PHYSICAL REVIEW A 73, 012345 共2006兲
Class of bound entangled states of N + N qubits revealed by nondecomposable maps
Marco Piani*
Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland
共Received 26 August 2005; published 31 January 2006兲
We introduce a family of linear maps which are positive but not completely positive. We exhibit examples
of nondecomposable maps and 2N 丢 2N, N 艌 2, bound entangled states—e.g., nondistillable bipartite states of
N + N qubits. Such states, as the standard Bell diagonal states, are diagonal in a maximally entangled basis and,
apart remaining positive under partial transposition, are not detected by the realignment criterion.
DOI: 10.1103/PhysRevA.73.012345
PACS number共s兲: 03.67.Mn, 02.10.Ud
I. INTRODUCTION
Entanglement appears to be a basic resource in the fields
of quantum information and quantum computation 共see 关1,2兴
and references therein兲. Even if there is a sound definition of
what an entangled state is 关3兴, it is in general difficult to
determine whether a given state is entangled or not.
There are different results in the literature regarding the
classification of states. One of the more interesting 关4,5兴 is
based on the use of linear maps which are positive 共P兲 关6–8兴
but not completely positive 共CP兲 关9–11兴: we shall refer to
them as PnCP maps. A map is P if it trasforms any state into
another positive operator. In the case of a bipartite system, a
state is entangled if and only if there exists a PnCP map such
that the operator obtained acting with the map on only one of
the two subsystems is not positive any more. The simplest
example of a PnCP map is the operation of transposition T
共with respect to a given basis兲. The action of trasposition on
one of the subsystems is called partial transposition 共PT兲.
Because of the structure of the set of positive maps 关7,12兴, in
the 共2 丢 2兲- and 共2 丢 3兲-dimensional cases PT can “detect”
all entangled states: states that remain positive under PT
共PPT states兲 are separable; states that develop negative eigenvalues under PT 共NPT states兲 are entangled. Unfortunately in higher dimensions PT is not a “complete” test any
more and there are PPT states which are entangled 关13兴.
The PnCP map approach to the problem of entanglement
characterization can also give information about the distillability of the state 共see 关14兴 for a review兲. A state is said to be
distillable if, having at disposal a large number of copies of
the state, it is possible to obtain some maximally entangled
states under the constraint of performing only local operations and using classical communication. It turns out that a
PPT entangled state 共PPTES兲 can not be distilled, so that its
entanglement can be considered “bound” 关15兴; however, it
can be useful for tasks that would be impossible to perform
classically 关16,17兴. It is still unknown whether all bound entangled states are PPT—i.e., whether the PPT property is a
necessary condition for nondistillability, besides being a sufficient one. If so, the set of bound entangled states would
correspond to the set of PPTES. On the other hand, it is
evident that, in order to identify PPT bound entangled states
*Electronic address: piani@ts.infn.it
1050-2947/2006/73共1兲/012345共10兲/$23.00
by means of linear maps, it is necessary to use PnCP that are
not decomposable—i.e., that cannot be written as the sum of
a CP map and a CP map composed with transposition.
It is therefore clear that the study of P maps is strictly
related to the study of entanglement, the link being provided
by the Choi-Jamiołkowsky isomorphism 关18,19兴. In this
work we contribute to the phenomenology of positive maps
关20–22兴, giving some general methods to construct classes of
PnCP maps. In one instance we test their decomposability by
finding at the same time examples of PPT 共and therefore
bound兲 entangled states of N + N qubits. Moreover, we prove
that such states are not detected by the so-called realignment
criterion 关23,24兴, so that the symmetry of the considered
states seems to “defeat” permutational separability criteria
关25–27兴.
In Sec. II we review some basic notions and results concerning the properties of positivity and complete positivity of
maps and their relation to entanglement. In Sec. III we provide a method to construct a class of PnCP maps. We are led
quite naturally to consider a set of states to test the decomposability properties of representative maps in such a class.
Therefore in Sec. IV we focus on states such that the condition of positivity under PT has a simple form. We exhibit
examples of 共2N 丢 2N兲-dimensional PPTES, thus proving at
the same time that the representative maps are not decomposable. Moreover, we prove that the only other independent
permutational criterion in the bipartite case, realignment,
does not detect such bound entangled states, either.
II. LINEAR MAPS AND ENTANGLEMENT
We start with some basic facts about positive maps and
entanglement, presented for finite d-dimensional systems Sd
described by the algebra of d ⫻ d matrices with complex entries M d共C兲. We shall denote by Sd the space of the states
共density matrices兲—that is, the convex set of positive ␳
苸 M d共C兲 of unit trace.
The action of any Hermiticity-preserving linear map
⌳ : M d共C兲 → M d共C兲 can be written as 关28兴
d2−1
M d共C兲 苹 X 哫 ⌳关X兴 =
␭kiFkXF†i ,
兺
k,i=0
共1兲
where Fk’s are d2 matrices d ⫻ d, forming an orthonormal
basis in M d共C兲 with respect to the Hilbert-Schmidt scalar
012345-1
©2006 The American Physical Society
PHYSICAL REVIEW A 73, 012345 共2006兲
MARCO PIANI
product, Tr共F†i Fk兲 = ␦ki, and C⌳ = 关␭ki兴 is a generic Hermitian
matrix. The map is also trace preserving if and only if
d2−1
␭kiF†i Fk = 1.
兺k,i=0
Remark 1. Expression 共1兲 does not depend on the choice
of the orthonormal basis of the matrices Fk’s. In fact, let 兵Gl其
be another orthonormal basis; there exists a d2 ⫻ d2 unitary
d2−1
UlkGl. Thus the
matrix U, Ulk = Tr共G†l Fk兲, such that Fk = 兺l=0
action of ⌳ can be written as
d2−1
⌳关X兴 =
兺 ␭lm⬘ GlXGm† ,
with ␭lm
⬘ = 共C⌳⬘ 兲lm = 共UC⌳U†兲lm. Therefore it is always possible to find an orthonormal basis 兵Gl其 such that C⌳ is diag2
onal and Eq. 共1兲 reads ⌳关X兴 = 兺dj=0−1␭ jG jXG†j , where the ␭ j’s
are the eigenvalues of C⌳. We will refer to such an orthonormal basis as a diagonal basis for ⌳.
Any linear map ⌳ that is used to describe a physical-state
transformation must preserve the positivity of all states ␳;
otherwise, the appearance of negative eigevalues in ⌳关␳兴
would spoil its statistical interpretation which is based on the
use of the state eigenvalues as probabilities. If a map preserves the positivity of the spectrum of all ␳, then we say it is
positive; however, it is not sufficient to make ⌳ fully physically consistent. Indeed, the system Sd may always be
thought to be statistically coupled to an ancilla n-level system Sn. One is thus forced to consider the action idn 丢 ⌳ over
the compound system Sn + Sd, where by idn we will denote in
the following the identity action on M n共C兲. It is not only ⌳
that should be positive, but also idn 丢 ⌳ for all n; such a
property is called complete positivity 关6,10,11兴. Complete
positivity is necessary because of the existence of entangled
states of the compound system Sn + Sd: namely, of states that
cannot be written as factorized linear convex combinations—
that is, as
ci 艌 0,
i
兺 ci = 1.
1
for all 共normalized兲 兩␾典 , 兩␺典 苸 Cd, with 兩␾*典 denoting the conjugate of 兩␾典 with respect to the fixed orthonormal basis in
C d;
共ii兲 completely positive if and only if 关6,30兴
共idd 丢 ⌳兲关P̂+d 兴 艌 0;
⌳关␳兴 =
d2−1
⌳关␳兴 =
where 兩j典, j = 1,2, . . . , d is any fixed orthonormal basis in Cd,
and its unnormalized version 兩␺ˆ +d 典 = 冑d兩␺+d 典; let us denote by
P+d ⬅ 兩␺+d 典具␺+d 兩 and P̂+d ⬅ dP+d 苸 Sd⫻d the corresponding 共normalized and unnormalized兲 projections onto them.
Theorem 1. A linear map ⌳ : M d共C兲 → M d共C兲 is
共i兲 positive if and only if 关19,29兴
共6兲
兺
j=0
d2−1
␭ jG j␳G†j
=
共冑␭ jG j兲␳共冑␭ jG j兲† ,
兺
j=0
with 冑␭ jG j identified as the Kraus operator K j.
Theorem 2. The following statements are equivalent 关5兴:
共i兲 ␳ 苸 Sd⫻d is entangled;
共ii兲 for some positive map ⌳ on M d共C兲
D⌳共␳兲 ª Tr共共idd 丢 ⌳兲关P̂+d 兴␳兲 ⬍ 0;
共7兲
共iii兲 for some positive map ⌳ on M d共C兲
␳⬘ = 共idd 丢 ⌳兲关␳兴
i
共3兲
K j␳K†j ,
兺
j=0
with 兺 jK†j K j = 1 if ⌳ is trace-preserving.
Remark 2. The properties of positivity and complete positivity depend on the coefficient matrix C⌳ of Eq. 共1兲. In
particular a linear map is CP if and only if C⌳ is positive
semidefinite 共we will simply say “positive” from now on兲. In
fact in this case it is possible to obtain the Kraus-Stinespring
of point 共iii兲 of theorem 1 diagonalizing the coefficient matrix and using the fact that the eingenvalues of C⌳ are positive:
is not a positive operator any more.
d
␺ij兩i典 丢 兩j典 in Cd⫻d there is
Remark 3. For any 兩␺典 = 兺i,j=1
A␺ in M d共C兲 such that 具j兩A␺兩i典 = ␺ij and therefore 兩␺典 = 共1d
ˆ d 典. Then, considering the spectral decomposition of
丢 A 兲兩␺
␺
+
any state ␳ 苸 Sd⫻d, it is clear that there is a CP map ⌳␳,
characterized by a coefficient matrix C⌳␳, such that ␳ = 共idd
丢 ⌳␳兲关P̂+兴.
Thus for any positive map ⌳ characterized by a
coefficient matrix C⌳, we have
d
D⌳共␳兲 = Tr共C⌳C⌳␳兲,
d
兩j典 丢 兩j典,
冑d 兺
j=1
共5兲
d2−1
共2兲
In fact, any PnCP map ⌳, when acting partially as id 丢 ⌳,
moves some entangled states out of the space of states; however, exactly for this reason, it may be used to detect entanglement 关4,5兴.
In the first of the following two theorems we collect some
useful results concerning positivity and complete positivity;
the second one links entanglement detection to positive
maps.
In the space of states Sd⫻d of the bipartite system Sd + Sd,
let us introduce the symmetric state
兩␺+d 典 =
共4兲
共iii兲 completely positive if and only if it can be expressed
in the Kraus-Stinespring form 关10,11兴
k,i=0
␳Ssep
= 兺 ci␳S共i兲 丢 ␳S共i兲,
n
d
n+Sd
具␾ 丢 ␺兩共idd 丢 ⌳兲关P̂+d 兴兩␾ 丢 ␺典 = 具␺兩⌳关兩␾*典具␾*兩兴兩␺典 艌 0
with the two coefficient matrices expressed in the same orthonormal basis.
Remark 4. We note two interesting properties of the symmetric state 兩␺+d 典:
共i兲 For all matrices A,B acting on Cd one has
A 丢 B兩␺+d 典 = 1d 丢 BAT兩␺+d 典 = ABT 丢 1d兩␺+d 典;
共ii兲 under partial transposition P+d gives rise to the flip
operator
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PHYSICAL REVIEW A 73, 012345 共2006兲
CLASS OF BOUND ENTANGLED STATES OF N + N…
F␮共i兲 = 共F␮共i兲兲†
d
Vª
兺 兩i典具j兩 丢 兩j典具i兩 =
共idd 丢 T兲关P̂+d 兴,
for all ␮ = 0 , . . . , d2i − 1, i = 1,2. If all the coefficients ␭␮共i兲 are
共2兲
positive apart from one, let us say ␭共2兲
k = −兩␭k 兩, and all the
positive coefficients are greater or equal to 兩␭共2兲
k 兩, then the
map ⌳ : M d1⫻d2共C兲 → M d1⫻d2共C兲,
i,j=1
which is such that
V兩␺ 丢 ␾典 = 兩␾ 丢 ␺典,
V共A 丢 B兲V = B 丢 A.
Unlike the case of CP maps, there is no general prescription on C⌳ ensuring that ⌳ preserves the positivity of ␳. For
instance, if C⌳ is not positive, then, by separating positive
and negative eigenvalues, one sees that every ⌳ can be written as the difference of two CP maps ⌳1,2 关31兴:
⌳关␳兴 =
␭ jG j␳G†j − 兺 兩␭ j兩G j␳G†j ,
兺
␭ 艌0
␭ ⬍0
j
共8兲
j
with G j a diagonal basis. However, no general rule is known
that may allow us to recognize the positivity of ⌳ by looking
at the eigenvalues ␭ j and at the matrices G j.
Given a set of positive maps P = 兵⌳1 , . . . , ⌳ p其 we can define a larger set of positive maps,
再冏兺 冏 冎
p
⍀共P兲 =
⌫i ⴰ ⌳i ⌫i CP ,
共9兲
i=0
with ⌳0 = id. Then, given a set of PnCP maps 兵⌳PnCP
其i, we
i
其兲
of
P
maps,
potencan construct a whole class ⍀共兵⌳PnCP
i
tially PnCP. It is quite evident that no map in ⍀共兵⌳PnCP
其 i兲
i
gives a stronger test for entanglement, in the sense of theorem 2, than the ensemble of tests performed with the single
’s. In particular, if a map is in ⍀共兵T其兲, it is said to be
⌳PnCP
i
decomposable and cannot provide a stronger test than PT.
According to a theorem by Woronowicz 关12兴, all P maps
M 2共C兲 → M 2共C兲 are decomposable, whence the transposition
detects all the entangled states in S2⫻2; in other words, 共id2
丢 T2兲关␳兴 is nonpositive if and only if ␳ is entangled. On the
contrary, when d 艌 3, there are PPT states which are entangled 共PPTES兲 关7,9,13,14兴. The entanglement in a PPTES
cannot be distilled by means of local operations and classical
communication 关15兴; therefore, it is referred to as bound entanglement.
The relation between nondecomposability of maps and
PPT entangled states is summarized in the following proposition.
Proposition 1. If ⌳ is positive on Sd, ␳ 苸 Sd⫻d is PPT, and
D⌳共␳兲 ⬍ 0, then ⌳ is not decomposable and ␳ is a PPTES.
⌳ = ⌳1 丢 idd2 + idd1 丢 ⌳2
共11兲
is positive.
Remark 5. Any map ⌳ satisfying the requirements of the
previous theorem is positive. It is, moreover, PnCP as soon
as its matrix of coefficients is not positive—i.e., as soon as
共2兲
the negative contribution in idd1 丢 ⌳2 due to ␭共2兲
k = −兩␭k 兩
⬍ 0 is not actually canceled by terms in ⌳1 丢 idd2.
Theorem 3 is suggested by a similar result regarding dynamical semigroups 关32,34兴. A dynamical semigroup is a set
of Hermiticity and trace-preserving linear maps ␥t, t 艌 0, on
Sd which obey a semigroup composition law ␥t ⴰ ␥s = ␥t+s, for
any t , s 艌 0. Semigroups are used to describe the dynamics of
a system immersed in an environment and weakly coupled to
it 关35–38兴. With the further assumption of continuity in t
共time兲 the semigroup has the form ␥t = exp共tL兲, where L is a
map called the generator which determines all the properties
of the semigroup. The issue of complete positivity in the
description of the evolution of dynamical systems is indeed
related to the existence of entangled states 关39–44兴. The generator of a factorized semigroup exp共tL兲 = exp共tL1兲
丢 exp共tL2兲 on Sd 丢 Sd is L = L1 丢 idd + idd 丢 L2, which is
1
2
2
1
similar to Eq. 共11兲.
IV. CLASS OF „2N ‹ 2N…-DIMENSIONAL BOUND
ENTANGLED STATES
We will now use the result of the previous section to
detect bound entangled states “canonically” related to maps
having the structure of theorem 3. We will take advantage of
the simple properties of Pauli matrices under product and
transposition.
We notice that 兵␴ˆ ␮ ª ␴␮ / 冑2其␮3 =0, with ␴0 the twodimensional identity matrix and
␴1 =
冉 冊
0 1
1 0
,
␴2 =
冉 冊
0 −i
i
0
,
␴3 =
冉 冊
1
0
0 −1
the Pauli matrices, is a Hermitian orthonormal basis in
M 2共C兲. Let L共k兲 be the the set
III. CLASS OF POSITIVE MAPS
L共k兲 ª 兵共␻1, . . . , ␻k兲兩␻i = 0,1,2,3,i = 1, . . . ,k其,
In the following theorem we will give a sufficient condition for the positivity of a class of maps on M d1共C兲
丢 M d 共C兲. The proof is presented in the Appendix.
2
Theorem 3. Let ⌳i be maps acting on M di共C兲, i = 1,2, in the
following way:
whose elements are k-dimensional 共integer兲 vectors ␻. Let
us take m 艌 n 艌 1 and let L be the lattice
d2i −1
⌳i关X兴 =
␭␮共i兲F␮共i兲XF␮共i兲;
兺
␮=0
i.e., they admit Hermitian diagonal bases
共10兲
L ª L共m兲 ⫻ L共n兲 = 兵共␣, ␤兲兩␣ 苸 L共m兲, ␤ 苸 L共n兲其,
共12兲
with 4N, N = m + n elements. In a geometric representation L
can be considered in an N-dimensional integer space as a
hypercube whose sides contain four points. Every index
among ␣1 , . . . , ␣m , ␤1 , . . . , ␤n is then a coordinate. We will
associate to the points of L the tensor products of Pauli matrices
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PHYSICAL REVIEW A 73, 012345 共2006兲
MARCO PIANI
␴␣␤ ª ␴␣ 丢 ␴␤ = 共␴␣1 丢 ¯
丢
␴ ␣m兲 丢 共 ␴ ␤1 丢 ¯
丢
␴␤n兲.
共13兲
stitute a natural extension of the standard Bell states to bipartite states of many qubits; they can be considered bipartite
tensor products of N Bell states of two qubits:
It is clear that
兵␴ˆ ␣ ª ␴␣/共冑2兲 兩␣ 苸 L
m
共m兲
兩␺␣␤典 = 兩␺␣1典 丢 ¯ 丢 兩␺␤n典,
其
with 兩␺␩典 ª 12 丢 ␴␩兩␺+2典.
We call lattice states 共LS’s兲 the states diagonal in the
兵兩␺␣␤典其 basis—that is, the mixtures ␳␲ belonging to the convex span of the projectors P␣␤:
and
兵␴ˆ ␤ ª ␴␤/共冑2兲n兩␤ 苸 L共n兲其
are orthonormal Hermitian bases, respectively, in M 2m共C兲
and M 2n共C兲, while
兵␴ˆ ␣␤ ª ␴ˆ ␣ 丢 ␴ˆ ␤兩共␣, ␤兲 苸 L其
共14兲
is an orthonormal Hermitian basis in M 2m共C兲 丢 M 2n共C兲
⯝ M 2N共C兲.
␳␲ ª
兺
共␣,␤兲苸L
␲␣␤ P␣␤,
␲␣␤ 艌 0,
Let us consider the map
⌳␤0 = ⌳1 丢 id2n + id2m 丢 ⌳2 ,
They are a possible generalization of the standard Bell diagonal states 关33兴.
C. Partial transposition of lattice states
共15兲
␳␲T2 ª id2N 丢 T关␳I兴 =
with
冉
兺
␤苸L共n兲\兵␤0其
V␣␤ ª 12N 丢 ␴␣␤V12N 丢 ␴␣␤ .
冊
␴ˆ ␤X␴ˆ ␤ − ␴ˆ ␤0X␴ˆ ␤0 ,
and ␤0 ⫽ 0m, denoting with 0k the k-dimensional null vector
共0 , . . . , 0兲 苸 L共k兲. Since m 艌 n, the map ⌳␤0 satisfies the hypothesis of theorem 3 and is therefore P. Note that id2k关X兴
= ␴0kX␴0k. In the basis 共14兲 the coefficient matrix C⌳␤ is
0
diagonal with eigenvalues
␭␣␤ =
冦
共␣, ␤兲 = 共0m,0n兲,
2,
␣ = 0 m ∧ ␤ ⫽ 0 n, ␤ 0 ,
1,
␣ ⫽ 0m ∧ ␤ = 0n ,
− 1, 共␣, ␤兲 = 共0m, ␤0兲,
0,
␣ ⫽ 0m ∧ ␤ ⫽ 0n .
1,
冧
共16兲
It is therefore clear that, because of our choice of ␤0, the map
⌳␤0 is PnCP. Accordingly to proposition 1, we will show that
it is also nondecomposable, exhibiting a PPT state ␳ such
that D⌳␤ 共␳兲 ⬍ 0.
0
B. Lattice states
共19兲
N
兩␺␣␤典 ª 12N 丢 ␴␣␤兩␺+2 典,
m+n
J␣␤ =
兺 共− 1兲a
a=0
such that
冉
兺
␣
共␥,␦兲苸L␣␤
冦
1
, 共␣, ␤兲 苸 I,
N
␲␣␤ =
I
0, 共␣, ␤兲 苸 I,
共21兲
冧
with I 債 L a subset of L and NI ª card共I兲—that is, states
1
兺 P␣␤ .
NI 共␣,␤兲苸I
共22兲
Such states are completely characterized by a set I 債 L. The
condition of positivity under PT 共21兲 becomes
N
P␣␤ P␥␧ = ␦␣␥␦␤␧ P␣␤ .
The states 兩␺␣␤典, 共␣ , ␤兲 苸 L, are 4N maximally entangled
N
N
states forming an orthonormal basis in C2 丢 C2 . They con-
冊
␲␥␦ 艌 0,
a
where L␣␤
債 L is the set of lattice points 共␥ , ␦兲 such that
exactly a indices among ␥1 , . . . , ␥m , ␦1 , . . . , ␦n are equal to
the corresponding indices among ␣1 , . . . , ␣m , ␤1 , . . . , ␤n.
The previous condition for positive partial transposition is
necessary and sufficient for the whole class of lattice states.
For the sake of simplicity we now focus on a subset of these
states. We will call equidistributed LS’s 共ELS’s兲 the LS’s
such that
␳I =
共17兲
共20兲
Analyzing the spectral decomposition of V␣␤ 共see lemma 2兲
we obtain the following condition for the PPT property of a
lattice state.
Theorem 4. A lattice state remains positive under partial
transposition if and only if, for any lattice site 共␣ , ␤兲 苸 L,
We construct orthogonal one-dimensional projectors
P␣␤ = 兩␺␣␤典具␺␣␤兩,
1
兺 ␲␣␤V␣␤ ,
2N 共␣,␤兲苸L
with
1
⌳1关X兴 = n 兺 ␴ˆ ␣X␴ˆ ␣ ,
2 ␣苸L共m兲
1
2m
␲␣␤ = 1. 共18兲
We now analyze the problem of which ␳␲ are PPT. We
start by operating the partial transposition on P␣␤ in Eq.
共17兲, obtaining
A. Representative class of PnCP maps
⌳2关X兴 =
兺
共␣,␤兲苸L
J␣␤ =
1
a 艌 0,
兺 共− 1兲aNI␣␤
NI a=0
a
a
a
兲 and I␣␤
ª I 艚 L␣␤
.
where NIa = card共I␣␤
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␣␤
PHYSICAL REVIEW A 73, 012345 共2006兲
CLASS OF BOUND ENTANGLED STATES OF N + N…
We now notice the following.
共i兲 共 Na 兲 is the number of different ways to choose a conditions among N—that is, the number of different sets A共a兲.
共ii兲 Given two sets A共a兲, A⬘共a兲, if A共a兲 ⫽ A⬘共a兲, then GA共a兲
and GA⬘共a兲 are disjoint, GA 艚 GA⬘ = 쏗.
共iii兲 Two sets of points GA共a兲, GA⬘共a兲 satisfying different
conditions A共a兲, A⬘共a兲 are mapped one onto the other by
suitable permutations of the indices. Thus, they must contain
the same number of points: card共GA兲 = ga for all A共a兲.
a
In summary, for each a, the set I␣␤
can be split into 共 Na 兲
disjointed sets GA共a兲 each containing ga points. We argue that
ga = 2N−a, a = 0 , . . . , N. This is certainly true for a = N, since
there is only one point satisfying N conditions. Let us suppose the statement is true for a = ã , . . . , N. Then gã−1 is given
by
N−共ã−1兲
gã−1 = 3
N−共ã−1兲
兺
i=1
−
N−共ã−1兲
= 3N−共ã−1兲 −
兺
i=0
冉
冉
N − 共ã − 1兲
i
N − 共ã − 1兲
i
冊
冊
g共ã−1兲+i
2N−共ã−1兲−i + 2N−共ã−1兲
共25兲
= 2N−共ã−1兲 .
FIG. 1. Geometric representation of IC for 共a兲 N = 2 and 共b兲 N
a
= 3. The different levels of gray mark points of different sets I␣␤
to
calculate J␣␤, with 共␣ , ␤兲 苸 IC. The black circle is the element of IC
N
whose coordinates are exactly 共␣ , ␤兲; i.e., it constitutes the set I␣␤
.
0
The open circles correspond to I␣␤.
It is in principle possible to construct all the ELS’s that
are PPT. A similar task has been accomplished in 关34兴 for the
case m = n = 1, N = 2—i.e., for the ELS’s ␳I 苸 S4⫻4. In the
present work, instead, we just show that for any N 艌 2 among
the ELS’s there is at least a PPTES.
We first need the following lemma.
Lemma 1. The ELS described by
IC = 兵共␣, ␤兲 苸 L兩␣i ⫽ 0 ∧ ␤ j ⫽ 0;i = 1, . . . ,m; j = 1, . . . ,n其,
共23兲
NIC = 3N, is positive under partial transposition and J␣␤
艌 1 / NIC for all 共␣ , ␤兲 苸 L.
Proof. In Fig. 1 we represent graphically the set IC for
N = 2 and N = 3 to make the proof easier to understand. Consider first the case 共␣ , ␤兲 苸 IC. Then,
N
J␣␤ =
冉冊
N
1
兺 共− 1兲a a ga ,
3N a=0
The relation between the ga’s written in the first line of Eq.
共25兲 is readily explained: a set of points satisfying ã − 1 conditions 共and no further兲 is given by the number of points
satisfying at least ã − 1 conditions minus all the disjoint sets
satisfying exactly i further conditions chosen among the remaining N − 共ã − 1兲. Therefore,
N
J␣␤ =
共26兲
For the state to be PPT, the condition J␣␤ 艌 0 must hold
for any choice of 共␣ , ␤兲 苸 L. We have already considered the
case 共␣ , ␤兲 苸 IC. We now show that this is the worst case, in
the sense that J␣␤ is the smallest possible. In fact, without
loss of generality, consider the case where k indices
␣i1 , . . . , ␣ik among the indices 共␣ , ␤兲 are equal to zero. Because of our choice of IC, no element 共␥ , ␦兲 of IC will satisfy
any of the corresponding k conditions—e.g., “␥i1 = ␣i1
= 0 ” , . . . , “ ␥ik = ␣ik = 0 . ” This amounts to consider N − k instead of N as the maximum number of conditions that one
element of IC can satisfy in the previous reasoning. We find
共24兲
N Ia =
␣␤
where the coefficient ga is such that 共 Na 兲ga is the number of
a
—that is, of points satisfying exactly a condipoints in I␣␤
tions, as expressed by the ␦’s appearing, for example, in Eq.
共A8兲. To show the validity of Eq. 共24兲, let us denote the
following.
共i兲 A共a兲 a set of a conditions of the form “␣i = ␥i”—that is,
equivalently, of a numbers chosen between 兵1,2, . . . , N其.
共ii兲 GA共a兲 債 IC the set of points satisfying conditions A共a兲
and no further ones.
冉冊
N
1
1
兺 共− 1兲a2N−a = 3N .
3N a=0 a
冦
冉 冊
N−k
a
0,
ga , 0 艋 a 艋 N − k,
N − k ⬍ a 艋 N,
冧
with ga = 2N−k−a3k. Therefore J␣␤ = 1 / 3N−k 艌 1 / 3N.
䊏
We are now able to construct bound entangled states in
S2N⫻2N.
Theorem 5. The state ␳IBE共␤0兲, with IBE共␤0兲
= IC 艛 兵共0m , ␤0兲其, ␤0 ⫽ 0n, and IC given by Eq. 共23兲, is a
PPTES.
Proof. The state is PPT because the sufficient and necessary conditon for positivity under PT of theorem 4 is satis-
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PHYSICAL REVIEW A 73, 012345 共2006兲
MARCO PIANI
fied. For this state NIBE共␤0兲 = 3N + 1. We use the result of
lemma 1 with the slight difference that now all the points in
IBE共␤0兲 have a weight 1 / NIBE共␤0兲 = 1 / 共3N + 1兲. Therefore for
any 共␣ , ␤兲 苸 L the elements in IC contribute at least with
1 / NIBE共␤0兲 to J␣␤. On the other hand, the element 共0m , ␤0兲
苸 IBE共␤0兲 contributes with ±1 / NIBE共␤0兲, the sign depending
on the number of identical indices between 共0m , ␤0兲 and
共␣ , ␤兲. Therefore,
J␣␤ 艌
1
NIBE共␤0兲
±
1
NIBE共␤0兲
艌 0.
We check now that ␳IBE共␤0兲 is also entangled. In fact, Eq. 共14兲
is a diagonal basis for the associated map ⌳␳␲ of any LS ␳␲,
as well as for the map ⌳␤0 of Eq. 共15兲. The eigenvalues of
C⌳␳ are exactly the weights ␲␣,␤, 共␣ , ␤兲 苸 L; in particular,
␲
for ␳IBE共␤0兲 they are
冦
1
, 共␣, ␤兲 苸 IBE共␤0兲,
3 +1
N
0,
共␣, ␤兲 苸 IBE共␤0兲,
冧
while those of ⌳␤0 are listed in Eq. 共16兲. Therefore, in the
case of ␳IBE共␤0兲, we have
1
1
D⌳共␳IBE共␤0兲兲 = N
兺 ␭␣␤ = − 共3N + 1兲 .
3 + 1 共␣,␤兲苸IBE共␤0兲
According to proposition 1, ␳IBE共␤0兲 is entangled and ⌳␤0 is
not decomposable.
䊏
Two examples of sets I describing PPT entangled ELS for
N = 2 and N = 3 are shown in Fig. 2.
Remark 6. Notice that local unitary operations preserve
the properties of any state as regards entanglement and positivity under PT. It is therefore quite evident that our choice of
I is just one among many possible ones. Generalizing the
reasoning in 关34兴, we note that, given two 2N ⫻ 2N unitary
matrices U , W such that W␴␣␤U† = ␴␥␦ up to a phase, we
have
共U* 丢 W兲P␣␤共UT 丢 W†兲
FIG. 2. Geometric representation of two sets IBE共␤0兲 identifying
PPT entangled equidistributed lattice states: 共a兲 m = 1, n = 1, N = 2,
␤0 = 共2兲 and 共b兲 m = 2, n = 1, N = 3, ␤0 = 共2兲. In both cases the black
circle corresponds to the element 共0m , ␤0兲.
nates. Therefore every set that can be transformed into a
␳IBE共␤0兲 by means of the “elementary” operations just described is a PPTES.
It is not necessary to consider sets as symmetric as ␳IBE共␤0兲
to find PPT entangled states using the class of PnCP maps
previously depicted, as the results in 关32,34兴 show. The problem is that, even if we have found a simple condition to
check if a ELS is PPT, this does not mean its application is
straightforward. In this framework, the class of states ␳IBE共␤0兲
appears to be a “canonical” family of PPTES, whose high
symmetry allows an easy check of the PPT condition.
N
= 关12N 丢 共W␴␣␤U†兲兴P+2 关12N 丢 共U␴␣␤W†兲兴
= P ␥␦ .
D. Realignment of lattice states
共27兲
Since the transformation is unitary and thus invertible, it induces a permutation among the elements of L, so that ␳␲⬘
= 共U* 丢 W兲␳␲共UT 丢 W†兲 is another LS with permuted eigenvalues. Let us indicate with ␩i, i = 1 , . . . , N, the elements of
the vector 共␣ , ␤兲. For example, U and W can be chosen such
that
共␩1, . . . , ␩i, . . . , ␩N兲 哫 关␩1, . . . ,p共␩i兲, . . . , ␩N兴,
with p : 兵0,1,2,3其 → 兵0,1,2,3其 a permutation, or such that
共␩1, . . . , ␩i, . . . , ␩ j, . . . , ␩N兲 哫 共␩1, . . . , ␩ j, . . . , ␩i, . . . , ␩N兲.
In the geometric picture the first operation amounts to exchanging two parallel 共N − 1兲-dimensional hyperplanes,
while the second one corresponds to exchanging two coordi-
Partial transposition, even if the most known and used
method to detect entanglement, it is certainly not the only
one. Another test is given, for example, by the realignment
criterion 关23,24兴. Essentially it corresponds to a linear map R
acting on the total state of both subsystems, such that it does
not increase the trace norm of product states; i.e., given the
norm 储X储 = Tr兩X兩, we have 储R共␴A 丢 ␴B兲储 艋 1. In 关25兴 it was
shown that both partial transposition and realignment are
part of a family of permutation criteria, further studied in
关26,27兴. In the bipartite setting the only independent tests in
this family are exactly partial transposition and realignment.
As regards the latter, we will consider the map defined by
R共兩i典具j兩 丢 兩k典具l兩兲 = 兩j典具l兩 丢 兩i典具k兩.
共28兲
The realignment criterion, as partial transposition, is a necessary separability criterion: if ␳ is separable, then the re-
012345-6
PHYSICAL REVIEW A 73, 012345 共2006兲
CLASS OF BOUND ENTANGLED STATES OF N + N…
aligned matrix ␳r = R共␳兲 satisfies 储␳r储 艋 1, so that if 储␳r储 ⬎ 1,
we know that the state is entangled.
Let us consider the action of map 共28兲 on maximally entangled states of the form 1 丢 U兩␺+d 典具␺+d 兩:
1
R共1 丢 U兩␺+d 典具␺+d 兩1 丢 U†兲 = U† 丢 UT .
d
共29兲
Therefore for the states 共17兲 we have
Lr共兩␺␣␤典具␺␣␤兩兲 =
1
1
T
N ␴␣␤ 丢 ␴␣␤ = N ␧␣␧␤␴␣␤ 丢 ␴␣␤ ,
2
2
共30兲
= 共−1兲␦␣,2
m
␧␣ = 兿i=1
␧ ␣i,
with
␧␣
aligned LS corresponds to
␳␲r ª R共␳␲兲 =
共similarly ␧␤兲, so that the re-
1
兺 ␲␣␤␧␣␧␤␴␣␤ 丢 ␴␣␤ .
2N 共␣,␤兲苸L
共31兲
As regards eigenvalues and eigenvectors of a general realigned LS we find
␳␲r 兩␺␥␦典 =
冉
冊
1
兺 ␩␣␥␩␤␦ 兩␺␥␦典,
2N 共␣,␤兲苸L
共32兲
m
␩␣i␥i and ␩␣␥ defined in Eq. 共40兲 共similarly
with ␩␣␥ = 兿i=1
␩␤␦兲.
For a general LS we have
储␳␲r 储 =
1
兺
2N 共␥,␦兲苸L
冏
兺
共␣,␤兲苸L
冏
␲␣␤␩␣␥␩␤␦ ,
共33兲
while for ELS the norm of the realigned state assumes the
following simple expression:
储␳Ir储 =
1
兺
2 NI 共␥,␦兲苸L
N
冏兺
共␣,␤兲苸I
冏
␩␣␥␩␤␦ .
共34兲
We focus on the canonical class of bound entangled states
I = IBE共␤0兲, and for definiteness we consider the case 共␤0兲 j
⫽ 0 for all j = 1 , . . . , n. As proven in the Appendix, we have,
in this case,
储␳Ir储 =
3N + 共− 1兲m
艋 1,
3N + 1
共35兲
so that the bound entangled state is not revealed by the relignment criterion.
V. CONCLUSIONS
A general class of positive but not completely positive
linear maps was found. The decomposability of a subset of
such maps has been studied, exploiting the characterization
of entanglement by means of linear maps. We have established the nondecomposability of such a subfamily and
states
found
examples
of
共2N 丢 2N兲-dimensional
共N 艌 2兲—e.g., states of a bipartite N + N qubits system—
which are PPT but nevertheless entangled. Such states belong to a family of lattice states, which are described by a
distribution probability on a lattice and are diagonal in a
maximally entangled basis. Moreover, the PPT bound entangled states we found are not revealed by the realignment
criterion. The states introduced are potentially interesting to
better understand the phenomenon of bound entanglement; in
particular, lattice states are a possible generalization of the
well-studied Bell diagonal states.
Theorem 3 provides a family of PnCP maps in dimensions
d = d1d2. In the particular case studied, both factor dimensions are powers of 2—i.e., d1 = 2m and d2 = 2n, with m , n
艌 1. The resulting map is applied to detect PPT bound entangled states among lattice states. As explicitly shown, for
lattice states the PPT condition is relatively easy to handle.
We exploit the fact that in dimension 2N a complete orthonormal basis for operators may be built out of Pauli matrices.
Besides being unitary, the latter are Hermitian, behave simply under transposition, and have easy-to-handle commutation rules.
As regards possible generalizations of the results, it is
definitely possible to consider PnCP maps for all product
dimensions d = d1d2. In particular notice that one may take
d1 = 1, so to have PnCP maps in prime dimensions. Unfortunately, in the latter case the structure 共11兲 simplifies too
much. It can be proved that the resulting maps are not useful
to detect PPT bound entanglement, because they turn out to
be decomposable.
It is possible to generalize lattice states for d ⫽ 2N, e.g.,
considering instead of 共normalized兲 Pauli matrices some
other 共orthonormal兲 basis in the space of square matrices of
dimension greater than two. However, as already recalled,
Pauli matrices enjoy nice mathematical properties. Such
properties make the evaluation of partial transposition of the
corresponding lattice states easy and allow to construct immediately the PnCP map needed to detect bound entanglement by following the prescription of theorem 3.
For general d = d1d2, it is difficult to study the behavior of
mixed states under partial transposition to impose the PPT
property, even if we confine ourselves to states diagonal in a
maximally entangled basis. Anyway, the fact that the maps
共15兲 are nondecomposable suggests that also other maps of
the form 共11兲 共and acting on dimensions different than 2N兲
are nondecomposable. Even if, as just explained, it may be
difficult to construct analytically examples of PPT bound entangled 共lattice兲 states, it is anyway possible to look numerically for PPT entangled states detected by the depicted maps.
Indeed, if some PPT state does not remain positive under the
partial action of a given positive map, we know at the same
time that the state is 共bound兲 entangled and the map is nondecomposable.
We plan to study the properties of the considered bound
entangled lattice states with respect to other criteria to detect
and possibly quantify entanglement. Indeed, such states constitute a possible test for the power of any 共new兲 separability
criterion; it is remarkable that they pass both permutational
criteria in the bipartite setting—i.e., partial transposition and
entanglement.
It would be also interesting to check whether the range
criterion 关45兴 is able to detect the non-full-rank states ␳IBE,
considering that only recently non-full-rank bound entangled
states satisfying such criterion have been found 关46兴.
012345-7
MARCO PIANI
PHYSICAL REVIEW A 73, 012345 共2006兲
In 关47兴 bound entangled states of 2N qubits, N 艌 2, were
also studied, but those states are considered to be multipartite; i.e., each qubit is held by a different party; the property
of multipartite activability 关48,49兴 was studied. In our case
there are only two parties, each holding N qubits, so that one
may consider only bipartite activation 关16兴.
Proof of theorem 4
Lemma 2. The spectral decompostion of V␣␤ is
兺
V␣␤ =
共␥,␦兲苸L
⌶␣␥⌶␦␤ P␥␦ ,
with
m
⌶␣␥ = 兿 ⌶␣i␥i,
ACKNOWLEDGMENTS
The author thanks F. Benatti and R. Floreanini for fruitful
discussions and encouragement. This work was initiated during composition of the author’s Ph.D. thesis at the Department of Theoretical Physics of the University of Trieste. The
work is supported by EC grants RESQ, Contract No. IST2001-37559, QUPRODIS, Contract No. IST-2001-38877,
and CNR-NATO.
i=1
N
V␣␤兩␺␥␦典
n
m
= 兿 ␧␣i␧␥i 兿 ␧␤ j␧␦ j12N 丢 关共␴␣␴␥␴␣兲 丢 共␴␤␴␦␴␤兲兴兩␺+2 典
N
j=1
i=1
m
n
i=1
j=1
= 兿 共␧␣i␧␥i␩␣i␥i兲 兿 共␧␤ j␧␦ j␩␤ j␦ j兲兩␺␥␦典,
Proof of theorem 3
where
Proof. We have to check that
␩␣␥ ª
D共␾, ␺兲 ª 具␺兩⌳关兩␾典具␾兩兴兩␺典 艌 0
for all 兩␾典, 兩␺典 苸 Cd1⫻d2, which can both be expanded on a
basis 兵兩i典 丢 兩j典其:
兩␾典 = 兺 兺 ⌽ij兩i典 丢 兩j典,
i=1 j=1
d1 d2
兩␺典 = 兺 兺 ⌿kl兩k典 丢 兩l典,
共A1兲
so that they are determined by the coefficient matrices ⌽ and
⌿. It is straighforward to find the following expression for
D共␾ , ␺兲:
D共␾, ␺兲 = 兺 ␭␮共1兲兩Tr共F␮共1兲⌽⌿†兲兩2 + 兺 ␭␯共2兲兩Tr共F␯共2兲共⌿†⌽兲T兲兩2 .
共A2兲
共2兲 d22−1
Since 兵F␮ 其␮=0 and 兵F␯ 其␮=0 are two orthonormal bases and
Tr共AT兲 = Tr共A兲, we have
Tr共⌽⌿†⌽⌿†兲 = 兺 关Tr共F␮共1兲⌽⌿†兲兴2 = 兺 兵Tr关F␯共2兲共⌿†⌽兲T兴其2
␮
兺
共␥,␦兲苸L
␮
+
兺 兩Tr关F␯共2兲共⌿†⌽兲T兴兩2 .
␯⫽k
兺
−
冋兺
␮
兩Tr共F␮共1兲⌽⌿†兲兩2 +
†
T 2
兩Tr关F共2兲
k 共⌿ ⌽兲 兴兩
册
艌 0,
for all 兩␾典, 兩␺典; therefore, ⌳ is P.
冉
兺
共␥,␦兲苸L
冊
␲␥␦⌶␣␥⌶␤␦ P␣␤
共A6兲
␲␥␦⌶␣␥⌶␤␦ 艌 0 ∀ 共␣, ␤兲 苸 L.
共A7兲
m
n
i=1
j=1
␲␥␦ 兿 共− 1兲␦˜␣i␥i 兿 共− 1兲␦˜␤ j␦ j 艌 0,
whence, since it must hold for all 共␣ , ␤兲, it must be
共A3兲
From the hypothesis of the theorem, the above inequality,
and Eq. 共A2兲 we find
D共␾, ␺兲 艌 兩␭共2兲
k 兩
共A5兲
˜ 兲,
˜ ,␤
We introduce the bijection L → L given by 共␣ , ␤兲 哫 共␣
˜ = 共˜␣1 , . . . , ˜␣m兲 and ␮
˜ ª 共␮ + 2兲 mod 共4兲, 0 艋 ␮ 艋 3. It
where ␣
then follows that ⌶␣␥ = 共−1兲␦␣,˜␥. The PPT condition can
therefore be written as
␥␦苸L
兩Tr共F␮共1兲⌽⌿†兲兩2
冎
and therefore ␳␲T1 is positive if and only if
and, using the triangle inequality, we have
艋兺
1
兺
2N 共␣,␤兲苸L
␳␲T1 =
␯
†
T 2
兩Tr共F共2兲
k 共⌿ ⌽兲 兲兩
␣ = 0 ∨ ␥ = 0 ∨ ␣ = ␥,
− 1, ␣ ⫽ 0 ∧ ␥ ⫽ 0 ∧ ␣ ⫽ ␥ ,
1,
and ␧␣ = 共−1兲␦␣,2. Setting ⌶␥␣ = ␧␣␧␥␩␣␥, the result follows by
direct inspection.
䊏
Proof of theorem 4. Using lemma 2, from Eq. 共19兲 we
have
␯
共1兲 d21−1
再
k=1 l=1
␮
N
Proof. From remark 4 and V兩␺+2 典 = 兩␺+2 典, it follows that
APPENDIX
d1 d2
⌶␣␥ = 共− 1兲␦兩␣−␥兩,2 .
兺 兩Tr关F␯共2兲共⌿†⌽兲T兴兩2
J␣␤ ª
␲␥␦ 兿 共− 1兲
n
␦␣ ␥
i i
i=1
共− 1兲␦
兿
j=1
␤ j␦ j
艌 0.
共A8兲
We now split the sum over ␥ , ␦ into different sums according
to the number of delta conditions that are satisfied; that is,
a
. Explicitly,
we isolate the contributions of different L␣␤
␯⫽k
m+n
J␣␤ =
共A4兲
䊏
兺
␥␦苸L
m
兺 兺
a=0 共␥,␦兲苸La
␣␤
m
n
i=1
j=1
␲␥␦ 兿 共− 1兲␦␣i␥i 兿 共− 1兲␦␤ j␦ j 艌 0.
The theorem follows noticing that
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PHYSICAL REVIEW A 73, 012345 共2006兲
CLASS OF BOUND ENTANGLED STATES OF N + N…
n
m
共␥, ␦兲 苸
a
L␣␤
⇒ 兿 共− 1兲
␦␣ ␥
i i
i=1
共− 1兲␦
兿
j=1
␤ j␦ j
the result of the sum over the remaining nonzero indexes.
The latter is equal to
= 共− 1兲a .
g共k,l兲 = 3m−k兵兩共− 1兲N−共k+l兲3k+l + 1兩f共n − l兲
䊏
+ 兩共− 1兲N−共k+l兲3k+l − 1兩关3n−l − f共n − l兲兴其,
Evaluation of the norm of the realigned canonical bound
entangled state
共A10兲
To evaluate the absolute value in Eq. 共34兲 we observe that
for I = IBE共␤0兲, 共␤0兲 j ⫽ 0 for all j = 1 , . . . , n, given the definition 共A5兲,
兺
共␣,␤兲苸I
冋兿 冉 兺 冊册冋兿 冉 兺 冊册
m
␩␣␥␩␤␦ =
3
␣i=1
i=1
␩ ␣i␥i
n
3
j=1
␤ j=1
␩␥ j␦ j
兩共− 1兲N−共k+l兲3k+l ± 1兩 = 3k+l ± 共− 1兲N−共k+l兲 .
+ ␩␣0␥␩␤0␦ = 共− 1兲N−共k+l兲3k+l + ␩␤0␦ ,
where k and l are the number of indexes ␥i and ␦ j equal to
zero in ␥ and ␦, respectively. Therefore we can split the
outer sum in Eq. 共34兲 according to k and l—i.e., according to
the ways to fix k indexes among m and l indexes among n
equal to zero:
储␳Ir储 =
1
兺 兩共− 1兲N−共k+l兲3k+l + ␩␤0␦兩
2 NI 共␥,␦兲苸L
n
冉 冊冉 冊
m
1
兺
兺
N
2 NI k=0 l=0 k
n
g共k,l兲,
l
f共a + 1兲 = 2关3a − f共a兲兴 + f共a兲 = 2 ⫻ 3a − f共a兲, 共A12兲
because for every fixed ␤ ⫽ 0 there are two possible choices
of ␦ ⫽ 0 such that ␩␤␦ = −1 and one such that ␩␤␦ = 1. Therefore,
f共a兲 = 关3a + 共− 1兲a兴/2.
共A9兲
共A13兲
Finally we find
储␳Ir储 =
where, for any choice of k + l indexes equal to zero, g共k , l兲 is
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