PHYSICAL REVIEW A 71, 012316 共2005兲
Non-full-rank bound entangled states satisfying the range criterion
Somshubhro Bandyopadhyay,* Sibasish Ghosh,† and Vwani Roychowdhury‡
Department of Electrical Engineering, University of California, Los Angeles, California 90095, USA
共Received 11 June 2004; published 12 January 2005兲
A systematic method for generating bound entangled states in any bipartite system, with ranks ranging from
five to full rank, is presented. These states are constructed by mixing separable states with unextendable
product basis–generated bound entangled states that are positive under partial transposition 共PPT兲. A subset of
this class of PPT bound entangled states, having less than full rank, is shown to satisfy the range criterion 关P.
Horodecki, Phys. Lett. A 232, 333 共1997兲兴.
DOI: 10.1103/PhysRevA.71.012316
PACS number共s兲: 03.67.Hk, 03.65.⫺w, 89.70.⫹c
I. INTRODUCTION
One of the recent fundamental advances in quantum information theory 关1兴, and in particular in the theory of quantum entanglement 关2兴, is the discovery of bound entangled
states 关3兴: the mixed entangled states from which no pure
entanglement can be obtained by local operations and classical communication, whatever be the number of copies of
the state being shared. Bound entangled 共BE兲 states have
been studied extensively in the recent past 关4–6兴, and the
primary focus has been on obtaining succinct characterizations of bound entangled states, on deriving appropriate tools
to identify bound entanglement, and on enumerating possible
applications of bound entangled states, if any, for quantum
information processing purposes. A comprehensive understanding of BE states, however, still remains elusive. For
example, while a few systematic procedures for constructing
BE states that are positive under partial transposition 共i.e.,
PPT BE states兲 have been presented 关4–6兴, the relative abundance and distribution of PPT BE states in the Hilbert space
is still not completely understood.
Perhaps the main difficulty in studying bound entangled
states is related to its identification. The problem is complicated by the fact that most bound entangled states are positive under partial transposition, like any separable state; the
existence of BE states that might be negative under partial
transposition 共NPT兲 has only been conjectured 关7兴. Thus a
major challenge in identifying and characterizing BE states
concerns itself with the question of whether a given PPT
state is separable or inseparable. In general, despite recent
efforts 关8兴, there are no succinct criteria or efficient computational tools that would determine a separable decomposition of any given PPT state, if it exists, or otherwise would
indicate that no such decomposition is possible. An ingenious technique to get around this hurdle is based on studying the range of the state under consideration 关3兴. Recall that
*Present address: Department of Chemistry, University of Toronto, 80 St. George St., Toronto, ON, Canada M5S 3H6. Electronic
address: som@ee.ucla.edu
†
Present address: Department of Computer Science, The University of York, Heslington, York, YO10 5DD, U.K. Electronic address: sibasish@cs.york.ac.uk
‡
Electronic address: vwani@ee.ucla.edu
1050-2947/2005/71共1兲/012316共6兲/$23.00
the range of a Hermitian operator is the space spanned by the
eigenvectors corresponding to the nonzero eigenvalues. The
range criterion 共RC兲 of separability can be stated as follows:
If a state ␳AB acting on a Hilbert space is separable, then
there exists a family of product vectors 兩␺i典A 丢 兩␾i典B such that
共a兲 they span the range of ␳AB, and 共b兲 the vectors 兩␺i典A
*
丢 兩␾i 典B span the range of ␳TB 共where the superscript TB represents the partial transposition operation with respect to
party B, and the asterisk denotes the complex conjugation in
the basis where the partial transposition was performed兲. In
particular, 兩␺i典 丢 兩␾i典 belongs to the range of ␳. It is to be
mentioned here that the separability problem 共i.e., to test
whether any given state of a composite system is separable兲
has been shown to be NP-hard 关9兴. Recently Doherty et al.
关10兴 provided a complete family of separability criteria for
detecting whether a given state of a composite system is
separable or entangled. This method is based on the possibility of extending the state to a state of a larger number of
parties, satisfying some symmetry conditions. For all separable states, such extensions are always possible, but if a
state is entangled, it will definitely lack this possibility—a
fact that can be detected after finitely many steps in the hierarchy of separability criteria. This extension method can be
cast as a semidefinite programming 关11兴. Also it has been
shown recently by Perez-Garcia 关12兴 that the cross-norm
characterization of separability, a necessary and sufficient
criterion for testing separability 共given by Rudolph 关13兴兲, can
be reduced to a linear programming problem, for fixed chosen error.
The RC is of course a necessary condition for separability,
but if a state violates the criterion it must be entangled. Most
systematic procedures for constructing PPT BE states presented so far are based on showing that the underlying PPT
states violate the RC. For example, the first systematic way
to construct PPT BE states was provided in Ref. 关5兴 based on
the concept of unextendable product bases 共UPB’s 兲, where
the BE states violate the RC in an extreme way, in the sense
that there are no product states in its range. The range criterion, however, cannot always be applied. If the given state is
of full rank then it trivially satisfies the range criterion. Indeed, in Ref. 关14兴 the authors constructed a class of full-rank
PPT states in 3 丢 3 quantum systems, which are entangled. In
fact, all classes of bound entangled states that have been
obtained so far either 共1兲 violate the RC and are of less than
full rank or 共2兲 are of full rank 共the only such known class is
the one in 3 丢 3, mentioned above兲.
012316-1
©2005 The American Physical Society
PHYSICAL REVIEW A 71, 012316 共2005兲
BANDYOPADHYAY, GHOSH, AND ROYCHOWDHURY
One question is immediate: Do PPT BE states exist that
do not have full rank but nevertheless satisfy the RC? Moreover, can one find a systematic procedure to obtain PPT BE
states that satisfy the RC?
The answers to the above questions are not immediately
clear. It is also not known whether there are full-rank PPT
BE states in any d 丢 d system. In our effort to identify PPT
BE states that satisfy the RC, the theory of nondecomposable
positive maps and entanglement witness turns out to be extremely useful. The witness operators to detect bound entanglement were first introduced for UPB-generated BE
states in Ref. 关15兴, and were developed further in Ref. 关16兴.
The present work addresses the construction and identification of both non-full-rank and full-rank PPT BE states that
satisfy the RC in any d 丢 d bipartite quantum system. First, in
Sec. II, we use the UPB states in 3 丢 3 from Ref. 关17兴 to
construct a class of PPT BE states that have rank 5 共while the
system has rank 9兲. We prove their inseparability from first
principles, i.e., by showing that a bound entangled state cannot be written as a convex combination of the pure product
states in its support, even though the support admits an orthogonal product basis. In Sec. III, we generalize the results
for 3 丢 3 and show that for any d 丢 d bipartite quantum system, there are PPT bound entangled states of rank r, where
d2 − 4 艋 r 艋 d2, satisfying the range criterion. In fact, we show
that a much larger set of BE states, which includes such
RC-satisfying BE states as a subset, can be constructed as
convex combinations of a UPB-generated BE state and a
separable state, which is a projector on the space spanned by
a subset of the UPB’s. A proof of inseparability of these
states is obtained by constructing an appropriate entanglement witness, which allows us to explicitly calculate welldefined ranges of the parameter used in convex combination,
such that all states in the range are PPT BE. This construction leads to another class of PPT BE states in any d 丢 d
bipartite system, with rank ranging from 5 to d2; only a subset of this is proven to satisfy the RC. Note that this construction also yields full-rank PPT BE states in any d 丢 d
bipartite quantum system.
兩 ␻ 5典 =
1
冑3
共兩0典 + 兩1典 + 兩2典兲 丢
Let
冉
1
␳BE =
I−
4
兩 ␻ 3典 =
1
1
冑2
共兩1典 − 兩2典兲,
冑2 共兩0典 − 兩1典兲 丢 兩2典,
兩␻2典 = 兩0典 丢
兩 ␻ 4典 =
1
1
冑2 共兩0典 − 兩1典兲,
冑2 共兩1典 − 兩2典兲 丢 兩0典,
5
兩␻i典具␻i兩
兺
i=1
冊
共1兲
共2兲
␳i共⍀兲 = ⍀兩␻i典具␻i兩 + 共1 − ⍀兲␳BE
共3兲
for any i共1 艋 i 艋 5兲 have the following properties.
共i兲 They are bound entangled states if and only if 0 艋 ⍀
⬍ 51 .
共ii兲 They satisfy the range criterion, i.e., the range of
␳i共⍀兲 is spanned by an orthogonal product basis 兵兩␺i典
丢 兩␾i典 : i = 1 , 2 , … , N其, and that of 关␳i共⍀兲兴TB is spanned by
the product basis 兵兩␺i典 丢 兩␾*i 典 : i = 1 , 2 , … , N其.
共iii兲 The range of ␳i共⍀兲 contains more pure product states
than its dimension: In fact, there are exactly six product pure
states in the range.
Let us first start with the case i = 1. Since
␳BE =
冉
1
I − 兩␻1典具␻1兩 −
4
one obtains
5
兩␻i典具␻i兩
兺
i=2
冋冉
5⍀ − 1
5共1 − ⍀兲 1
␳1共⍀兲 =
I−
兩␻1典具␻1兩 +
4
4
5
冊
5
兩␻i典具␻i兩
兺
i=2
共4兲
冊册
.
共5兲
We show below that one can find five mutually orthogonal
pure product states in the range of the rank-5 projector
冉
5
I−
冉
兩␻i典具␻i兩
兺
i=2
1
I−
5
II. NON-FULL-RANK BE STATES IN 3 ‹ 3 SATISFYING
THE RC
兩␻1典 = 兩2典 丢
冑3 共兩0典 + 兩1典 + 兩2典兲.
be the associated bound entangled state. We now show that
the states
and therefore the state
We first show inseparability of a set of non-full-rank PPT
states in 3 丢 3 that satisfy the range criterion in a direct way.
The proof relies on the fact that the state cannot be written as
a convex combination of the pure product states in its support even though the support can be spanned by an orthogonal set of pure product states, and there are more product
states than the dimension of the support. Let
兵兩␻1典 , 兩␻2典 , 兩␻3典 , 兩␻4典 , 兩␻5典其 be the UPB in 3 丢 3 constructed
in Ref. 关17兴:
1
5
冊
兩␻i典具␻i兩
兺
i=2
冊
共6兲
共7兲
is separable. This also implies that for all ⍀, such that 51
艋 ⍀ 艋 1 , ␳1共⍀兲 is a convex combination of separable states
and hence is a separable state itself.
The first part involves counting explicitly the number of
pure product states in the support of ␳1共⍀兲; we show that
there are only six of them. The proof of inseparability will
then follow by showing that ␳1共⍀兲 cannot be expressed as a
convex combination of the product states in its support when
0 艋 ⍀ ⬍ 51 .
Let HS be the subspace spanned by the UPB, and let
4
be a set of pairwise orthonormal vectors spanning
兵兩␹i典其i=1
the orthogonal subspace H⬜
S , which is the range of the state
␳BE. Let A be the new subspace spanned by the vectors
4
and 兩␻1典. The support of the density operators ␳1共⍀兲
兵兩␹i典其i=1
is therefore nothing but the subspace A.
012316-2
NON-FULL RANK-BOUND ENTANGLED STATES …
PHYSICAL REVIEW A 71, 012316 共2005兲
Any pure product state in 3 丢 3 can be written as
5
␩i⬘兩␩i典具␩i兩 = ␤兩␻1典具␻1兩,
兺
i=1
兩␺典 = 共␣兩0典 + ␤兩1典 + ␥兩2典兲 丢 共␣⬘兩0典 + ␤⬘兩1典 + ␥⬘兩2典兲, 共8兲
where the coefficients are complex and satisfy the normalization conditions
兩␣兩2 + 兩␤兩2 + 兩␥兩2 = 兩␣⬘兩2 + 兩␤⬘兩2 + 兩␥⬘兩2 = 1.
共9兲
If 兩␺典 苸 A, we must have 具␺ 兩 ␻i典 = 0, for i = 2,…,5. Using the
orthogonality and normalization conditions one can show
that there can be only six pure product states in A, including
兩␻1典. The five other pure product states
兩 ␩ 1典 =
兩 ␩ 2典 =
兩 ␩ 3典 =
兩 ␩ 4典 =
1
冑6
1
冑2 共兩1典 − 兩2典兲 丢 兩1典,
1
1
冑2 共兩1典 + 兩2典兲 丢 冑2 共兩0典 − 兩1典兲,
共2兩0典 − 兩1典 − 兩2典兲 丢
1
1
冑2 共兩0典 + 兩1典兲,
1
冑3 共兩0典 + 兩1典 + 兩2典兲 丢 冑6 共兩0典 + 兩1典 − 2兩2典兲,
兩 ␩ 5典 =
1
冑6 共兩0典 + 兩1典 − 2兩2典兲 丢 兩2典
共10兲
are mutually orthogonal and form a basis in A. Let us write
兩␩i典 = 兩␺i典 丢 兩␾i典 for i = 1,2,…,5. From Eq. 共5兲 we see that
兩␺i典 丢 兩␾i典 = 兩␺i典 丢 兩␾*i 典, as for each i = 1,2,…,5, 兩␾i典 is a real
state. Since in this case ␳1共⍀兲 = 关␳1共⍀兲兴TB and 兵兩␺i典 丢 兩␾i典 : i
= 1 , 2 , … , 5其 spans the range of ␳1共⍀兲, therefore 兵兩␺i典
*
丢 兩␾i 典 : i = 1 , 2 , … , 5其 spans the range of 关␳1共⍀兲兴TB. Thus we
see that ␳1共⍀兲 satisfies the range criterion for all 0 艋 ⍀ 艋 1.
Next, let us consider the case where 0 ⬍ ⍀ ⬍ 51 and let us
suppose that the state ␳1共⍀兲 is separable. Then it must be
expressed by the convex combination of the pure product
states in its support, which implies
5
␳1共⍀兲 = ⍀兩␻1典具␻1兩 + 共1 − ⍀兲␳BE = 兺 ␩i兩␩i典具␩i兩 + ␻1兩␻1典具␻1兩,
i=1
共11兲
where ␻1 , ␩i 艌 0 , i = 1,…,5. Substituting the expression for
␳BE from Eq. 共5兲, and noting that
5
I−
where ␤ ⬎ 0 and at least one ␩k⬘ ⫽ 0 共1 艋 k 艋 5兲. Since
具␩i 兩 ␩ j典 = ␦ij and 具␩i 兩 ␻1典 ⫽ 0, for all i = 1,…, 5, we get
5
␩⬘i 兩␩i典具␩i兩␩k典 = ␩⬘k 兩␩k典 = ␤共具␻1兩␩k典兲兩␻1典,
兺
i=1
which is a contradiction. Thus the states ␳1共⍀兲 are bound
entangled if and only if 0 艋 ⍀ ⬍ 1 / 5.
The above-mentioned results about ␳1共⍀兲 equally hold
good for all other values of i 苸 兵1 , 2 , … , 5其. That it is true for
i = 2,3,4 follows from the symmetry of the four elements
兩␻1典 , 兩␻2典 , 兩␻3典, and 兩␻4典 of the UPB of Eq. 共1兲 with respect
to each other. Thus, for example, in order to study the properties of ␳2共⍀兲, we need to interchange 兩␻1典 and 兩␻2典, which
can be achieved 共up to some unimportant global phases兲 by
performing the following interchange on both the systems:
兩0典 ↔ 兩2典; in order to study the properties of ␳3共⍀兲, we need
to interchange 兩␻1典 and 兩␻3典, which can be achieved 共up to
some unimportant global phases兲 by performing first the
swap operation, followed by the following interchange on
the first system: 兩0典 ↔ 兩2典; in order to study the properties of
␳4共⍀兲, we need to interchange 兩␻1典 and 兩␻4典, which can be
achieved 共up to some unimportant global phases兲 by performing first the swap operation, followed by the following
interchange on the second system: 兩0典 ↔ 兩2典. Finally, in the
same way, as described above, one can show that there exist
exactly six product states, namely, 兩1典 丢 兩1典, 共1 / 冑2兲共兩0典
+ 兩1典兲 丢 兩2典, 兩0典 丢 共1 / 冑2兲共兩0典 + 兩1典兲, 兩2典 丢 共1 / 冑2兲共兩1典 + 兩2典兲,
共1 / 冑2兲共兩1典 + 兩2典兲 丢 兩0典, and 共1 / 冑3兲共兩0典 + 兩1典 + 兩2典兲 丢 共1 / 冑3兲
⫻共兩0典 + 兩1典 + 兩2典兲 = 兩␻5典, within the range of ␳5共⍀兲. Therefore,
the above-mentioned analysis for ␳1共⍀兲 equally holds good
for all other ␳i共⍀兲’s.
III. BE STATES SATISFYING THE RC IN d ‹ d
We next generalize the preceding results for the case of
d 丢 d. A direct proof of inseparability from the first principles, however, seems difficult to obtain, and instead we
construct an entanglement witness to show inseparability. Let
H be a finite-dimensional Hilbert space of the form HA
丢 HB. For simplicity, we assume that dim HA = dim HB = d.
n
be an UPB with cardinality 兩S兩 = n.
Let S = 兵␻i = ␺Ai 丢 ␸Bi 其i=1
Let the projector on HS, the subspace spanned by the UPB,
be denoted by
n
PS =
5
兩␻i典具␻i兩 = 兺 兩␩i典具␩i兩,
兺
i=2
i=1
共12兲
兩␻i典具␻i兩.
兺
i=1
共14兲
Then the state proportional to the projector 共P⬜
S , say兲 on
is given by
H⬜
S
one obtains
5⍀ − 1
1−⍀
兩␻1典具␻1兩 +
4
4
=
兺
兩␩i典具␩i兩
兺 ␩i兩␩i典具␩i兩 + ␻1兩␻1典具␻1兩.
If ⍀ ⬍ 51 , then we get
␳BE =
共13兲
P⬜
1
共I − PS兲 = S ,
D−n
D−n
共15兲
where D = d2. Thus ␳BE is bound entangled.
Let G be a subset of S, where 1 艋 兩G兩 艋 n = 兩S兩. Let PG be
the projector onto the Hilbert space HG spanned by G. By
012316-3
PHYSICAL REVIEW A 71, 012316 共2005兲
BANDYOPADHYAY, GHOSH, AND ROYCHOWDHURY
following the same construction as in the previous section,
we consider PPT states of the following form:
␳G共⍀兲 =
1−⍀
⍀
PG +
共I − PS兲.
兩G兩
D−n
共16兲
That is, we consider a class of PPT states by mixing a
subset of the UPB’s with ␳BE, and then show that there always exists a ␮ ⬎ 0, such that the states defined in Eq. 共16兲
are bound entangled for all 0 ⬍ ⍀ ⬍ ␮. In order to show the
inseparability of the states under consideration, we consider
the following witness operator that was first stated in Ref.
关16兴 to detect entanglement of the edge states:
W = PS − ␭I,
共17兲
where ␭ is chosen as the value specified in the following
result.
n
be a UPB. Then
Lemma 1. Let S = 兵␻i = ␺Ai 丢 ␸Bi 其i=1
n
␭ = min
具␾A␾B兩␻i典具␻i兩␾A␾B典
兺
i=1
n
= min
円具␾A兩␺Ai 典円2円具␾B兩␸Bi 典円2
兺
i=1
共18兲
over all pure states 兩␾A典 苸 HA , 兩␾B典 苸 HB exists and is strictly
larger than 0 关15兴.
For highly symmetric UPB’s, like the one given in Eq.
共1兲, it is comparatively easy to calculate the value of ␭. In
Ref. 关15兴, it was also noted that a tight lower bound on ␭ can
be explicitly calculated because of the high symmetry of
some of the UPB’s. In fact, this lower bound has been calculated in Ref. 关15兴 for the highly symmetric pyramid UPB
of
3 丢 3. It is now straightforward to verify that the operator in
Eq. 共17兲 is a witness operator. First of all note that the operator is Hermitian. Next, for any product state,
兩␾A, ␾B典 苸 H,
具␾A, ␾B兩W兩␾A, ␾B典 艌 0,
冉
␳G共⍀兲 =
冊
⍀
PG − ␭␳G共⍀兲 = 共⍀ − ␭兲.
兩G兩
Thus, Tr关W␳G共⍀兲兴 ⬍ 0 when 0 ⬍ ⍀ ⬍ ␭, and hence ␳G共⍀兲
is inseparable for all 0 ⬍ ⍀ ⬍ ␭. Note that the rank of ␳G共⍀兲
is simply 共D − n兲 + 兩G兩. Therefore, the rank of this particular
class of PPT BE states ranges from D − n + 1 to D for a UPB
with n elements. Since n 艋 共D − 4兲 and 兩G兩 艌 1 , 5
艋 rank关␳G共⍀兲兴 艋 D. Unfortunately not much can be said as
to whether the states ␳G共⍀兲 in general satisfy or violate the
RC. However, as we show next, a subset of these BE states
satisfy the RC in any dimension.
Definition 1. A UPB is said to be real 共alternatively, a
UPB is said to be with real elements兲 if all the coefficients of
⍀
PG + 共1 − ⍀兲␳BE
兩G兩
共20兲
共0 ⬍ ⍀ ⬍ ␭兲 satisfy the range criterion for all G, such that
兩G兩 艌 共n − 4兲.
Proof. Let HS−G be the Hilbert space spanned by the elements remaining in the UPB S, after G has been taken out
⬜
be the orthogonal complement. Since 兩G兩
from S. Let HS−G
艌 n − 4, then 兩S − G兩 艋 4. From Theorem 3 of Ref. 关17兴, it is
⬜
can be spanned by an orthogonal
sufficient to note that HS−G
set of pure product states. Since S is a UPB with only real
elements therefore the projectors PG , PS, and I − PS are invariant under partial transposition. Hence the state ␳G共⍀兲 is
invariant under partial transposition. Now note that the range
⬜
that admits an
of ␳G共⍀兲 is nothing but the subspace HS−G
orthonormal product basis 兵兩␺i典 丢 兩␾i典 : i = 1 , 2 , … , N其, where
N = D − 共n − 兩G兩兲. Thus we can write the projector I − PS−G on
⬜
as
HS−G
N
I − PS−G =
D−n
兩␩i典具␩i兩 =
兺
1−⍀
i=1
再
␳G共⍀兲 +
冋
册冎
1−⍀ ⍀
−
PG
D − n 兩G兩
共21兲
where 兩␩i典 = 兩␺i典 丢 兩␾i典 for i = 1 , 2 , … , N. Taking partial transposition 共with respect to the second subsystem兲, we have
N
兩␺i典具␺i兩 丢 兩␾*i 典具␾*i 兩
兺
i=1
=
再
冋
册冎
1−⍀ ⍀
D−n
−
␳G共⍀兲 +
PG
1−⍀
D − n 兩G兩
N
共19兲
where the equality is achieved by the product state for which
具␾A , ␾B兩PS兩␾A , ␾B典 = ␭, and from Lemma 1 we know such a
product state exists. Therefore, for all separable states
␴ , Tr共W␴兲 艌 0.
Now if we consider the state in Eq. 共16兲, then we get
Tr关W␳G共⍀兲兴 = Tr
each of the elements of the UPB, with respect to the standard
basis, are real.
Theorem 1. If S be a UPB with real elements in d 丢 d and
兩S兩 = n, then the bound entangled states
=
兩␺i典具␺i兩 丢 兩␾i典具␾i兩
兺
i=1
共22兲
as S is a real UPB. This implies that 兵兩␺i典 丢 兩␾*i 典 : i
⬜
= 1 , 2 , … , N其 also spans HS−G
, and hence it also spans the
TB
range of 关␳G共⍀兲兴 关=␳G共⍀兲兴. Therefore ␳G共⍀兲 satisfies the
range criterion.
䊏
Thus, the class of BE states ␳G共⍀兲 satisfy the RC and
have ranks 共D − 4兲 艋 rank关␳G共⍀兲兴 艋 D, i.e., the above construction provides classes of bound entangled states satisfying the RC of less than full rank, as well as with full rank in
any dimension.
Note that the condition in Theorem 1, which states that
the underlying UPB consists of real elements, is crucial because it guarantees the invariance of the state under partial
transposition. Thus, a natural question is how to construct
real UPB’s for any n 艌 2d − 1, where 2d − 1 is the lower
bound on the dimension of any UPB in d 丢 d. It was proved
in Ref. 关18兴 that if there is a UPB with minimum dimension
then it can be realized with real elements. Unfortunately the
proof is existential and not constructive. Following a suggestion by Smolin 关19兴, here we show that for any bipartite
012316-4
NON-FULL RANK-BOUND ENTANGLED STATES …
PHYSICAL REVIEW A 71, 012316 共2005兲
system we can have a real UPB with dimension D − 4. We
first construct it in 4 丢 4 and as we will see the construction
can be trivially generalized to any d 丢 d.
Consider the real UPB in 3 丢 3 as provided in Ref. 关17兴,
and enumerated in Eq. 共1兲. Let us now add the following
states: 兵兩03典 , 兩13典 , 兩23典 , 兩33典 , 兩30典 , 兩31典 , 兩32典其 to the above set.
Thus we have now a set S of 12 pairwise orthogonal pure
product states of 4 丢 4. It is now impossible to find a product
state 共a兩0典 + b兩1典 + c兩2典兲 丢 共a⬘兩0典 + b⬘兩1典 + c⬘兩2典兲 in the orthogonal subspace H⬜
S of S, because S contains the UPB of Eq. 共1兲.
So any pure product state 共if there is any兲 in H⬜
S must be of
the
form
共a兩0典 + b兩1典 + c兩2典 + d兩3典兲 丢 共a⬘兩0典 + b⬘兩1典 + c⬘兩2典
+ d⬘兩3典兲, where at least one of d and d⬘ is nonzero. But at the
same time, this latter product state must have to be orthogonal
to
each
of
the
product
states
兩03典 , 兩13典 , 兩23典 , 兩33典 , 兩30典 , 兩31典 , 兩32典, which, in turn, implies
that both d and d⬘ must be zero. Hence S is a UPB. As one
can also see, the construction can be trivially generalized to
d 丢 d, and after a proper counting, the number of elements
turns out to be D − 4.
Finally, we note that it is surprising that only one witness
is sufficient to show inseparability of such a wide range of
PPT BE states. Naturally we would like to know if the witness is optimal in the sense whether it is the best witness to
detect inseparability for our class of states. For example,
given a separable state ␳sep, we would like to know the maximum value of ⍀ for which the mixed state, ⍀␳sep + 共1
− ⍀兲␳BE remains a BE state. For the 3 丢 3 states discussed in
Sec. II, we were able to exactly find the value of ⍀ below
which the state is a BE state, where ␳sep is a product state.
Construction of a witness that will be optimal in this sense
seems to be a difficult problem. However, we show that in
detecting entanglement of our class of states, the witness in
Eq. 共17兲 is not unique. In fact there can be infinitely many of
them. Before we give an example of another witness let us
prove a helpful lemma that bounds the inner product between
a pure entangled state and any product state.
Lemma 2. Let 兩⌿典 be a pure entangled state written in the
Schmidt form:
k
兩⌿典 =
␥ j兩j典A兩j典B
兺
i=1
共23兲
where the Schmidt rank k , 2 艋 k 艋 d. Let 兩␥兩2 = max兵兩␥ j兩2其.
Then for all normalized product states 兩␾A典 丢 兩␾B典,
円具⌿兩␾A 丢 ␾B典円2 艋 兩␥兩2 .
共24兲
Schmidt coefficient. Consider now the Hermitian operator
W = PS −
␭
兩⌽典具⌽兩.
兩␥兩2
共26兲
Then from Lemmas 1 and 2 it follows that for all product
states 兩␾A典 丢 兩␾B典 苸 HA 丢 HB , Tr共W兩␾A典具␾A兩 丢 兩␾B典具␾B兩兲 艌 0.
Consider the states defined by Eq. 共16兲. It follows that
Tr共W␳兲 =
⍀关兩␥兩2共D − n兲 + ␭兴 − ␭
.
兩␥兩2共D − n兲
共27兲
This is negative when
⍀⬍
␭
.
兩␥兩 共D − n兲 + ␭
2
共28兲
Let us note that the choice of any pure state 兩⌽典 that
belongs to H⬜
S works for our construction. However, we also
wish to maximize the range over which the state is bound
entangled. For example, the above-mentioned entanglement
witness W = PS − 共␭ / 兩␥兩2兲兩⌽典具⌽兩 will be better than the entanglement witness given in Eq. 共17兲 关so far as detection of
the bound entanglement in the state of Eq. 共16兲 is concerned兴, provided 兩␥兩2 ⬍ 共1 − ␭兲 / 共D − n兲. This can be done by
doing a minimization over the set of all 兩␥兩2 and thereby
choosing the corresponding pure state. We leave the construction of such witnesses as a future research problem.
IV. CONCLUDING REMARKS
We have studied PPT BE states for bipartite quantum systems, and have provided a systematic method of obtaining
bound entangled states in any bipartite system with ranks
ranging from 5 to full rank. We have also constructed a class
of entanglement witness that detects the inseparability of our
class of PPT states. We have also shown that a subset of our
class having less than full rank satisfies the range criterion.
This enabled us to provide a qualitative classification of PPT
BE states based on rank and satisfaction/violation of the
range criterion. For a very specific class of states 共i.e., in
3 丢 3兲 we have been able to prove the inseparability from
first principles by showing that the bound entangled state
cannot be written as a convex combination of the product
states in its support even though the support admits an orthogonal product basis and more product states than the dimension of the support.
Proof. We can write
兺 ␥ j具␾A兩j典具␾B兩j典兩
2
艋 兩␥兩2兩兺 具␾A兩j典具␾B兩j典兩 艋 兩␥兩2 .
円具⌿兩␾A 丢 ␾B典円2 = 兩
2
ACKNOWLEDGMENTS
共25兲
The last inequality in Eq. 共25兲 is obtained by using the
Schwartz inequality and the facts that 兺円具␾A 兩 j典円2
艋 1 , 兺円具␾B 兩 j典円2 艋 1.
䊏
,
where
Let 兩⌽典 be a pure entangled state belonging to H⬜
S
S is a UPB. Let 兩␥兩2 be the absolute square of its largest
Part of this work was done when S.B. was with the Electrical Engineering Department, UCLA, and S.G. was visiting
this department. This work was sponsored in part by the U.S.
Army Research Office/DARPA under Contract/Grant No.
DAAD 19-00-1-0172, and in part by the NSF under Contract
No. CCF-0432296.
012316-5
PHYSICAL REVIEW A 71, 012316 共2005兲
BANDYOPADHYAY, GHOSH, AND ROYCHOWDHURY
关1兴 M. A. Nielsen and I. L. Chuang, Quantum Computation and
Quantum Information 共Cambridge University Press, Cambridge, U.K., 2000兲.
关2兴 For an excellent introduction, see A. Peres, Quantum Theory:
Concepts and Methods 共Kluwer Academic Publishers, Dordrecht, 1993兲.
关3兴 P. Horodecki, Phys. Lett. A 232, 333 共1997兲.
关4兴 D. Bruss and A. Peres, Phys. Rev. A 61, 030301共R兲 共2000兲.
关5兴 C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A.
Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 共1999兲.
关6兴 P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev.
Lett. 82, 1056 共1999兲.
关7兴 W. Dür, J. I. Cirac, M. Lewenstein, and D. Bruß, Phys. Rev. A
61, 062313 共2000兲; D. P. DiVincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, ibid. 61, 062312
共2000兲; S. Bandyopadhyay and V. P. Roychowdhury, ibid. 68,
022319 共2003兲.
关8兴 See, for example, D. Bruß, J. I. Cirac, P. Horodecki, F. Hulpke,
B. Kraus, M. Lewenstein, and A. Sanpera, J. Mod. Opt. 49,
1399 共2002兲, for a general review.
关9兴 L. Gurvits, in Proceedings of the 35th ACM Symposium on
Theory of Computing 共ACM Press, New York, 2003兲, pp. 10–
19.
关10兴 A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Phys. Rev. A
69, 022308 共2004兲.
关11兴 L. Vandenberghe and S. Boyd, URL http://www.stanford.edu/
boyd/cvxbook.html
关12兴 D. Perez-Garcia, Phys. Lett. A 330, 149 共2004兲.
关13兴 O. Rudolph, J. Phys. A 33, 3951 共2000兲.
关14兴 K. G. H. Vollbrecht and M. M. Wolf, Phys. Rev. Lett. 88,
247901 共2002兲.
关15兴 B. M. Terhal, Linear Algebr. Appl. 323, 61 共2000兲.
关16兴 M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Phys.
Rev. A 62, 052310 共2000兲.
关17兴 D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M.
Terhal, Commun. Math. Phys. 238, 379 共2003兲.
关18兴 N. Alon and L. Lovász, J. Comb. Theory, Ser. A 95, 169
共2001兲.
关19兴 J. A. Smolin 共private communication兲.
012316-6