PHYSICAL REVIEW A 74, 022308 共2006兲
Class of positive partial transposition states
Dariusz Chruściński and Andrzej Kossakowski
Institute of Physics, Nicolaus Copernicus University, Grudziadzka 5/7, 87-100 Toruń, Poland
共Received 10 February 2006; published 7 August 2006兲
We construct a class of quantum bipartite d 丢 d states which are positive under partial transposition 共PPT
states兲. This class is invariant under the maximal commutative subgroup of U共d兲 and contains as special cases
many well-known examples of PPT states. States from our class provide criteria for testing the indecomposability of positive maps. Such maps are crucial for constructing entanglement witnesses.
DOI: 10.1103/PhysRevA.74.022308
PACS number共s兲: 03.67.⫺a, 03.65.Ud
The interest on quantum entanglement has dramatically
increased over the last two decades due to the emerging field
of quantum information theory 关1兴. It turns out that quantum
entangled states may be used as basic resources in quantum
information processing and communication, such as quantum
cryptography, quantum teleportation, dense coding, error
correction, and quantum computation.
A fundamental problem in quantum information theory is
to test whether a given state of a composite quantum system
is entangled or separable. Several operational criteria have
been proposed to identify entangled states 关2兴. The most famous Peres-Horodecki criterion 关3,4兴 is based on the partial
transposition: if a state ␳ is separable then its partial transposition 共1 丢 ␶兲␳ is positive. States which are positive under
partial transposition are called PPT states. Clearly each separable state is necessarily PPT but the converse is not true. It
was shown by Horodecki et al. 关5兴 that the PPT condition is
both necessary and sufficient for separability for 2 丢 2 and
2 丢 3 systems.
Now, since all separable states belong to a set of PPT
states, the structure of this set is of primary importance in
quantum information theory. Unfortunately, this structure is
still unknown, that is, one may check whether a given state is
PPT but we do not know how to construct a general quantum
state with PPT property. There are several well-known examples of PPT states. One class contains PPT states which
are separable, e.g., Werner 关6兴 and isotropic states 关7兴. Other
examples present PPT states that are entangled. Actually
there is a systematic method of construction of PPT entangled states which is based on a concept of unextendible
product bases 关8兴 共see also Ref. 关9兴兲. Other examples of PPT
entangled states were constructed in Refs. 关4,10–15兴. PPT
states also play a crucial role in mathematical theory of positive maps and, as is well known, these maps are very important in the study of quantum entanglement. Recently, the
mathematical structure of quantum states with positive
partial transposition were studied in Refs. 关16,17兴.
In the present paper we propose a class of bipartite d
丢 d PPT states. This class is important for the following
reasons. 共i兲 It contains many of the abovementioned examples of PPT states. 共ii兲 We claim that this is the most
general class of PPT states available at the moment. Moreover, unlike other examples it fully uses complex parametrization of density operators. 共iii兲 Finally, it may be used to
study important properties of positive maps, e.g., to test
whether a given positive map is indecomposable and atomic.
1050-2947/2006/74共2兲/022308共4兲
As is well known indecomposable positive maps are crucial
in constructing entanglement witnesses.
The defining property of this class is very simple: it contains bipartite states invariant under the maximal commutative subgroup of U共d兲, i.e., d-dimensional torus Td = U共1兲
⫻ ¯ ⫻ U共1兲. This commutative subgroup is generated by d
mutually commuting operators
t̂k = 兩k典具k兩,
共1兲
k = 1, . . . ,d,
where 兩k典 denotes an orthonormal base in Cd. Now, any
vector x 苸 Rd gives rise to the following element from Td:
共2兲
Ux = e−ix·t̂ ,
where t̂ = 共t̂1 , . . . , t̂d兲. Evidently, UxUy = Ux+y.
In the present paper we consider two classes of bipartite
states each of which is invariant under a representation of Td
on Cd 丢 Cd.
共1兲 Werner-like state, or Ux 丢 Ux-invariant states
U x 丢 U x␳ = ␳ U x 丢 U x .
共2兲 Isotropiclike state, or
Ux 丢 U*x-invariant
Ux 丢 U*x␳ = ␳Ux 丢 U*x ,
共3兲
states
共4兲
for all x 苸 Rd. U*x denotes complex conjugation of Ux in a
fixed basis.
Clearly, these two classes are related by a partial transposition, i.e., a bipartite operator Ô is Ux 丢 U*x invariant iff
共1 丢 ␶兲Ô is Ux 丢 Ux invariant.
The most general state which is Ux 丢 U*x invariant has the
following form:
d
␳=
兺
d
aij兩ii典具jj兩 +
i,j=1
兺
cij兩ij典具ij兩.
共5兲
i⫽j=1
Now, since ␳† = ␳ the matrix â = 储aij储 has to be hermitian and
d2 − d coefficients cij have to be real. Moreover, ␳ is positive
iff
â = 储aij储 艌 0
and cij 艌 0.
共6兲
Finally, normalization Tr ␳ = 1 leads to
Tr â + 兺 cij = 1.
共7兲
i⫽j
Consider now the partial transposition of ␳:
022308-1
©2006 The American Physical Society
PHYSICAL REVIEW A 74, 022308 共2006兲
DARIUSZ CHRUŚCIŃSKI AND ANDRZEJ KOSSAKOWSKI
d
d
兺 cij兩ij典具ij兩.
兺 aij兩ij典具ji兩 + i⫽j=1
共1 丢 ␶兲␳ =
共8兲
i,j=1
Note, that the above formula may be rewritten as follows:
冉
Clearly, W p belongs to a class 共13兲 with
d
共1 丢 ␶兲␳ = 兺 aii兩ii典具ii兩 + 兺 X̂ij ,
i=1
共9兲
bij =
i⬍j
X̂ij = aij兩ij典具ji兩 + a*ij兩ji典具ij兩 + cij兩ij典具ij兩 + c ji兩ji典具ji兩.
X̂ij兩ij典 = cij兩ij典 + aij兩ji典,
X̂ij兩ji典 = a*ij兩ij典 + c ji兩ji典,
艌 0,
cijc ji − 兩aij兩2 艌 0.
共11兲
I=
共12兲
兺 bij兩ij典具ji兩 + i⫽j=1
兺 cij兩ij典具ij兩.
共13兲
i,j=1
Now, positivity of ˜␳ is equivalent to cij 艌 0 and
cijc ji − 兩bij兩2 = 0.
共14兲
1−␭
␭
I 丢 I + 兺 兩ii典具jj兩
d2
d i,j=1
再
␭/d,
␭/d + 共1 − ␭兲/d , i = j.
d
␳bc = a 兺 兩ii典具ii兩 + b
i=1
and cij 艌 0.
where
兺
i⬍j=1
d
兩␺−ij典具␺−ij兩
+c
兩␺+ij典具␺+ij兩,
兺
i⬍j=1
bij =
再
共c − b兲/2, i ⫽ j,
a,
i=j
冎
and cij = 共c + b兲 / 2. Now, cij 艌 0 implies c + b 艌 0 whereas
b̂ 艌 0 gives
共16兲
Examples. Now we show that many well-known examples of
PPT states belong to our class.
共1兲 Werner state 关6兴.
W p = 共1 − p兲Q+ + pQ− ,
d
where 兩␺±ij典 = 共兩ij典 ± 兩ji典兲 / 冑2. Note, that the unit trace condition 共7兲 enables one to compute a in terms of b and c:
a = 1 / d − 共b + c兲 / 共2d − 2兲. Clearly, ␳bc belongs to a Werner
class 共13兲 with
共15兲
is Ux 丢 U*x invariant and, therefore, ˜␳ is PPT iff
b̂ = 储bij储 艌 0
冎
共19兲
d
兺 bij兩ii典具jj兩 + i⫽j=1
兺 cij兩ij典具ij兩,
i,j=1
i ⫽ j,
2
On the other hand, partial transposition
共1 丢 ␶兲˜␳ =
共18兲
Positivity of â together with cij 艌 0 imply −1 / 共d2 − 1兲 艋 ␭
艋 1. PPT condition 共12兲 leads to ␭ 艋 1 / d + 1 which reproduces well known result for PPT property of isotropic states
关7兴.
共3兲 The authors of Ref. 关11兴 considered the following
two-parameter family:
d
d
冎
belongs to a class 共5兲 with cij = 共1 − ␭兲 / d2 and
aij =
There is an evident example of isotropiclike PPT states. Let
␭ជ = 共␭1 , . . . , ␭d兲 be a normalized complex vector. Consider a
state from the class 共5兲 with aij = ␭i␭*j and cij = 兩␭i␭*j 兩. Evidently â 艌 0, cij 艌 0, and cijc ji − 兩aij兩2 = 0. Hence each complex
vector ␭ជ gives rise to a PPT state.
Similarly one may analyze a general Werner-like
Ux 丢 U*x-invariant state
˜␳ =
x− + x+ , i = j,
d
共10兲
which is equivalent to the following condition:
d
i ⫽ j,
Condition 共14兲 implies p 苸 关0 , 1兴. To check condition 共16兲 for
PPT let us observe that the spectrum of b̂ consists of only
two points ␭1 = x+ with multiplicity d − 1 and ␭2 = dx− + x+.
Therefore b̂ is positive iff ␭2 艌 0 which is equivalent to
p 艋 1 / 2 and hence it reproduces well known result for PPT
property of Werner states 关6兴.
共2兲 Isotropic state 关7兴.
and hence X̂ij兩ij典 艌 0 iff
a*ij cij
x− ,
1−p
p
±
.
d2 + d d2 − d
x± =
Since two operators 兺iaii兩ii典具ii兩 and 兺i⬍jX̂ij live in a mutually
orthogonal subspaces of Cd 丢 Cd, the positivity of 共1 丢 ␶兲␳
implies separately 兺iaii兩ii典具ii兩 艌 0, which is equivalent to
aii 艌 0, and 兺i⬍jX̂ij 艌 0. Now, for any pair i ⬍ j an operator
X̂ij acts on a two-dimensional subspace of Cd 丢 Cd spanned
by 兩ij典 and 兩ji典:
冋 册
再
and cij = x+, where
where the operator X̂ij is given by
cij aij
冊
d
1
Q =
I 丢 I ± 兺 兩ij典具ji兩 .
d共d ± 1兲
i,j=1
±
共17兲
1 − d共d − 1兲b 艌 0,
2 − d共d − 2兲b − d2c 艌 0,
which reproduce results of Ref. 关11兴.
共4兲 Horodecki et al. 关12兴 considered the following 3 丢 3
state:
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PHYSICAL REVIEW A 74, 022308 共2006兲
CLASS OF POSITIVE PARTIAL TRANSPOSITION STATES
2
5−␣
␣
␴␣ = P+ + ␴+ +
␴ −,
7
7
7
2 艋 ␣ 艋 5,
ui = 兩ii典,
共20兲
1
zi = 兩i + 1,i典 + s兩i,i + 1典,
s
where P+ denotes a projector onto the canonical maximally
entangled state and
with s ⬎ 0 and i = 1 , . . . , d. Define the following family of
positive d2 ⫻ d2 matrices
1
␴+ = 共兩01典具01兩 + 兩12典具12兩 + 兩20典具20兩兲,
3
Bs = 兺 共兩ui典具ui兩 + 兩zi典具zi兩兲.
d
Observe, that Eq. 共22兲 belongs to a Werner-like class 共13兲
with
1
␴− = 共兩10典具10兩 + 兩21典具21兩 + 兩02典具02兩兲.
3
Clearly, Eq. 共20兲 belongs to isotropiclike class 共5兲 with aij
= 2 / 21, c01 = c12 = c20 = ␣ / 21 and c10 = c21 = c02 = 共5 − ␣兲 / 21.
One easily finds that PPT condition 共12兲 reproduces well
known fact that Eq. 共20兲 is PPT for ␣ 艋 4. Recently, a oneparameter family 共20兲 was generalized for d 丢 d systems as
follows 关15兴:
d
d
aj
a1
␳ = P+ + 兺 兺 兩i,i + j − 1典具i,i + j − 1兩,
d
i=1 j=2 d
共21兲
where the positive numbers ai satisfy 兺iai = 1. Clearly, it belongs to an isotropiclike family 共5兲. If ai+1ad−i+1 艌 a21 then the
state is PPT.
共5兲 Bound entangled states considered in Ref. 关13兴 belong
to our class 共5兲 with aij = 1.
共6兲 Størmer state. Størmer 关18兴 analyzed an
un-normalized 3 丢 3 positive PPT matrix with aij = 1 and
cij = 2␮,
i ⬍ j;
cij = 1/2␮,
i ⬍ j;
cij = ␣2c−1
ji ,
␥2 + d − 1
,
d
and the remaining bij and cij vanish. Note, that PPT condition 共16兲 is trivially satisfied.
共8兲 PPT states which do not belong to our class. It turns
out that apart from bound entangled states constructed via
unextendible product bases 关8兴 almost all other examples of
PPT states do belong to our class. We are aware of only few
exceptions: one is the family of O 丢 O-invariant states 关19兴
共see also Ref. 关24兴兲 and the second one is the celebrated
family of 3 丢 3 states which are nonseparable but PPT
constructed by Horodecki 关4兴. Note that the d-dimensional
torus U共1兲 ⫻ ¯ ⫻ U共1兲 does not allow for an orthogonal
subgroup and hence states with orthogonal symmetry has to
be considered separately. Now, the Horodecki state ␳a may
be rewritten as ␳a = ␳a⬘ + ␳a⬙, where
冉兺
with ␥ ⬎ 0. Now, conditions for positivity 共6兲 are trivially
satisfied. Moreover, due to ␭␭⬘ 艌 1, the PPT condition 共12兲 is
also satisfied which shows that Ha’s family is PPT.
Another example constructed in Ref. 关16兴 is a family of
共un-normalized兲 3 丢 3 positive PPT matrices but the construction may be generalized to an arbitrary d as follows. Let
兺
兩ij典具ij兩
i⫽j=1
冊
and
␳a⬙ =
␥−2 + d − 1
,
d
3
兩ii典具jj兩 +
i,j=1
i ⬎ j,
ci,i 丣 1 = ␭⬘ ,
␭⬘ =
ci 丣 1,i = s−2 ,
3
and the remaining cij = 1. In the above formula “ 丣 ” denotes
addition modulo d, and
␭=
ci,i 丣 1 = s2,
␳a⬘ = ␣a
where ␣ ⬎ 0 is a normalization constant. One has cijc ji
= 兩aij兩2, and Eq. 共12兲 implies that the corresponding state is
PPT.
共7兲 Ha 关16兴 performed very sophisticated construction of a
one-parameter family of d2 ⫻ d2 共un-normalized兲 positive
matrices and showed that this family remains positive after
performing partial transposition. It turns that Ha’s family is a
special example of an isotropiclike class 共5兲 with aij = 1 and
ci 丣 1,i = ␭,
bii = bi,i 丣 1 = bi 丣 1,i = 1,
i ⬎ j.
This example may be immediately generalized for d 丢 d case
as follows: aij = ␣ and
cij ⬎ 0,
共22兲
i=1
␣冑
1 − a2共兩31典具33兩 + 兩33典具31兩兲,
2
with ␣ = 1 / 共8a + 1兲 being a normalization constant. Note, that
␳a⬘ is an isotropiclike matrix and does belong to Eq. 共5兲.
However, ␳a⬙ is not invariant under the maximal commutative
subgroup of U共3兲. Note, that ␳a⬙ is invariant only under the
one-parameter subgroup generated by t̂2 = 兩2典具2兩. It shows
that Horodecki state ␳a is also symmetric but with respect to
smaller symmetry group.
Separability. A state from a class 共5兲 is separable iff
there exists a separable state ␴ such that ␳ = P␴, where P
denotes a projector operator projecting an arbitrary state
onto the class 共5兲, i.e., P␴ belongs to Eq. 共5兲 with “pseudofidelities” aij = Tr共␴兩ii典具jj兩兲 and cij = Tr共␴兩ij典具ij兩兲. Taking
␴ = 兩␣ 丢 ␤典具␣ 丢 ␤兩 one finds the following sufficient condition
for separability:
aij = ␣*i ␣ j␤*i ␤ j,
cij = 兩␣i兩2兩␤ j兩2 ,
共23兲
where ␣i = 具i 兩 ␣典 and ␤i = 具i 兩 ␤典.
Positive maps. PPT states are also important in the study
of positive maps 关16,18,20兴 共see also Ref. 关12兴 for a useful
review兲. It has been shown 关5兴 that there exists a strong
connection between the classification of the entanglement of
022308-3
PHYSICAL REVIEW A 74, 022308 共2006兲
DARIUSZ CHRUŚCIŃSKI AND ANDRZEJ KOSSAKOWSKI
quantum states and the structure of positive linear maps: a
state ␳ living in Cd 丢 Cd is separable iff 共1 丢 ⌽兲␳ 艌 0 for all
positive maps ⌽ : M d → M d, where M d denotes the set of d
⫻ d complex matrices. Now, let Vk be the cone of positive
matrices A 苸 共M d 丢 M d兲+ such that Schmidt number Sch共A兲
艋 k 关21兴. Now, one says that A belongs to a cone Vl iff A is
PPT and SN关共1 丢 ␶兲A兴 艋 l. It is clear that V1 = V1 defines a
cone of separable elements. Recall that a positive map
⌽ : M d → M d is k positive iff 共1 丢 ⌽兲 is positive when restricted to Vk. Similarly, ⌽ is k copositive iff 共1 丢 ⌽ ⴰ ␶兲 is
positive on Vk. Clearly, k-positive map is necessarily l positive for l ⬍ k 共the same is true for k copositivity兲. If k = d then
the d-positive 共d-copositive兲 map is called completely
positive 共completely copositive兲.
Concerning the PPT states the crucial role is played by so
called indecomposable maps: a linear positive map is indecomposable iff it cannot be decomposed into the sum of
completely positive and completely copositive maps. The
most basic class of indecomposable maps consists of socalled atomic ones 关22兴—⌽ is atomic iff it cannot be decomposed into the sum of two-positive and two-copositive maps.
Atomic maps posses the “weakest” positivity property and
hence may be used to detect the bipartite states with the
“weakest” entanglement, i.e., states from V2 艚 V2. Conversely, PPT states may be used to check the atomic property
of positive maps. Suppose that we are given an indecomposable positive map ⌽. If for some A 苸 V2 艚 V2 one finds that
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共1 丢 ⌽兲A is not positive then ⌽ is necessarily atomic. Now, it
would be interesting to know when PPT states from our class
belong to V2 艚 V2. Consider, e.g., a state ␳ from an isotropiclike class 共5兲. Note, that if ␳ is PPT then, due to Eq. 共10兲, ␳
necessarily belongs to V2. Hence, it is enough to check when
␳ 苸 V2. It is clear that ␳ 苸 V2 iff there is ␴⬘ 苸 V2 such that
␳ = P␴⬘. Taking ␴⬘ = 21 兩␣ 丢 ␤ + ␺ 丢 ␾典具␣ 丢 ␤ + ␺ 丢 ␾兩 one finds
the following sufficient condition for ␳ to be an element from
a cone V2:
1
aij = 共␣*i ␤*i 关␣ j␤ j + ␺ j␾ j兴 + ␺*i ␾*i 关␣ j␤ j + ␺ j␾ j兴兲,
2
1
cij = 共␣i␤ j关␣*i ␤*j + ␺*i ␾*j 兴 + ␺i␾ j关␣*i ␤*j + ␺*i ␾*j 兴兲,
2
where ␣i = 具i 兩 ␣典 and similarly for ␤i, ␺i and ␾i. Interestingly,
any Werner-like state from 共13兲 belongs to V2. Hence it suffices to check wether it belongs to V2. One may easily derive
sufficient conditions for bij and cij in analogy to the above
conditions for aij and cij. It would be interesting to generalize PPT states analyzed in this paper to multipartite case
following the construction presented recently in Ref. 关23,24兴.
This work was partially supported by the Polish
State Committee for Scientific Research Grant “Informatyka
i inżynieria kwantowa” No. PBZ-Min-008/P03/03.
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
关20兴
关21兴
关22兴
关23兴
关24兴
022308-4
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