兩i 丢 i典
共6兲
is the maximally entangled state for a d 丢 d system, d ⱖ 2.
In particular, 共nontrivial兲 witnesses are isomorphic to
PnCP maps. An example of PnCP map is transposition, that
fails already to be 2-positive, and is associated to V = d共idd
+
丢 T兲关Pd 兴, that is the swap operator: V 兩 ␾ 丢 ␹典 = 兩␹ 丢 ␾典. In the
same way as there is always an entanglement witness that
detects a bipartite entangled states ␳AB, there is also a PnCP
map ⌳ such that
共idA 丢 ⌳B兲关␳AB兴 ⱖ 0
共7兲
is not satisfied 关9兴.
Every nondecomposable witness is associated to a nondecomposable map 关30兴. A map ⌳ is decomposable if it can be
CP
CP
written as ⌳ = ⌳CP
1 + ⌳2 ⴰ T, where ⌳1共2兲 is a completely positive map and ⴰ stands for composition. Indeed, Eq. 共7兲 is
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PHYSICAL REVIEW A 75, 012305 共2007兲
MARCO PIANI AND CATERINA E. MORA
satisfied for all PPT states and decomposable maps. Moreover, every 共nontrivial兲 Schmidt-number k witness is associated to a 共k − 1兲-positive but not k-positive map 关28,31兴.
d 22
..
.
III. BASIC SET OF STATES
3
We start by considering a bipartite system with associated
Hilbert space HAB = HA 丢 HB, with
H A = H A1 丢 H A2,
H B = H B1 丢 H B2
2
共8兲
1
and HAi = HBi = Cdi.
We focus on states
共1兲
j
共2兲
共1兲
共2兲
␳AB共␺ , ␺ 兲 = ␳1共␺ 兲 丢 ␳2共␺ 兲,
共9兲
␳i共␺共i兲兲 = ␳AiBi = Ni共1 − P␺共i兲兲AiBi ,
共10兲
with
where Ni = 1 / 共d2i − 1兲 are normalization factors. Each pure
state 兩␺共i兲典 ⬅ 兩␺共i兲典AiBi is given by
ri
兩␺共i兲典 = 兺 ␮共i兲
j 兩j Ai 丢 j Bi典,
1
2
FIG. 1. Graphical representation of the choice of pij 共white and
black兲 and wij 共patterns兲 in 共1兲 and 共15兲, respectively. White corresponds to the separable part of ␳ p共␺共1兲 , ␺共2兲兲, whereas the vertical
and horizontal patterns correspond to the positive part of the witness W⑀共␺共1兲 , ␺共2兲兲. Black and the diagonal pattern stand for p11
= p ⬎ 0 and w11 = −⑀ ⬍ 0, respectively 共see the main text for details兲.
共11兲
␳⌫i A共␺共i兲兲 = Ni„1 − 共兩␺共i兲典具␺共i兲兩兲⌫A….
共12兲
From what we have seen about partial transposition of pure
states in Sec. II B, it is clear that ␳⌫i A共␺共i兲兲 has full rank if and
only if ␺共i兲 is entangled. It follows that, if both ␺共1兲 and ␺共2兲
⌫A
are entangled, ␳AB
共␺共1兲 , ␺共2兲兲 is strictly positive. This implies
that a change of ␳AB共␺共1兲 , ␺共2兲兲 small enough cannot spoil the
positivity of the partial transpose.
This leads us to define the class of states of interest consisting of the convex combination 共1兲. Such a class has already appeared in literature 关18,19兴, and it was proved that
some states in the class are entangled even if PPT, by showing that they allow to perform tasks that are impossible under
LOCC. Here, in a different vein, we will look for entanglement witnesses to prove that, for almost all choices of entangled states ␺共1兲 and ␺共2兲, the state ␳ p共␺共1兲 , ␺共2兲兲 defined in
共1兲 is entangled as soon as p ⬎ 0. Therefore, if p is chosen to
be small enough, ␳ p共␺共1兲 , ␺共2兲兲 is a PPT entangled state. We
remark that if one of the two states ␺共i兲 is separable, while
the other is entangled, then ␳ p共␺共1兲 , ␺共2兲兲 is always NPT, for
p ⬎ 0; ␳ p共␺共1兲 , ␺共2兲兲 can be made PPT entangled, for some
choice of p ⬎ 0, only if both pure states ␺共i兲 are entangled.
IV. CANONICAL WITNESS
We now construct a suitable entanglement witness. If we
2
共i兲
共i兲
共i兲
di
complete ␺共i兲 to a basis 兵␺共i兲
1 ⬅ ␺ , ␺2 , . . . , ␺d2 其 of C , for
i
both i = 1 , 2, we see that ␳ p共␺共1兲 , ␺共2兲兲 is diagonal in the basis
共2兲
兵␺共1兲
i 丢 ␺ j 其:
d 12
i
d21 d22
j=1
where 共11兲 and ri are the corresponding Schmidt decomposition and Schmidt number, respectively. The states ␳i共␺共i兲兲
are Ai : Bi separable 关33兴, thus ␳AB共␺共1兲 , ␺共2兲兲 is A : B separable. The partial transposition of ␳i共␺共i兲兲 with respect to Ai is
...
3
␳ p共␺共1兲, ␺共2兲兲 = 兺 兺 pij P␺共1兲 丢 P␺共2兲
i
i=1 j=1
j
共13兲
with pij ⱖ 0. Note that, since the tensor product structure is
along the A1B1 : A2B2 cut, and not along the A1A2 : B1B2 cut,
the expression 共13兲 is not related to the A : B separability
property given by 共2兲. We will consider witnesses diagonal in
the same basis, i.e.,
d21 d22
W = 兺 兺 wij P␺共1兲 丢 P␺共2兲 ,
i=1 j=1
i
j
共14兲
so that Tr(W␳ p共␺共1兲 , ␺共2兲兲) = 兺ij pijwij. Of course, the operator
共14兲 is not a trivial witness only if wij ⬍ 0 for some 共i , j兲. In
particular, if we define
W⑀共␺共1兲, ␺共2兲兲 = P␺共1兲 丢 共1 − P␺共2兲兲 + 共1 − P␺共1兲兲 丢 P␺共2兲
− ⑀ P␺共1兲 丢 P␺共2兲
= P␺共1兲 丢 1 + 1 丢 P␺共2兲 − 共2 + ⑀兲P␺共1兲 丢 P␺共2兲 ,
共15兲
with ⑀ ⱖ 0, we have
Tr关W⑀共␺共1兲, ␺共2兲兲␳ p共␺共1兲, ␺共2兲兲兴 = − p⑀ .
共16兲
Indeed, ␳ p共␺共1兲 , ␺共2兲兲 and W⑀共␺共1兲 , ␺共2兲兲 are diagonal in the
same basis and their supports are orthogonal except for the
unidimensional subspace spanned by ␺共1兲 丢 ␺共2兲. A graphical
representation of both the state and the witness decompositions 共in terms of pij and wij兲 for the choices 共1兲 and 共15兲 is
given in Fig. 1.
We have to prove that, at least for some choices of ␺共i兲,
there exists ⑀ ⬎ 0 such that W⑀共␺共1兲 , ␺共2兲兲 is a non-trivial entanglement witness. Indeed, as soon as ⑀ ⬎ 0, W⑀共␺共1兲 , ␺共2兲兲 is
012305-4
PHYSICAL REVIEW A 75, 012305 共2007兲
CLASS OF POSITIVE-PARTIAL-TRANSPOSE BOUND…
not a positive semidefinite operator. We proceed by finding
the conditions for which it is positive on separable states:
具␣A 丢 ␤B 兩 W⑀共␺共1兲 , ␺共2兲兲 兩 ␣A 丢 ␤B典 ⱖ 0, for all factorized 共not
necessarily normalized兲 兩␣A 丢 ␤B典. Let us therefore consider
vectors
d1 d2
兩␣典 = 兺 兺 ␣ij兩iA1 丢 jA2典
i=1 j=1
d1 d2
兩␤典 = 兺 兺 ␤ij兩iB1 丢 jB2典,
␣␮共2兲␤T = ␮共1兲 = G共1兲
1 ,
where c is a complex constant of proportionality. Only in this
case, in fact, ␣␮共2兲␤T and ␤T␮共1兲␣ are orthogonal to all the
other elements of the two matrix ONB. If condition 共19兲 is
satisfied, the first two terms on the right-hand side of 共17兲
must be equal. Thus, one finds 兩c 兩 = 1, and, finally, taking into
account Hermiticity and positivity of ␮共i兲, one obtains c = 1.
We have reduced the problem of determining the existence of a nontrivial witness of the form 共15兲 to that of verifying whether, for given states ␺共1兲 and ␺共2兲, there exist matrices ␣ and ␤, which solve the system of matrix equations
i=1 j=1
where ␣ = 关␣ij兴, ␤ = 关␤ij兴 are complex d1 ⫻ d2 rectangular matrices, and where we have taken the bases 兵兩iAk典其, 兵兩iBk典其 in the
Hilbert spaces HAk, HBk, k = 1 , 2 to be the ones corresponding
to the Schmidt decomposition 共11兲 of ␺共1兲 and ␺共2兲. We find
具␣A 丢 ␤B兩W⑀共␺共1兲, ␺共2兲兲兩␣A 丢 ␤B典 = Tr关共␤T␮共1兲␣兲†共␤T␮共1兲␣兲兴
+ Tr关共␣␮共2兲␤T兲†共␣␮共2兲␤T兲兴 − 共2 + ⑀兲兩Tr共␣␮共2兲␤T␮共1兲兲兩2 ,
共17兲
with ␮共i兲 = 共␮共i兲兲† = 共␮共i兲兲T the positive diagonal matrix of the
Schmidt coefficients of ␺共i兲.
Let us consider a matrix orthonormal basis 共ONB兲 in M d,
d2
such that the matrices are orthoi.e., a set of matrices 兵Fi其i=1
normal with respect to the Hilbert-Schmidt inner product:
Tr共F†i F j兲 = ␦ij. For any matrix ONB 兵Fi其 and any matrix X,
we have X = 兺iTr共F†i X兲Fi, and 兺i 兩 Tr共F†i X兲兩2 = Tr共X†X兲. As
Tr共␮共i兲2兲 = 1, each ␮共i兲 can be considered as an element of a
共i兲
to an
matrix ONB in M di. Let us complete each G共i兲
1 =␮
ONB
2
di
兵G共i兲
j 其 j=1.
and, similarly,
␮共2兲
i
␤ ji ,
␮共1兲
j
共21a兲
共21b兲
From 共21a兲, we have that if ␤ ji = 0 then also 共␣ 兲ij = 0; from
共2兲
共21b兲, we find that, if ␤ ji ⫽ 0, then ␮共1兲
j = ␮i and, therefore,
from 共21a兲, 共␣−1兲ij = ␤ ji. In conclusion, we have ␣−1 = ␤T.
Therefore, a solution to Eqs. 共20兲 exists only if ␮共1兲 and ␮共2兲
are connected by a similarity transformation
d22
2
兩Tr共␤ ␮ ␣␮ 兲兩 ⱕ 兺 兩Tr共␤T␮共1兲␣G共2兲
j 兲兩
共2兲 2
共20b兲
−1
= Tr关共␣␮共2兲␤T兲†共␣␮共2兲␤T兲兴, 共18a兲
共1兲
␤T␮共1兲␣ = ␮共2兲 .
共2兲 2
2
␤ ji共␮共1兲
j 兲 = ␤ ji共␮i 兲 .
j=1
T
共20a兲
共␣−1兲ij =
d21
2
兩Tr共␣␮ ␤ ␮ 兲兩 ⱕ 兺 兩Tr共␣␮共2兲␤TG共1兲
j 兲兩
共1兲 2
␣␮共2兲␤T = ␮共1兲
First, we note that this is possible only if ␮共2兲 and ␮共1兲 have
the same rank, i.e., only if the states ␺共1兲 and ␺共2兲 have the
same Schmidt number r = r1 = r2. It is sufficient to focus on
this case. We further observe that without loss of generality,
we can consider the r nonvanishing Schmidt coefficients of
␺共i兲 to appear in the first r diagonal entries of ␮共i兲, for i
= 1 , 2. Therefore, we can consider all the matrices entering
共20兲 to be r ⫻ r square matrices, even if the initial dimensions
d1 and d2 were different. Moreover, they are all invertible,
since we are considering the case the rank of both Schmidt
coefficient matrices ␮共i兲 is r. We can therefore rewrite 共20兲 as
␮共2兲␤T = ␣−1␮共1兲 and ␤T␮共1兲 = ␮共2兲␣−1. Taking into account
that both matrices ␮共i兲 are diagonal and strictly positive 共i.e.,
all the r Schmidt coefficients are not null兲, we arrive at the
following relations:
Then,
共2兲 T
␤T␮共1兲␣ = c␮共2兲 = cG共2兲
1 , 共19兲
␣␮共2兲␣−1 = ␮共1兲
j=1
= Tr关共␤T␮共1兲␣兲†共␤T␮共1兲␣兲兴. 共18b兲
Inequalities 共18兲 correspond to P␺共1兲 丢 P␺共2兲 ⱕ 1 丢 P␺共2兲 and
P␺共1兲 丢 P␺共2兲 ⱕ P␺共1兲 丢 1, respectively. Yet having cast them in
the form 共18兲 allows us to argue about the necessary and
sufficient conditions on ␺共1兲 and ␺共2兲 to have a nontrivial
witness W⑀共␺共1兲 , ␺共2兲兲, i.e., to have ⑀ ⬎ 0.
Positivity on factorized states imposes ⑀ = 0 if and only if
there are matrices ␣ and ␤ such that the inequalities 共18兲 are
both saturated at the same time, i.e., both sums in 共18兲 reduce
to just the first term, and this term does not vanish. Indeed,
under these conditions, 共17兲 is equal to −⑀ 兩 Tr共␣␮共2兲␤T␮共1兲兲兩2
and strictly negative as soon as ⑀ ⬎ 0. The two sums reduce
to the first term if and only if there are ␣ and ␤ such that 关34兴
共22兲
and have the same eigenvalues. In such case, we have that
共17兲 reduces to −⑀ Tr关共␮共i兲兲2兴 = −⑀, so that we must choose
⑀ = 0 to have positivity on separable states.
We have shown that the witness W⑀共␺共1兲 , ␺共2兲兲 defined in
共15兲 can always be chosen to be nontrivial, i.e., with ⑀ ⬎ 0,
except in the case where ␺共1兲 and ␺共2兲 have essentially the
same Schmidt decomposition. Note that without loss of generality, we can consider the Schmidt coefficients to be or共i兲
dered as ␮共i兲
k ⱖ ␮k+1, for i = 1 , 2. Thus, we have always a witness except in the case ␮共1兲 = ␮共2兲 关indeed, the similarity
transformation 共22兲 is actually a permutation兴. Correspondingly, we have proved that, for almost all pairs of pure entangled states ␺共i兲, i = 1 , 2, the state ␳ p共␺共1兲 , ␺共2兲兲 is entangled
as soon as p ⬎ 0.
012305-5
PHYSICAL REVIEW A 75, 012305 共2007兲
MARCO PIANI AND CATERINA E. MORA
V. PARTIAL TRANSPOSITION AND REALIGNMENT
p
Let us now consider more in detail the behavior of the
class of states ␳ p共␺共1兲 , ␺共2兲兲 under the operations of partial
transposition and realignment 关35兴. Partial transposition for
such states has already been studied in 关18兴. For completeness, we reproduce here those results and extend the analysis
by comparing the entanglement detection power of partial
transposition and realignment. Moreover, we observe that no
element in the class is a candidate to be an NPT bound state,
i.e., as soon as the states are NPT, they are provably distillable.
With regard to PT, we have that the eigenvalues of
␳⌫p A共␺共1兲 , ␺共2兲兲 are 共1 − p兲N1N2共1 − ␭共1兲兲共1 − ␭共2兲兲 + p␭共1兲␭共2兲,
where the ␭共i兲s run over eigenvalues of 共兩␺共i兲典具␺共i兲 兩 兲⌫, i
= 1 , 2. Let us recall that a state ␳AB is distillable if and only if
there exist a number of copies n and a Schmidt rank 2 state
␾2 such that 具␾2 兩 共␳⌫A兲 丢 n 兩 ␾2典 ⬍ 0 关36兴. It is easy to see that
the minimum eigenvalue of ␳⌫p A共␺共1兲 , ␺共2兲兲 is of the form
共j兲 共j兲
共i兲 2 共j兲 共j兲
2
共1 − p兲N1N2„1 − 共␮共i兲
k 兲 …共1 + ␮m ␮n 兲 − p共␮k 兲 ␮m ␮n ,
共23兲
with m ⫽ n and 共i , j兲 苸 兵共1 , 2兲 , 共2 , 1兲其, i.e., it corresponds to a
−
典A jB j. Therefore, as
Schmidt rank 2 eigenvector 兩kAikBi典 丢 兩␺mn
soon as the state is NPT, we prove that it is also distillable by
considering n = 1 and taking as ␾2 the eigenvector corresponding to the minimal negative eigenvalue. On the other
hand, by choosing p small enough, it always possible to
make the smallest eigenvalue positive, if the first term in 共23兲
is not null, i.e., if both states ␺共i兲 are entangled. More precisely, it can be shown 关18兴 that the necessary and sufficient
condition for the state to be PPT is
再
共2兲 共2兲
2
关1 − 共␮共1兲
p
1 兲 兴共1 + ␮1 ␮2 兲
ⱕ min
,
2 共2兲 共2兲
共1 − p兲N1N2
共␮共1兲
1 兲 ␮1 ␮2
共1兲 共1兲
2
关1 − 共␮共2兲
1 兲 兴共1 + ␮1 ␮2 兲
2 共1兲 共1兲
共␮共2兲
1 兲 ␮1 ␮2
冎
共24兲
.
In particular, to calculate the smallest eigenvalue of the partially transposed state, it is sufficient to consider only the two
biggest Schmidt coefficients of ␺共1兲 and ␺共2兲. We will denote
by p⌫ the largest value of p for which ␳ p共␺共1兲 , ␺共2兲兲 is PPT. In
Fig. 2, we plot the dependence of p⌫ on the Schmidt coefficients of the two pure states in the case d1 = d2 = 2 关i.e., when
␳ p共␺共1兲 , ␺共2兲兲 is a state of four qubits兴.
The condition to determine when the realignment criterion detects entanglement is not trivial to handle analytically.
Thus, we will restrict ourselves to the case in which the two
pure states ␺共1兲 and ␺共2兲 are maximally entangled. In this
case, we have
冉兺
d1
R„␳ p共⌿d+ ,⌿d+ 兲… = 共1 − p兲N1N2
1
2
冉兺
i=1
d2
丢
i,j=1
兩ii典具jj兩 −
兩ii典具jj兩 −
1
d2
冊
+p
1
d1
冊
1
d 1d 2
. 共25兲
The condition 储R(␳ p共⌿d+ , ⌿d+ 兲) 储 ⬎ 1 is thus satisfied only for
1
2
μ 11
μ 11
0.15
μ 11
3 5
3 4
9 10
0.1
0.05
0.7
0.8
0.9
1
μ1
2
FIG. 2. Dependence of the threshold probability p⌫ on ␺共2兲 for
fixed choices of ␺共1兲 in the case d1 = d2 = 2. The state ␳ p共␺共1兲 , ␺共2兲兲 is
共2兲
entangled for all choices of ␮共1兲
1 ⫽ ␮1 for p ⬎ 0; when 0 ⱕ p ⬍ p⌫
the state is PPT, whereas it is NPT if p⌫ ⬍ p ⱕ 1. The point in which
the minimum on the right-hand side of 共24兲 changes from one element to the other is clear from the sharp change in the behavior of
共2兲
the curve, and it coincides with the point ␮共1兲
1 = ␮1 , note that, for
such a point, with the methods introduced in this work we are not
able to say that the state ␳ p共␺共1兲 , ␺共2兲兲 is entangled when PPT 共see
main text and Table I兲. As expectable, the threshold value of p⌫
goes to 0 as one of the two pure states becomes separable.
d d −2
p ⬎ d21共d22−2兲 , where we have assumed without loss of general1
2
ity, d2 ⱖ d1. Note that this value is always greater than p⌫
1
= 1+共d1+1兲共d
: thus, in the case in which the pure states ␺共1兲
2−1兲
and ␺共2兲 are maximally entangled, realignment is always less
sensitive than PT.
In Sec. VI, we will provide analytical examples of states,
which have a structure similar 关see Eq. 共26兲兴 to that of
␳ p共␺共1兲 , ␺共2兲兲, detected as entangled by realignment but not
by partial transposition. In Fig. 3, we show that realignment
and partial transposition are inequivalent 共i.e., there are entangled states detected by one criterion but not by the other
one兲 also in the class ␳ p共␺共1兲 , ␺共2兲兲. The plot of Fig. 3 is
relevant also for another reason: it shows that states for
which it is not possible to construct a nontrivial witness
W⑀共␺共1兲 , ␺共2兲兲 共i.e., states for which ␮共1兲 = ␮共2兲兲 may be entangled.
R ρ AB
1
1.005
1
0.995
0.99
0.72
0.76
0.8
0.84
μ1
FIG. 3. Comparison of the detection power of realignment and
partial transposition in the d1 = d2 = 2 case. We take ␺共1兲 = ␺共2兲 = ␺,
where ␺ is a pure 共entangled兲 state of two qubits, characterized by
its larger Schmidt coefficient ␮1. We consider 储R共␳AB兲储1, for ␳AB
= ␳ p⌫共␺ , ␺兲, i.e., for the state at the border of PPT states. For most of
the range 1 / 冑2 ⱕ ␮1 ⱕ 1 partial transposition is more sensitive than
realignment, i.e., 储R共␳AB兲储1 ⬍ 1 even if a slight change of p makes
the state NPT entangled. The plot shows that realignment is more
sensitive than partial transposition for ␺ almost maximally entangled, i.e., 储R共␳AB兲储1 ⬎ 1 even if the state is PPT.
012305-6
PHYSICAL REVIEW A 75, 012305 共2007兲
CLASS OF POSITIVE-PARTIAL-TRANSPOSE BOUND…
TABLE I. Relation between the entanglement properties of the
two pure states ␺共1兲 , ␺共2兲, and those of ␳ p共␺共1兲 , ␺共2兲兲. When both
␺共1兲 , ␺共2兲 are entangled, and do not have the same Schmidt coefficients 共i.e., they are not equivalent up to local unitaries兲,
␳ p共␺共1兲 , ␺共2兲兲 is PPT entangled in the interval 0 ⬍ p 艋 p⌫. If both the
pure states are entangled, but ␮共1兲 = ␮共2兲, the techniques 共witnesses兲
adopted in this work do not help. There are choices of ␺共1兲 , ␺共2兲
such that the mixed state ␳ p共␺共1兲 , ␺共2兲兲 is separable as soon as, decreasing p, it is PPT 共see Sec. VI兲, as well as other choices such that
the corresponding mixed states can be PPT entangled 共see Fig. 3兲.
␺共1兲, ␺共2兲
A. Maximally entangled pure states ␺„i…
We now focus on an even more specific class of states.
Recalling the definition 共6兲 of maximally entangled state P+d ,
we define the states
M
␳ p共d1, . . . ,dM 兲 = 共1 − p兲 丢
i=1
␳ p共d兲 = 共1 − p兲
Separable for all 0 ⱕ p ⱕ 1
NPT entangled for all 0 ⱕ p ⱕ 1
␮共1兲 ⫽ ␮共2兲: PPT entangled for 0 ⬍ p ⬍ p⌫
␮共1兲 = ␮共2兲: No general statement
For the sake of clarity, in Table I we summarize the relation between the entanglement properties of the two pure
states ␺共1兲 , ␺共2兲, and those of ␳ p共␺共1兲 , ␺共2兲兲.
VI. GENERALIZATION TO MORE THAN TWO STATES
␺„i…
It is possible to straightforwardly generalize the construction of the states ␳ p共␺共1兲 , ␺共2兲兲 to the case in which one considers more than two pure states ␺共i兲.
M
and a probability p, we deGiven a set of states 兵␺共i兲其i=1
fine
M
␳ p共兵␺共i兲其兲 = 共1 − p兲 丢
i=1
1 − P␺共i兲
d2i − 1
d2i − 1
M
+ p 丢 Pd+ .
i=1
i
共27兲
1 − P+d
d2 − 1
+ pP+d .
共28兲
Isotropic states can be considered a subclass of the class of
states we are studying, with M = 1. It is remarkable that isotropic states ␳ p共d兲 are either distillable or separable: no phenomenon of bound entanglement 共either PPT or NPT, if existing兲 is present in such class, while it is sufficient to go to
M = 2 to have it.
It is worth noting that ␳ p共d , d兲 is separable for all values
of p for which it is PPT, i.e., for 0 ⱕ p ⱕ 1 / d2 关37兴, and
indeed, we are not able to construct a witness of the form
共15兲 for it, since in this case ␺共1兲 and ␺共2兲 have the same
Schmidt coefficients; they are equal. On the other hand, a
witness as in 共15兲 exists for ␳ p共d1 , d2兲 in the case d2 ⬎ d1
ⱖ 2.
With regard to the sensitivity of partial transposition, for
␳ p共d1 , . . . , dM 兲, we have
p⌫ =
M
+ p 丢 P␺共i兲 .
i
Compare them to the isotropic states for a d 丢 d system
␳ p共␺共1兲 , ␺共2兲兲
Both separable
One entangled
Both entangled
1 − Pd+
1
M−1
,
1 + 共d M − 1兲 兿 共di + 1兲
i=1
共26兲
i=1
To prove that for M ⱖ 3, the state is entangled for p ⬎ 0 as
soon as one of the ␺共i兲 is entangled, it is sufficient to use the
class of witnesses we studied for M = 2.
Indeed, for M ⱖ 3 it is always possible to split any set of
M
into two nonempty disjoint sets,
natural numbers 兵r共i兲其i=1
m
which without loss of generality can be indicated as 兵r共i兲其i=1
m 共i兲
M
共i兲 M
共i兲
and 兵r 其i=m+1, and such that 兿i=1r ⫽ 兿i=m+1r . Let us conM
are the Schmidt
sider the case in which the numbers 兵r共i兲其i=1
M
ranks of the states in 兵␺共i兲其i=1. For the sake of testing entanglement, it is possible to consider two states 兩˜␺共1兲典
m
M
␺共i兲 and 兩˜␺共2兲典 = 丢 i=m+1
␺共i兲 of different Schmidt rank
= 丢 i=1
共which is a multiplicative quantity under tensoring兲. Thus, if
at least one state ␺共i兲 is entangled, we can construct a nontrivial entanglement witness W⑀共˜␺共1兲 , ˜␺共2兲兲, ⑀ ⬎ 0, as in 共15兲
such that Tr关␳ p共兵␺共i兲其W⑀共˜␺共1兲 , ˜␺共2兲兲兴 = −p⑀. Note that, if all
states ␺共i兲 are separable, then also ␳ p共兵␺共i兲其兲 is separable, i.e.,
there is no entanglement to be detected.
Similarly to the case M = 2, it is possible to prove that the
smallest eigenvalue of a state 共26兲 corresponds to a Schmidt
rank 2 eigenvector, so that as soon as the state is NPT we
know also that it is distillable. Moreover, it is possible to find
a p ⬎ 0, such that the state ␳ p共兵␺共i兲其兲 is PPT entangled, if and
M
only if the states 兵␺共i兲其i=1
are all entangled.
taking, without loss of generality, d1 ⱕ d2 ⱕ . . . ⱕ d M . We
have seen that, if M = 2 and we consider the case in which
both pure states ␺共i兲 are maximally entangled in dimension
di, PT is always more sensitive than realignment. To study
the more general case M ⬎ 2, we restrict ourselves, for simplicity, to the case in which all the dimensions di coincide
共di = d ⱖ 2兲 and ␺共i兲 = ⌿+d for all i = 1 , . . . M,
共29兲
In this case, we have that ␳ p共d ; M兲 is PPT for p ⱕ p⌫
1
= 1+共d−1兲共d+1兲
M−1 . With regard to realignment, we have
M
冉冊
M
1
兩1 − p − p共1 − d2兲 j兩. 共30兲
储R共␳ p共d;M兲兲储1 = M 兺
d j=0 j
Although it is not trivial to find an analytical solution in p of
the inequality 储R(␳ p共d ; M兲)储1 ⱕ 1, it is possible to see that
there are cases in which the realignment criterion is more
sensitive than PT. Indeed, this happens for d = 2 and M ⱖ 3
odd. To verify this, it is sufficient to plug in 共30兲 the corresponding value of p⌫, i.e., p = p⌫ = 1+31M−1 . By definition, for
such value of p, the state is PPT and the condition
储R(␳ p⌫共d ; M兲)储1 ⬎ 1 is satisfied for all odd values of M ⱖ 3,
whereas 储R(␳ p⌫共d ; M兲)储1 = 1 for M = 1 and M even. Numeri-
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PHYSICAL REVIEW A 75, 012305 共2007兲
MARCO PIANI AND CATERINA E. MORA
cal results indicate that d = 2 and M ⱖ 3 odd is the only case
in which realignment detects PPT entangled states of the
form ␳ p共d ; M兲, but we could not verify this analytically.
VII. MORE ON WITNESSES
tangled, it is possible to consider two states 兩˜␺共j兲典, j = 1 , 2 of
different rank obtained from ␺共i兲s by tensoring. It is therefore
clear that for M ⱖ 3, as soon as the problem is not trivial 共i.e.,
not all the states ␺共i兲 are factorized兲, it is always possible to
consider a witness W̃⑀共˜␺共1兲 , ˜␺共2兲兲 of the form 共31兲.
We proceed now to some remarks with regard to the witnesses we analyzed.
B. Canonical witnesses for ␳p„ˆ␺„i…‰…
A. Simplified witnesses
Both for W = W⑀共␺共1兲 , ␺共2兲兲 关Eq. 共15兲兴 and W
= W̃⑀共␺共1兲 , ␺共2兲兲 关Eq. 共31兲兴, we have not only
Tr关W␳0共␺共1兲 , ␺共2兲兲兴 = 0, but, more strongly,
We have seen that the necessary and sufficient condition
to have a nontrivial entanglement witness W⑀共␺共1兲 , ␺共2兲兲, with
⑀ ⬎ 0, is that the states ␺共i兲 have different Schmidt coefficients. When the Schmidt ranks of the states ␺共i兲 are different, i.e., without loss of generality, r1 ⬍ r2, it is possible to
detect the entanglement of ␳ p共␺共1兲 , ␺共2兲兲 by means of a witness with a structure even simpler than that of W⑀共␺共1兲 , ␺共2兲兲.
In such a case, in fact, it is possible to consider nontrivial
共⑀ ⬎ 0兲 witnesses of the form
共31兲
For this choice,
具␣A 丢 ␤B兩W̃⑀共␺共1兲, ␺共2兲兲兩␣A 丢 ␤B典 = Tr关共␤T␮共1兲␣兲†共␤T␮共1兲␣兲兴
− 共1 + ⑀兲兩Tr共␮共2兲␤T␮共1兲␣兲兩2 .
i.e., the witnesses 关38兴 are orthogonal to the separable part,
which corresponds to ␳0共␺共1兲 , ␺共2兲兲, of a state ␳ p共␺共1兲 , ␺共2兲兲.
Indeed, we have W̃⑀共␺共1兲 , ␺共2兲兲 ⱕ W⑀共␺共1兲 , ␺共2兲兲 关compare 共15兲
and 共31兲兴 and
W⑀共␺共1兲, ␺共2兲兲 = 1 − 共1 − P␺共1兲兲 丢 共1 − P␺共2兲兲 − 共1 + ⑀兲P␺共1兲
丢
W̃⑀共␺共1兲, ␺共2兲兲 = P␺共1兲 丢 关1 − 共1 + ⑀兲P␺共2兲兴
= P␺共1兲 丢 1 − 共1 + ⑀兲P␺共1兲 丢 P␺共2兲 .
W␳0共␺共1兲, ␺共2兲兲 = ␳0共␺共1兲, ␺共2兲兲W = 0,
共32兲
Following the same reasoning we used for W⑀共␺共1兲 , ␺共2兲兲, we
see that the quantity 共32兲 can be made negative for any ⑀
⬎ 0 if and only if 共without loss of generality兲 there exist ␣
and ␤ such that
␮共2兲 = ␤T␮共1兲␣ .
P␺共2兲 .
Moreover, ␳0共␺共1兲 , ␺共2兲兲 is exactly defined as the state corresponding 共via normalization兲 to the projector 共1 − P␺共1兲兲 丢 共1
− P␺共2兲兲.
In the case of M ⱖ 3 states ␺共i兲, we argued 关see Sec. VI,
paragraph following Eq. 共26兲兴 that, as soon as one state ␺共i兲 is
entangled, there exists a nontrivial entanglement witness of
the form W⑀共˜␺共1兲 , ˜␺共2兲兲 that detects the entanglement of
␳ p共兵␺共i兲其兲. The states ˜␺共i兲, i = 1 , 2 were taken to be tensor
products of two disjoint subsets of 兵␺共i兲其, so that 兩˜␺共1兲典
M
丢 兩˜
␺共2兲典 = 丢 i=1
兩 ␺共i兲典. We can instead consider a witness of the
form
M
共1兲
This is possible if and only if the rank of ␮ is greater than
that of ␮共2兲.
We may better understand this result by considering that
W̃⑀共␺共1兲, ␺共2兲兲 = 共P␺共1兲 丢 1兲 ⴰ 兵1 丢 关1 − 共1 + ⑀兲P␺共2兲兴其
ⴰ 共P␺共1兲 丢 1兲
共33兲
and that
共P␺共1兲 丢 1兲兩␣A 丢 ␤B典 = 兩␺共1兲典 丢 兩␥典,
共34兲
with 兩␥典 = 兺i␮共1兲
i 共兺l␣il 兩 l典兲 丢 共兺k␤ik 兩 k典兲. It is clear that, by the
right choice of ␣ and ␤, ␥—though, in general, not
normalized—can be made proportional to any state whose
Schmidt rank is not greater than the one of ␺共1兲. In particular,
if ␺共2兲 has the same Schmidt rank that ␺共1兲 has, it is possible
to choose ␣ and ␤ such that
具␣␤兩W̃⑀共␺共1兲, ␺共2兲兲兩␣␤典 = − ⑀兩c兩2 ,
with 兩␥典 = c 兩 ␺共2兲典 and 兩c 兩 ⬎ 0. Therefore, in this case, W̃ is
positive on separable states if and only if ⑀ = 0.
Note that in Sec. VI, when analyzing the multi-state case
for M ⱖ 3, we argued that as soon as one state ␺共i兲 is en-
共i兲
W⑀共兵␺ 其兲 = 1 −
M
丢 共1 − P␺共i兲兲 − 共1 + ⑀兲 丢 P␺共i兲 .
i=1
共35兲
i=1
We have W⑀共兵␺共i兲其兲 ⱖ W⑀共˜␺共1兲 , ˜␺共2兲兲, but W⑀共兵␺共i兲其兲 has the
same expectation value −⑀ p with respect to the states
␳ p共兵␺共i兲其兲. Moreover, it can be considered as a modification
of the projector onto the subspace orthogonal to the support
of the separable part ␳0共兵␺共i兲其兲 of the state, with the modifiM
cation −共1 + ⑀兲 丢 i=1
P␺共i兲 tailored to “intercept” the entangled
part of the state. Note that the witness W⑀共兵␺共i兲其兲 depends
only on ⑀ and on the set 兵␺共i兲其, not on the choice of two
subsets of 兵␺共i兲其, unlike W⑀共˜␺共1兲 , ˜␺共2兲兲.
VIII. MULTIPARTITE CASE
Now we consider the multipartite case, i.e., the states ␺共i兲,
i = , 1 . . . , M, are states of N parties. From the results presented in Sec. V, we know that the state ␳ p共兵␺共i兲其兲 can be
made PPT, with respect to a given bipartite cut S1 : S2, for
some strictly positive p only if all the states ␺共i兲 are entangled with respect to that cut. Therefore, for this to happen
for any possible bipartite cut, all the states ␺共i兲 must be
N-partite entangled.
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CLASS OF POSITIVE-PARTIAL-TRANSPOSE BOUND…
As regards witnesses, we are able to provide a nontrivial
共i.e., not positive semidefinite兲 witness that detects bipartite
S1 : S2 entanglement, if 共i兲 M = 2 and the states ␺共1兲 and ␺共1兲
have different Schmidt coefficients with respect to the cut, or
共ii兲 M ⱖ 3 and at least one state ␺共i兲 is entangled with respect
to the cut. As we discussed in Sec. VII B, it is always possible to consider witnesses W⑀共兵␺共i兲其兲 of the form 共35兲, for
every cut. In the construction of such witnesses, the only
parameter dependent from the cut is ⑀. If, for a given cut, one
of the above-mentioned conditions 共i兲 and 共ii兲 is valid, then it
is possible to take ⑀ ⬎ 0 and detect bipartite entanglement by
means of the corresponding witness. Let us consider
We are interested in operators that are entanglement witnesses, i.e., such that they are positive on separable states.
We correspondingly take l = 1 and consider a state ␴
= 兩␾典具␾ 兩 = P␾ of Schmidt number k and a Schmidt-rank m
witness W, with m ⱖ k + 1. We compose them to give an operator P␾ 丢 W, which is then positive on separable states by
construction, according to the Lemma III.1 of 关39兴. If 共i兲 ␾ is
entangled 共i.e., k ⱖ 2兲 and 共ii兲 ␺W, of Schmidt rank strictly
greater than k, is such that 具␺W 兩 W 兩 ␺W典 ⬍ 0, then P␾ 丢 W is a
nondecomposable entanglement witness. Indeed, we can
consider p ⬎ 0 such that ␳ p共␾ , ␺W兲 is PPT—both ␾ and ␺W
are entangled—and have
˜⑀ = min max兵⑀兩具␣S 丢 ␤S 兩W⑀共兵␺共i兲其兲兩␣S 丢 ␤S 典 ⱖ 0其.
1
2
1
2
Tr关P␾ 丢 W␳ p共␾, ␺W兲兴 = p具␺W兩W兩␺W典 ⬍ 0.
S1:S2
If ˜⑀ ⬎ 0, then W˜⑀共兵␺共i兲其兲 is a nontrivial witness for genuine
multipartite entanglement. Thus, we get immediately that
␳ p共兵␺共i兲其兲 is N-partite entangled for every p ⬎ 0, since its entanglement is detected by a witness that is positive with respect to all biseparable states.
Thus, it is possible to decide to construct a state that is
PPT with respect to some desired bipartitions, and NPT with
respect to the remaining ones. To do so, it is sufficient to
opportunely choose the states 兵␺共i兲其 and p. Indeed, the mixed
state is NPT with respect to a bipartition S1 : S2 for all p ⬎ 0 if
and only if there is a state ␺共i兲
i that is S1 : S2 separable. If the
states satisfy 共i兲 or 共ii兲 for every cut, the mixed state
␳ p共兵␺共i兲其兲 for sure contains N-partite entanglement 共for p
⬎ 0兲, because there is a witness that detects it.
IX. TENSORLIKE WITNESSES, k-POSITIVE MAPS, AND
NONDECOMPOSABILITY
Building on the considerations of Sec. VII A, here we
discuss the possibility of obtaining nondecomposable witnesses able to detect PPT bound entangled states by composing through tensor product a decomposable witness 共unable
to detect PPT bound entanglement兲 with a positive operator
共without loss of generality a state兲.
Lemma III.1 of 关39兴 says that, if a state ␴ on HA1 丢 HB1
has Schmidt number k, and ␩ is an operator on HA2 丢 HB2,
which is positive with respect to states of Schmidt number
kl, then ␴ 丢 ␩ is positive with respect to states ␶ on HA
丢 HB of Schmidt number l, i.e., Tr关共␴ 丢 ␩兲␶兴 ⱖ 0. The proof
of such Lemma 关39兴 is a generalization of the reasoning we
have adopted in Sec. VII A. In particular, it is sufficient to
consider the case ␴ = 兩␾典具␾兩 and ␶ = 兩␺典具␺兩, and note that
With these considerations and exploiting the ChoiJamiołkowski isomorphism, it is immediate to state a theorem relating the properties of k-positivity, complete positivity, and decomposability of maps.
Theorem. A linear map ⌳ which is k-positive, k ⱖ 2, is
completely positive if and only if idk 丢 ⌳ is decomposable.
Proof. The only if part is trivial: if ⌳ is CP, then idk 丢 ⌳ is
trivially decomposable—it is CP itself. To prove the if part,
let us suppose that ⌳ is not CP and show that idk 丢 ⌳ is
nondecomposable. Indeed, if ⌳ is not CP, then, even though
the corresponding witness W⌳ is positive on Schmidt rank k
states 关28,31兴, there exists a Schmidt rank m state ␺W⌳, m
⬎ k, such that 具␺W⌳ 兩 W⌳ 兩 ␺W⌳典 ⬍ 0. Thus, as remarked before,
the witness P+k 丢 W is nondecomposable and the same holds
for its isomorphic map idk 丢 ⌳.
䊏
Note that in the theorem we could have used idl 丢 ⌳, with
any 2 ⱕ l ⱕ k, instead of idk 丢 ⌳. The results just exposed
imply that, for k ⱖ 2, as soon as we know that a PnCP map ⌳
is k-positive or that a nonpositive witness W is positive on
Schmidt-number k states, we know that, for example, idk
+
丢 ⌳ and Pk 丢 W are, respectively, a positive nondecomposable map and a nondecomposable witness, without caring
about the decomposability of ⌳ or W.
Now, we provide a simple example illustrating how to
pass from provably decomposable witnesses to nondecomposable witnesses through tensoring. Obviously, the example
could be recast immediately in terms of maps.
Let us consider witnesses of the form
W⑀共␺兲 = 1 − 共1 + ⑀兲兩␺典具␺兩.
If we consider expectation values with respect to pure states,
we obtain
Tr关兩␺典具␺兩共兩␾典具␾兩 丢 ␩兲兴 = Tr2兵Tr1关兩␺典具␺兩共兩␾典具␾兩 丢 1兲兴␩其
具␸兩W⑀共␺兲兩␸典 = 1 − 共1 + ⑀兲兩具␺兩␸典兩2 .
= 具␥兩␩兩␥典,
with 共兩␾典具␾ 兩 丢 1兲 兩 ␺典 = 兩␾典 丢 兩␥典. Considering that ␺ has
Schmidt rank l, and that the action of 共兩␾典具␾ 兩 丢 1兲 on a separable states can create at most a state of Schmidt rank k 关see
共34兲兴, we conclude that the state ␥ has at most Schmidt rank
kl. Note that ␥ is not normalized, in general.
Suppose ␺ has Schmidt rank r and Schmidt decomposition
r
␮i 兩 ii典. It can be proved by Lagrange multipliers that
兩␺典 = 兺i=1
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k
max
兵␸ s.t. SR共␾兲ⱕk其
兩具␺兩␸典兩 = 兺 ␮2i ,
2
i=1
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MARCO PIANI AND CATERINA E. MORA
where the maximum is taken with respect to states ␸, which
have at most Schmidt rank k ⱕ r, and the Schmidt coefficients of ␺ are ordered as ␮i ⱖ ␮i+1. Thus, if we want the
witness W⑀共␺兲 to be positive on states of Schmidt rank k, we
must have
k
⑀ⱕ
1 − 兺 ␮2i
i=1
k
兺
i=1
.
␮2i
Note that if k = r, we must put ⑀ = 0. Correspondingly, for k
⬍ r 共in the case r ⱖ 2兲, let us define the Schmidt number k
+ 1 witness
1
W k共 ␺ 兲 = 1 −
k
兩␺典具␺兩.
␮2i
兺
i=1
Among Wk共␺兲’s, for fixed ␺, the witness able to detect the
largest number of entangled states is of course W1共␺兲. Yet,
W1共␺兲 cannot detect any PPT bound entangled state, as we
now prove.
Every pure state ␺ in Cd 丢 Cd can be written as 兩␺典
= 冑d共A 丢 1兲 兩 ⌿+d 典, with Tr共A†A兲 = 1. The Schmidt coefficients
of ␺ are given by the singular values of A. In particular, for
the largest Schmidt coefficient, we have ␮1 = 储A储⬁ = 冑储AA†储⬁,
where 储X储⬁ is the operator norm of X. We have 共兩␺典具␺ 兩 兲⌫
= A 丢 1VA† 丢 1, with V the swap operator. Note that V ⱕ 1.
The witness W1共␺兲 does not detect a state ␳ as entangled
if and only if 具␺ 兩 ␳ 兩 ␺典 ⱕ ␮21. If ␳ = ␳AB is PPT, ˜␳ = ␳⌫ is a
normalized state and
具␺兩␳兩␺典 = Tr共␳⌫共兩␺典具␺兩兲⌫兲
= Tr关˜␳共A 丢 1兲V共A† 丢 1兲兴 ⱕ Tr关˜␳共AA† 丢 1兲兴
= Tr共˜␳AAA†兲 ⱕ max具␾兩AA†兩␾典 = ␮21 ,
␾
共36兲
where ˜␳A = TrB共˜␳兲. A straightforward proof can also be obtained by considering that the reduction criterion 关7兴 is
weaker than the PT criterion. Thus, for every PPT state ␳
= ␳AB, we have ␳ ⱕ ␳A 丢 1. For the state ␺ under consideration:
具␺兩␳兩␺典 ⱕ 具␺兩␳A 丢 1兩␺典 = Tr共␳AAA†兲 ⱕ ␮21 .
Witnesses Wk共␺兲, N ⱖ 2 are even worse in detecting PPT
entangled states. Yet, for every 2 ⱕ k ⱕ r − 1 and for any state
␾ with Schmidt rank 2 ⱕ l ⱕ k, P␾ 丢 Wk共␺兲 is a nondecomposable entanglement witness.
X. CONCLUSIONS
We have been able to prove that any state ␳ p共兵␺共i兲其兲 is
M
entangled as soon as p ⬎ 0, for any set 兵␺共i兲其i=1
, when at least
共i兲
one state ␺ is entangled, except in the case with only two
pure states ␺共i兲 with the same Schmidt coefficients. In the
latter case, the state could be entangled as well, but an en-
tanglement witness different from 共15兲 would be required to
prove it.
The structure of the states ␳ p共兵␺共i兲其兲 is very simple: all the
entanglement appears to be concentrated in an eigenvector of
the mixed state, while the separable part, in the suitable region of parameters, plays the role of a “cover,” which prevents the detection by partial transposition 共and, hence, distillation兲. We remark the resemblance of the class of states
with isotropic states, most evident when considering the special case of maximally entangled pure states 兵␺共i兲 = ⌿d+ 其. Ini
deed, isotropic states were involved in the analysis that
signed the first appearance of the class of states ␳ p共␺共1兲 , ␺共2兲兲
in literature 关18兴.
Our analysis differs from the one appearing in 关18,19兴
because we focus on the properties of the states rather than
on what they allow one to do, and we construct witnesses to
detect entanglement both in the bipartite and multipartite settings. Moreover, we generalize the states to the case of the
possible choice of many pure states, i.e., from the states
␳ p共␺共1兲 , ␺共2兲兲 to the states ␳ p共兵␺共i兲其兲. It is clear that the states
␳ p共兵␺共i兲其兲 can be modified, both in their separable and entangled parts, to provide larger classes of PPT entangled
states. Indeed, the key point toward the construction of PPT
entangled states is that the separable part of ␳ p共兵␺共i兲其兲 is not
only positive semidefinite under partial transposition, but
strictly positive, so that slightly changing it does not affect
the positivity condition. For example, using states similar to
the ones studied in this paper, it is possible to prove that
many of the positive maps that were conjectured to be nondecomposable in 关24兴, are actually so 关40兴. One remarkable
property of the structure 共26兲 is that states ␳ p共兵␺共i兲其兲 are completely characterized by a set of pure states 兵␺共i兲其 and a mixing parameter p. Moreover, they are separable for p = 0 and
entangled—for almost any choice of 兵␺共i兲其 with at least one
␺共i兲 entangled—for p ⬎ 0. We hope that the variety of parameters at disposal through the choice of the set of pure states
兵␺共i兲其 共and of the mixing parameter p兲 could lead to the study
of interesting cases and/or effects both in the bipartite and in
multipartite setting. Moreover, the simplicity of the structure
of this class of states suggests that they might be used to
offer the first experimental verification of the existence, and
properties, of bound entanglement. Preliminary studies of the
robustness of these states under the action of noise, for some
proper choice of the two pure states ␺共1兲 and ␺共2兲, confirms
the fact that experimental construction of these states should
be possible with current technology. A detailed analysis of
the noise tolerance of the states ␳ p, together with a study of
the possible experimental realization of these states will be
presented elsewhere.
Note that all the results regarding witnesses can be directly translated into results regarding maps through the
Choi-Jamiołkowski isomorphism, so that, in this paper, we
provide many examples of nondecomposable maps useful to
detect the entanglement of PPT states. Moreover, through the
analysis of states ␳ p共兵␺共i兲其兲 and of the corresponding witnesses, we could provide a relationship among the properties
of k-positivity, complete positivity, and decomposability: any
map that is k-positive, for k ⱖ 2, but not completely positive,
can be extended to a nondecomposable map. Thus, it seems
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CLASS OF POSITIVE-PARTIAL-TRANSPOSE BOUND…
that further analysis of the property of k-positivity could not
only be useful to study the Schmidt-number property of
states, following 关28兴, but the entanglement property itself.
We thank H. J. Briegel, O. Guehne, M. Horodecki, P.
Horodecki, R. Horodecki, A. Kossakowski, and W. A. Majewski for discussions. We gratefully acknowledge support
by Austrian Science Foundation 共FWF兲, EU 关RESQ 共IST
2001 37559兲, OLAQUI, IP SCALA兴.
关1兴 M. A. Nielsen and I. L. Chuang, Quantum Computation and
Quantum Information 共Cambridge University Press, Cambridge, England, 2000兲.
关2兴 G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki,
M. Rotteler, H. Weinfurter, R. Werner, and A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments 共Springer, New York, 2001兲.
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关34兴 The problem can be cast in the form 共19兲 without loss of generality because there is no normalization constraint on 兩␣典 and
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关35兴 Given a composite system A1A2 : B1B2, the realignment operation can be split in realignement with respect to subsystems 1
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split into partial transpostion with respect to A1 and A2兲:
R共兩iA1mA2 jB1nB2典具kA1 pA2lB1qB2 兩 兲 = R1共兩iA1 jB1典具kA1lB1 兩 兲
丢 R2共兩mA nB 典具pA qB 兩 兲 = 兩iA mA kB pB 典具j A nA lB qB 兩.
2
2
2
2
1
2
1
2
1
2 1
2
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关40兴 M. Piani 共unpublished兲
012305-11