Every dXd bipartite system is shown to have a large family of undistillable
states with nonpositive partial transpose (NPPT). This family subsumes the
family of conjectured NPPT bound entangled Werner states. In particular, all
one-copy undistillable NPPT Werner states are shown to be bound entangled.Comment: 4 pages, Revte

Simon, Rajiah

English

arXiv

Every dXd bipartite system is shown to have a large family of undistillable states with nonpositive partial transpose (NPPT). This family subsumes the family of conjectured NPPT bound entangled Werner states. In particular, all one-copy undistillable NPPT Werner states are shown to be bound entangled

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NPPT Bound Entanglement Exists
Rajiah Simon∗
The Institute of Mathematical Sciences, C. I. T. campus, Taramani, Chennai - 600113, India
(Dated: August 30, 2006)
Every d×d bipartite system is shown to have a large family of undistillable states with nonpositive
partial transpose (NPPT). This family subsumes the family of conjectured NPPT bound entangled
Werner states. In particular, all one-copy undistillable NPPT Werner states are shown to be bound
entangled.
PACS numbers: 03.67.Mn, 03.67.-a, 03.67.Pp
Introduction: Maximally entangled bipartite states
form an indispensable resource in several quantum in-
formation processing situations [1, 2, 3, 4]. In reality,
however, readily available states may be mixed and less
than maximally entangled due to a variety of reasons in-
cluding influence of the environment. Thus distillation,
the process by which some copies of (nearly) maximally
entangled pure states are extracted from several copies
of partially entangled mixed states using local quantum
operations and classical communication (LOCC), is of
singular importance.
Rapid progress has been achieved in this regard, be-
ginning with the works of Popescu [5], Bennett et al. [6],
Deutsch et al. [7], and Gisin [8, 9]. An important question
that presented itself at an early stage of this development
was this: Can every entangled state be distilled? The
Horodecki family answered this question affirmatively in
a significant particular case: Every inseparable state of
a pair of qubits can, given sufficiently many copies, be
distilled into a singlet [10].
Subsequently they proved a result that applies to arbi-
trary m×n bipartite systems [11]: A state that does not
violate the Peres-Horodecki PPT (positive partial trans-
pose) criterion [12, 13] can not be distilled. Since in-
separable PPT states can not be distilled, they are said
to possess bound (i.e., undistillable) entanglement. The
first examples of such undistillably entangled PPT states
were constructed by Horodecki [14], and several families
of PPT bound-entangled states have been presented since
then. The role PPT bound-entanglement could play as a
quantum resource has also been studied.
The Problem of NPPT Bound Entanglement:
Given a bipartite state ρ, let (ρTB )⊗n = (ρ⊗n)TB denote
the partial transpose of the state of n identical copies.
Then the necessary and sufficient condition for distilla-
bility of ρ is that the inequality [11]
〈ψ| (ρTB)⊗n |ψ〉 ≥ 0, (1)
be violated by a Schmidt rank two vector |ψ〉 in the n-
copy Hilbert space, for some n. Thus in order to be dis-
tillable, the state should be NPPT (i.e., the state should
have non-positive partial transpose). What has remained
an open problem is this: Are there undistillable NPPT
states? Does undistillability imply PPT [15]?
Clearly, violation of the above inequality for a partic-
ular n = n0 implies violation for all n > n0. Given an
NPPT state ρ, if it satisfies this inequality for all Schmidt
rank two states, for a particular n, we say that the state
is n-copy undistillable or, equivalently, that the partial
transpose ρTB is n-copy 2-positive. Thus ρ is undistill-
able if ρTB is n-copy 2-positive for all n.
While Horodecki et al. pointed out in a subsequent
work [16] that NPPT bound entangled states, if they exist
in nature, should be found among the one-copy undistil-
lable NPPT Werner states [17], it may be fair to say that
this problem attained its present state of fame only when
two leading groups produced independently, and about
the same time, analytical and numerical evidence [18, 19]
for its existence in the context of the Werner family of
states [More recent numerical attempt has been under-
taken in Ref. [20]]. As the analytic part of the evidence
it was shown, for every n, that there is a corresponding
range of parameter values over which the NPPT Werner
state remains provably n-copy undistillable. But the
range itself becomes rapidly smaller with increasing n,
and was not proved to remain nonzero as n → ∞. It
should be added that no one-copy undistillable NPPT
Werner state was shown to be n-copy distillable either.
Similar evidence in the multipartite case has been con-
sidered by explicit construction of n-copy undistillable
NPPT states, for every n ≥ 1 [21].
It should be noted for completeness that some systems
are known not to support NPPT bound entangled states
at all. These include 2 × n dimensional systems [10, 19]
and all bipartite Gaussian states [22].
Shor et al. [23] have used the conjectured existence
of NPPT bound entanglement to prove nonadditivity
of bipartite distillable entanglement. And Eggeling et
al. [24] have related the existence of NPPT bound entan-
glement to the connection between the sets of separable
superoperators and PPT-preserving channels. Further,
Vollbrecht and Wolf [25] have shown that an additional
resource in the form of infinitesimal amount of PPT
bound-entanglement can render any one-copy undistill-
able NPPT state one-copy distillable.
In a more recent paper Watrous [26] has constructed,
for every n ≥ 1, states which are n-copy undistillable, and
yet are distillable. This is a surprising and important
2result, for it shows that n-copy undistillability, even if
n is very large, does not by itself prove undistillability.
The burden this finding places on numerical evidence for
NPPT bound entanglement is evident.
We conclude this brief summary of the present status
of the conjectured existence of NPPT bound entangle-
ment by noting that the conjecture itself seems to enjoy
the confidence of researchers in quantum information the-
ory, even though the evidence presented so far has been
assessed differently by different authors [23, 24, 26].
In this Letter we prove that any d × d bipartite sys-
tem with d ≥ 3 has a fairly large family of NPPT states
which are undistillable. As will be seen, this family is
much larger than the Werner family of conjectured NPPT
bound entangled states, but it turns out to be convenient
to begin our proof with the Werner family.
Proof of Existence of NPPT Bound Entangle-
ment: We may define the one-parameter family of
Werner state ρα in d × d dimensions through the par-
tial transpose of ρα:
ρTBα = Id− dαP. (2)
Here P is the projection on the standard maximally en-
tangled state:
P = |Ψ〉〈Ψ|, |Ψ〉 = 1√
d
d∑
k=1
|k, k〉. (3)
These states as defined are not normalized to unit trace,
but this does in no way affect our considerations below.
Nonnegativity of ρα forces on α the restriction −1 ≤
α ≤ 1. This allowed range for α divides into three inter-
esting regions [18, 19]:
− 1 ≤ α ≤ 1
d
, PPT, separable,
1
2
< α ≤ 1, NPPT, one−copy distillable,
1
d
< α ≤ 1
2
, NPPT, one−copy undistillable. (4)
Clearly, it is the last mentioned range which is of interest
for the issue on hand. In this range ρα is NPPT, yet ρα
TB
is 2-positive. Since ρα being NPPT implies that ρ
⊗n
α is
NPPT, for all n, the issue really is whether (ρα
TB )⊗n is
2-positive as well. We show that it indeed is, for all n.
Let us denote by S(1) the collection of all pure and
mixed states of Schmidt rank (number) ≤ 2 [27]. This is a
convex set whose extremals are all pure states of Schmidt
rank 1 or 2. Let θ = (θ1, θ2, · · · , θd) be a d-tuple of angles,
and consider the subgroup of U(d), the d-dimensional
unitary group, consisting of diagonal matrices Uθ:
(Uθ)kl = δkle
iθk . (5)
This is the standard maximal abelian subgroup UA of
U(d). Now average each element σ of S(1) over the local
group UA ⊗ U∗A:
σ ∈ S(1) → σ′ =
∫
dθ Uθ ⊗ U∗θ σ U †θ ⊗ UTθ . (6)
Here dθ = dθ1dθ2 · · ·dθd, with θk running over the in-
terval [0, 2pi], independently for every k. This group-
averaging process is analogous to the diagonal twirl op-
eration [18]. The resulting images of elements of S(1) con-
stitute a new convex set Ω(1), where the superscript on
S and Ω reflects our intention to extend these consider-
ations to several copies of the Werner state.
The reduction in complexity achieved by going from
S(1) to Ω(1) should be appreciated. For instance, whereas
S(1) has a CP d−1 × CP d−1 worth of product states
among its extremals, the product states among the ex-
tremals of Ω(1) are precisely d2 in number; these are the
standard computational basis states |k, l〉. Similarly the
pure states of Schmidt rank = 2 among the extremals of
Ω(1) necessarily have one of the forms α1|11〉 + α2|22〉,
β2|22〉+ β3|33〉, and γ1|11〉+ γ3|33〉; each of these three
sets is a Bloch-sphere in worth. And, therefore the maxi-
mally entangled rank-two states are precisely three circles
worth in number, being of the form 1√
2
(|11〉 + eiδ1 |22〉),
1√
2
(|22〉+ eiδ2 |33〉), or 1√
2
(|33〉+ eiδ3 |11〉).
Now 2-positivity of ρTBα is equivalent to the demand
that Tr(ρTBσ) ≥ 0, for all σ ∈ S(1). In view of the
UA ⊗ U∗A symmetry of ρTB , this is equivalent to the de-
mand that Tr(ρTBσ) ≥ 0, for all σ ∈ Ω(1) and in view
of the convexity of Ω(1), it is sufficient for the extremals
of Ω(1) to meet this demand. This convexity argment is
in essence a recognition of the fact that the real-valued
expression Tr(ρTBα sigma) is linear in σ.
We thus find that the minimum of Tr(ρTBσ) over Ω(1)
equals (1 − 2α), showing that ρTB is 2-positive for all
α ≤ 12 . The minimum is achieved by 1√2 (|11〉 + |22〉),
1√
2
(|22〉 + |33〉), and 1√
2
(|33〉 + |11〉), and by no other
states.
We may note in passing that the local symmetry of
ρTB is not UA⊗U∗A, but the full group U(d)⊗U(d)∗. We
wish to go beyond the Werner states later in this Letter,
and for this reason we have based our analysis on the
subgroup UA ⊗ U∗A, rather than on the full group.
We now move on to consider ρα ⊗ ρα, two copies of
the Werner state ρα. The set S(2) is constructed as
the collection of (d2 × d2)-dimensional states of Schmidt
number ≤ 2. From the convex set S(2) we obtain
Ω(2) by averaging each σ ∈ S(2) over the local group
(UA ⊗ U∗A) ⊗ (UA ⊗ U∗A). It is to be understood that
the first (UA ⊗ U∗A) factor acts on the Hilbert space of
the first copy and the second on that of the second copy,
independently.
The extremals of Ω(2) are readily enumerated. The
rank-1 extremals of Ω(2) are necessarily of the form
|k, l〉1 ⊗ |i, j〉2. They are d4 in number. The rank two
3states are of two types.
Type I : (α|kk〉1 + β|ll〉1)⊗ |ij〉2,
|kl〉1 ⊗ (α|ii〉2 + β|jj〉2);
Type II : α|kk〉1 ⊗ |ii〉2 + β|ll〉1 ⊗ |jj〉2. (7)
At the risk of sounding repetitive, we emphasize that this
is a complete enumeration of the extremals of Ω(2).
As with the single copy case 2-positivity of ρα ⊗ ρα is
equivalent to the demand Tr[(ρTBα ⊗ ρTBα )σ] ≥ 0, for all
σ ∈ Ω(2), which in turn is equivalent to the demand that
this condition be met by all the extremal states of Ω(2).
The type-I rank-2 states being products across the two
copies, cannot bring out genuinely two-copy properties if
any, and thus we are left with only the type-II states to
examine:
〈ΨII |ρTBα ⊗ ρTBα |ΨII〉
= |α|2〈kk|ρTBα |kk〉〈ii|ρTBα |ii〉
+|β|2〈ll|ρTBα |ll〉〈jj|ρTBα |jj〉
+α∗β〈kk|ρTBα |ll〉〈ii|ρTBα |jj〉
+αβ∗〈ll|ρTBα |kk〉〈jj|ρTBα |ii〉. (8)
Now, 2-positivity of ρTB is equivalent to the Schwartz
inequality
〈kk|ρTBα |ll〉 ≤ [〈kk|ρTBα |kk〉〈ll|ρTBα |ll〉]
1
2 , (9)
for rank-1 states. Use of this inequality in Eq.(8) proves
〈ΨII |ρTBα ⊗ ρTBα |ΨII〉 ≥ 0. (10)
That is one-copy 2-positivity of the (UA ⊗U∗A) invariant
operator ρTB implies its two-copy 2-positivity. And we
have proved
Theorem 1: All one-copy undistillable Werner states,
that is all ρα’s in the entire range
1
d
≤ α ≤ 12 , are two-
copy undistillable.
To move on to the n-copy case, assume that ρTBα is
(n − 1)-copy 2-positive. That is 〈ψ(ρTBα )⊗(n−1)|ψ〉 ≥ 0
for all rank-2 states of the (n−1)-copy Hilbert space. We
wish to prove that this implies (ρTBα )
⊗(n)
is 2-positive.
To this end, form Ω(n) by averaging each σ ∈ S(n) over
the local group (UA ⊗ U∗A)⊗n, with one (UA⊗U∗A) factor
acting on the Hilbert space of each copy independently.
Again, the extremals of Ω(n) consist of rank-1 and rank-2
pure states. The rank-1 states are of the form |i1, j1〉 ⊗
|i2, j2〉 ⊗ ....|in, jn〉. That is, these are tensor products of
computational basis states, one picked from each copy,
and thus are d 2n in number.
The rank-2 extremal (or pure) states in Ω(n) are of two
types, as in the two-copy case:
Type I : |ΨI〉 = |ψ〉 ⊗ |i, j〉,
Type II : |ΨII〉 = |φ1〉 ⊗ |i, i〉+ |φ2〉 ⊗ |j, j〉, (11)
where |ψ〉 is a (n − 1)-copy rank-2 state, and |φ1〉, |φ2〉
are (n − 1)-copy rank-1 states. Since type-I states have
a product structure across the copies, nonnegative ex-
pectation values of (ρTBα )
⊗n in respect of type-I rank-2
states follows directly from the assumed (n − 1)-copy 2-
positivity, the additional copy offering nothing new in
the type-I case. The same (n − 1)-copy 2-positivity is
equivalent to the validity of the Schwartz inequality for
(n − 1)-copy rank-1 states, and this in turn implies the
nonnegativity of the expectation values of (ρTBα )
⊗n for
type-II states. We have thus proved, by induction,
Theorem 2: The one-copy 2-positive (ρTBα )
⊗n’s in the
entire parameter range 1
d
≤ α ≤ 12 are n-copy 2-positive,
for all n. That is, these one-copy undistillable NPPT
Werner states ρα are n-copy undistillable, for all n, and
hence are bound entangled.
Finally, it should be evident that the only property
of ρα, apart from its NPPT and one-copy 2-positivity
properties, used in our analysis is its (UA⊗UA) invariance
or, equivalently, the (UA ⊗ U∗A) invariance of its partial
transpose. It follows that our conclusions apply to all
states with these properties. That is,
Theorem 3: Every one-copy undistillable NPPT state
in d × d dimensions is bound entangled if it possesses
(UA ⊗ UA) symmetry.
This is the main result of this Letter. It shows that
the family of NPPT bound entangled states in d × d di-
mensions, for any d ≥ 3, is much larger than what might
have been anticipated. This point is worth illustrating.
It is easily seen that in 3× 3 dimensions the most gen-
eral (UA ⊗ U∗A) invariant ρTB has the form
ρTB =


ρ11 0 0 0 −z12 0 0 0 −z∗31
0 ρ12 0 0 0 0 0 0 0
0 0 ρ13 0 0 0 0 0 0
0 0 0 ρ21 0 0 0 0 0
−z∗12 0 0 0 ρ22 0 0 0 −z23
0 0 0 0 0 ρ23 0 0 0
0 0 0 0 0 0 ρ31 0 0
0 0 0 0 0 0 0 ρ32 0
−z31 0 0 0 −z∗23 0 0 0 ρ33


Clearly, ρ has to be positive semidefinite in order to be
a valid density matrix. This demand is equivalent to the
conditions: (i) all the diagonal elements ρij ≥ 0, and
ρijρji ≥ |zij |2 for all i 6= j. The NPPT requirement
demands that the 3× 3 submatrix

 ρ11 −z12 −z
∗
31
−z∗12 ρ22 −z23
−z31 −z∗23 ρ33


should be nonpositive, and the 2-positivity demand is
equivalent to the three inequalities ρ11ρ22 ≥ |z12|2,
ρ22ρ33 ≥ |z23|2, and ρ33ρ11 ≥ |z31|2. The NPPT demand
and the 2-positivity demands thus involve only the six
parameters ρ11, ρ22, ρ33 and z12, z23, z31. The phases of
4the complex z-parameters can be tuned by (local) change
of phases of the basis vectors on the A and B sides (to
be precise it is sufficient to carry out the changes on one
side only), but the argument of z12 z23 z31 is invariant
under such gauge transformations.
Thus our family of NPPT bound entangled states in
3× 3 involves, when normalized to unit trace, 12 param-
eters; eight coming from the diagonals ρij , three coming
from the magnitudes of the z-parameters, and one gauge-
invariant phase. These are canonical parameters, and do
not take into consideration parameters arising from local
unitary transformations.
Acknowledgement: I would like to thank Sibasish
Ghosh, Solomon Ivan, Guruprasad Kar, and Anirban
Roy for many discussions on the problem of NPPT bound
entanglement.
∗ Electronic address: simon@imsc.res.in
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NPPT Bound Entanglement Exists

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