JOURNAL OF MATHEMATICAL PHYSICS 47, 052101 共2006兲
Quantum entanglement and geometry of determinantal
varieties
Hao Chena兲
Department of Computing and Information Technology, Fudan University, Shanghai
200433, People’s Republic of China
共Received 24 November 2005; accepted 15 March 2006; published online 3 May 2006兲
Quantum entanglement was first recognized as a feature of quantum mechanics in
the famous paper of Einstein, Podolsky, and Rosen. Recently it has been realized
that quantum entanglement is a key ingredient in quantum computation, quantum
communication, and quantum cryptography. In this paper, we introduce algebraic
sets, which are determinantal varieties in the complex projective spaces or the
products of complex projective spaces, for the mixed states on bipartite or multipartite quantum systems as their invariants under local unitary transformations.
These invariants are naturally arised from the physical consideration of measuring
mixed states by separable pure states. Our construction has applications in the
following important topics in quantum information theory: 共1兲 separability criterion, it is proved that the algebraic sets must be a union of the linear subspaces if
the mixed states are separable; 共2兲 simulation of Hamiltonians, it is proved that the
simulation of semipositive Hamiltonians of the same rank implies the projective
isomorphisms of the corresponding algebraic sets; 共3兲 construction of bound entangled mixed states, examples of the entangled mixed states which are invariant
under partial transpositions 共thus PPT bound entanglement兲 are constructed systematically from our new separability criterion. © 2006 American Institute of Physics.
关DOI: 10.1063/1.2194629兴
I. INTRODUCTION
A bipartite pure quantum state 兩␺典 苸 H = HAm 丢 HBn, where HAm , HBn are finite dimensional Hilbert
spaces of dimensions m and n, and the tensor inner product is used on H, is called entangled if it
cannot be written as 兩␺典 = 兩␺A典 丢 兩␺B典 for some 兩␺A典 苸 HAm and 兩␺B典 苸 HBn. A mixed state 共or a density
matrix兲 ␳, which is a semipositive definite operator on H with trace 1, is called entangled if it
cannot be written as
␳ = ⌺ i p i P 兩␺A典 丢 P 兩␺B典 ,
i
i
共1兲
for some set of states 兩␺Ai 典 苸 HAm , 兩␺Bi 典 苸 HBn, and pi 艌 0. Here Pv for a state 共unit vector兲 v means
the 共rank 1兲 projection operator to the vector v. If the mixed state ␳ can be written in the form of
共1兲, it is called separable 共see Refs. 28, 39, and 41兲.
For the mixed state ␳ on H = HAm 丢 HBn, we have the following partial transposition ␳PT and the
partial trace trB共␳兲 on HAm 关trA共␳兲 on HBn can be defined similarly兴 defined as follows:
具ij兩␳PT兩kl典 = 具il兩␳兩kj典,
a兲
Electronic mail: chenhao@fudan.edu.cn
0022-2488/2006/47共5兲/052101/19/$23.00
47, 052101-1
© 2006 American Institute of Physics
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052101-2
J. Math. Phys. 47, 052101 共2006兲
Hao Chen
具i兩trB共␳兲兩k典 = ⌺ j具ij兩␳兩kj典,
共2兲
where 兵兩1典 , . . . , 兩m典其 , 兵兩1典 , . . . , 兩n典其 , 兵兩11典 , . . . 兩1n典 , . . . 兩m1典 , . . . , 兩mn典其 are orthogonal bases of HA⬘ HBn
and H, respectively. We observe that the partial transposition of a separable mixed state is again a
separable mixed state, this implies that the partial transposition of a separable mixed state is semi
positive definite. This is the famous Peres PPT criterion of separability 共see Ref. 38兲.
For multipartite quantum systems, there are similar definitions of entangled and separable
mixed states 共see Refs. 28, 39, and 41兲. We restrict ourselves to the tripartite case. Let ␳ be a
mixed state on H = HAm 丢 HBn 丢 HCl. ␳ is called separable if it can be written as
␳ = ⌺ i p i P 兩␺A典 丢 P 兩␺B典 丢 P 兩␺C典 ,
i
i
i
共3兲
for some set of states 兩␺Ai 典 苸 HAm , 兩␺Bi 典 苸 HBn , 兩␺Ci 典 苸 HCl, and pi 艌 0. If the mixed state ␳ cannot be
written in the form of 共3兲, it is called entangled. Sometimes we also consider the separability
relative to the cut A : BC 共B : AC, etc.兲, that means ␳ is considered as a mixed state on the bipartite
quantum system H = HAm 丢 共HBn 丢 HCl兲.
For a n-party quantum system H = HAm1 丢 ¯ 丢 HAmn, local unitary transformations 共acting on a
1
n
mixed state ␳ by U␳U†, where † is the adjoint兲 are those unitary transformations of the form U
= UA1 丢 ¯ 丢 UAn, where UAi is a unitary transformation on HAmi for i = 1 , . . . , n. We can check that
i
all eigenvalues 共spectra兲 of ␳ 共global spectra兲 and, trAi ¯Ai 共␳兲 of mixed states ␳, where i1 , . . . , il
1
l
苸 兵1 , . . . , n其, 共local spectra兲 are invariant under local unitary transformations, and the invariants in
the examples of Refs. 33 and 34 are more or less spectra-involved. It is clear that separability 共or
being entangled兲 is an invariant property under local unitary transformations. For a mixed state ␳,
to judge whether it is entangled or separable and decide its entangled class 共i.e., the equivalent
class of all entangled 共or separable兲 mixed states which are equivalent to ␳ by local unitary
transformations兲 is a fundamental problem in the study of quantum entanglement.28,41 Thus for the
purpose to quantify entanglement, any good measure of entanglement must be invariant under
local transformations.33,34,28,41 Another important concept is the distillable mixed state, which
means that some singlets can be extracted from it by local operations and classical communication
共LOCC兲 共see Ref. 28兲. A mixed state ␳ on H is distillable if and only if for some t, there exists
projections PA : HA丢 t → H2 and PB : HB丢 t → H2, where H2 is a two-dimensional Hilbert space, such
that the mixed state 共PA 丢 PB兲␳ 丢 t共PA 丢 PB兲† is an entangled state in H2 丢 H2 共see Ref. 28兲. A mixed
state which cannot be distilled is called bound entangled mixed state.
The phonomenon of quantum entanglement lies at the heart of quantum mechanics since the
famous Einstein, Podolsky, and Rosen16 paper 共see Refs. 8, 28, and 39兲. Its importance lies not
only in philosophical consideration of the nature of quantum theory, but also in applications where
it has emerged recently that quantum entanglement is the key ingredient in quantum computation20
and communication4 and plays an important role in cryptography 共Ref. 19兲. These new applications of quantum entanglement have stimulated tremendous studies of quantum entanglements of
both pure and mixed states from both theoretical and experimental view, for surveys we refer to
Refs. 8, 28, 32, 39, and 41.
To find good necessary conditions of separability 共separability criterion兲 is a fundamental
problem in the study of quantum entanglement.28,41 Bell-type inequalities28 and entropy criterion28
are well-known numerical criterion of separable states. In 1996, Peres38 gave a striking simple
criterion which asserts that a separable mixed state ␳ necessarily has 共semi兲 positive definite
partial transposition 共PPT兲, which has been proved by Horodeckis25 also a sufficient condition of
separability in 2 ⫻ 2 and 2 ⫻ 3 systems. The significance of PPT property is also reflected in the
facts that PPT mixed states satisfy Bell inequalities44 and cannot be distilled,28,26 thus the first
several examples of the PPT entangled mixed states24 indicated the new phenomenon that there is
bound entanglement in nature.26 These examples were constructed from the so-called range criterion of Horodecki 共see Refs. 24 and 28兲. However, constructing PPT entangled mixed states 共thus
bound entanglement兲 is an exceedingly difficult task,6 and the only known systematic way of such
construction is the context of unextendible product base 共UPB兲 in Ref. 6, which works in both the
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052101-3
Quantum entanglement and geometry of determinantal
J. Math. Phys. 47, 052101 共2006兲
bipartite and multipartite case and is also based on Horodecki’s range criterion. The most recent
disorder criterion of separability in Ref. 37, which is stronger than entropy criterion, was proved
by the mathematics of majorization.
It has been realized that the entanglement of tripartite pure quantum states is not a trivial
extension of the entanglement of bipartite pure quantum states.21 Recently Bennett et al.,5 studied
the exact and asymptotic entanglement measure of multipartite pure states, which showed essential
difference to that of bipartite pure states. On the other hand Carteret et al.,12 proved a generalization of Schmidt decomposition for pure multipartite states. Basically, the understanding of
multipartite quantum entanglement for both pure and mixed states, is much less advanced.
It is clear that any separability criterion for bipartite mixed states, such as Peres PPT
criterion38 and Horodecki range criterion,24,28 can be applied to multipartite mixed states for their
separability under various cuts. For example, from Peres PPT criterion, a separable multipartite
mixed state necessarily has all its partial transpositions semipositive. In Ref. 27, Horodeckis
studied the separability criterion of multipartite mixed states by linear maps. Classification of
triqubit mixed states was studied in Ref. 2.
There have been many interesting works 共Refs. 10, 11, 30, 35, and 43兲 for understanding
quantum entanglement and related problems from the view of representation theory and topology.
The physical motivation of this paper is as follows. We consider the following situation. Alice
and Bob share a bipartite quantum system HAm 丢 HBn, and they have a mixed state ␳. Now they want
to understand the entanglement properties of ␳. It is certain that they can prepare any separable
pure state 兩␾1典 丢 兩␾2典 separately. Now they measure ␳ with this separable pure state, the expectation value is 具␾1 丢 ␾2兩␳兩␾1 丢 ␾2典. If Alice’s pure state 兩␾1典 is fixed, then 具␾1 丢 ␾2兩␳兩␾1 丢 ␾2典 is a
Hermitian bilinear form on Bob’s pure states 共i.e., on HBn兲. We denote this bilinear form by
具␾1兩␳兩␾1典. Intuitively the degenerating locus VAk共␳兲 = 兵兩␾1典 苸 P共HAm兲 : rank共具␾1兩␳兩␾1典兲 艋 k其, where
P共HAm兲 is the projective space of all pure states in HAm, should contain the physical information of
␳ and it is almost obvious that these degenerating locus are invariant under local unitary transformations. For a multipartite quantum system, a similar consideration leads to some Hermitian
bilinear forms on some of its parts and similarly we can consider the degenerating locus of these
Hermitian bilinear forms. We prove that these degenerating locus are algebraic sets, which are
determinantal varieties, for the mixed states on both bipartite and multipartite quantum systems.
They have the following properties.
共1兲 When we apply local unitary transformations on the mixed state the corresponding algebraic sets are changed by local 共unitary兲 linear transformations, and thus these invariants can be
used to distinguish inequivalent mixed states under local unitary transformations.
共2兲 The algebraic sets must be linear 共a union of some linear subspaces兲 if the mixed state is
separable, and thus we give a new separability criterion.
共3兲 The algebraic sets are independent of eigenvalues and only measure the positions of
eigenvectors of the mixed states.
共4兲 These algebraic sets can be calculated easily.
From our construction here, we establish a connection between quantum entanglement and
algebraic geometry. Actually from our results below, we can see that if the Fubini-Study metric of
the projective complex space is used, the metric properties of these algebraic sets are also preserved when local unitary transformations are applied on the mixed state. Hence we establish a
connection between quantum entanglement and both the algebraic geometry and complex differential geometry. Any algebraic geometric or complex differential geometric invariant of the algebraic set of the mixed state is an invariant of the mixed state under local unitary transformations.
The determinantal varieties 共Ref. 23 Lecture 9 and Ref. 3 Chap. II兲 have been studied from
different motivations such as geometry of curves,3,18 geometry of determinantal varieties,17 Hodge
theory,22 commutative algebra,15 and even combinatorics.1 It is interesting to see that it can be
useful even in quantum information theory. We refer to Refs. 23 and 3 for the standard facts about
determinantal varieties.
The paper is organized as follows. We define the algebraic sets of the mixed states and prove
their basic properties including the separability criteria based on these algebraic sets in Sec. II. In
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052101-4
Hao Chen
J. Math. Phys. 47, 052101 共2006兲
Sec. III, we indicate briefly how numerical invariants of the bipartite or multipartite mixed states
under local unitary transformations can be derived from these algebraic sets. As an easiest example, Schmidt rank of a pure state on bipartite quantum systems, a classical concept in quantum
entanglement, is shown to be the codimension of the algebraic set. Many examples of entangled
mixed states corresponding to the famous determinantal varieties in algebraic geometry, such as
Segre varieties, rational normal scrolls and generic determinantal varieties are constructed in Sec.
IV. We also show that a well-known theorem of Eisenbud17 can help us to construct many entangled bipartite mixed states of low ranks. In Sec. V we introduce a family of the generalized
Smolin states, which is a natural extension of Smolin’s physical construction40 from algebraicgeometric view. In Secs. VI and VII, it is proved that these algebraic sets are nonempty for low
rank mixed states, and indicate how a finer result with the same idea 共Theorem 9⬘兲 can be
potentially used to treat the entanglement properties of high rank mixed states. Based on the
algebraic sets introduced in Sec. II, a necessary condition about the simulating Hamiltonians
efficiently using local unitary transformations is given in Sec. VIII. In Sec. IX, we give a continuous family of bipartite mixed states, tripartite pure states, and bipartite Hamiltonians with the
property that the eigenvalues 共spectra兲 of them and their partial traces are constant, however, their
entanglement properties are distinct. This offers strong evidence that it is hopeless to characterize
the entanglement properties by only using eigenvalue spectra. In Sec. X, we illustrate by an
explicit example that our separability criterion can be used to construct PPT entangled mixed
states 共thus bound entanglement兲 systematically.
II. INVARIANTS AND SEPARABILITY CRITERIA
Now we define the invariants and give the coordinate form. Let H = HAm 丢 HBn, 兵兩ij典其, where i
= 1 , . . . , m and j = 1 , . . . , n, be an orthogonal base and ␳ be a mixed state on H. We represent the
matrix of ␳ in the bases 兵兩11典 , . . . 兩1n典 , . . . , 兩m1典 , . . . , 兩mn典其, and consider ␳ as a blocked matrix ␳
= 共␳ij兲1艋i艋m,1艋j艋m with each block ␳ij a n ⫻ n matrix corresponding to the 兩i1典 , . . . , 兩in典 rows and
the 兩j1典 , . . . , 兩jn典 columns. For any pure state 兩␾1典 = r1兩1典 + ¯ + rm兩m典 苸 P共HAm兲 the matrix of the
Hermitian linear form 具␾1兩␳兩␾1典 with the basis 兩1典 , . . . , 兩n典 is ⌺i,jrir*j ␳ij. Thus the degenerating
locus are as follows.
Definition 1: We define
VAk共␳兲 = 兵兩␾1典 苸 P共HAm兲:rank共具␾1兩␳兩␾1典兲 艋 k其 = 兵共r1, . . . ,rm兲 苸 CPm−1:rank共⌺i,jrir*j ␳ij兲 艋 k其,
共4兲
where k = 0 , 1 , . . . , n − 1. Similarly VBk共␳兲 債 CPn−1, where k = 0 , 1 , . . . , m − 1, can be defined.
Example 1: Let ␳ = 共1 / mn兲Imn be the maximally mixed state, we easily have VAk共␳兲
= 兵共r1 , . . . , rm兲 : rank共⌺rir*i In兲 艋 k其 = 쏗 for k = 0 , 1 , . . . , n − 1, where ⴱ is the complex conjugate.
Example 2: Let H = HA2 丢 HBn, T1 , T2 be 2 n ⫻ n matrices of rank n − 1 such that the n ⫻ 共2n兲
matrix 共T1 , T2兲 has rank n. Let T⬘ be a 2n ⫻ 2n matrix with 11 block T1, 22 block T2, 12 and 21
blocks 0. Its rows correspond to the orthogonal base 兩11典 , . . . , 兩1n典 , 兩21典 , . . . , 兩2n典 of H. Let ␳
= 共1 / D兲TT† 共where D is a normalizing constant兲 be a mixed state on H. It is easy to check that ␳
is of rank 2n − 2 and VAn−1共␳兲 = 兵共r1 , r2兲 : r1r2 = 0其.
Let H = HAm 丢 HBn 丢 HCl. Take an orthogonal base of H, 兩ijk典, where, i = 1 , . . . , m, j = 1 , . . . , n, and
k = 1 , . . . , l, and ␳ is a mixed state on H. We represent the matrix of ␳ in the base
兵兩111典 , . . . 兩11l典 , . . . , 兩mn1典 , . . . , 兩mnl典其 as ␳ = 共␳ij,i⬘ j⬘兲1艋i,i⬘艋m,1艋j,j⬘艋n, and ␳ij,i⬘ j⬘ is a l ⫻ l matrix.
k
共␳兲 = 兵共r11 , . . . , rmn兲
Consider H as a bipartite system as H = 共HAm 丢 HBn兲 丢 HCl, then we have VAB
†
mn−1
: rank共⌺rijri⬘ j⬘␳ij,i⬘ j⬘兲 艋 k其 defined as above. This set is actually the degenerating locus of
苸 CP
m
the Hermitian bilinear form 具␾12兩␳兩␾12典 on HCl for the given pure state 兩␾12典 = ⌺m,n
i,j rij兩ij典 苸 P共HA
n
丢 HB兲. When the finer cut A : B : C is considered, it is natural to take 兩␾12典 as a separable pure state
兩␾12典 = 兩␾1典 丢 兩␾2典, i.e., there exist 兩␾1典 = ⌺ir1i 兩i典 苸 P共HAm兲 and 兩␾2典 = ⌺ jr2j 兩j典 苸 P共HBn兲 such that rij
= r1i r2j . In this way the tripartite mixed state ␳ is measured by tripartite separable pure states
k
共␳兲 as follows. It is the degenerating locus
兩␾1典 丢 兩␾2典 丢 兩␾3典. Thus it is natural that we define VA:B
l
of the bilinear form 具␾1 丢 ␾2兩␳兩␾1 丢 ␾2典 on HC.
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052101-5
J. Math. Phys. 47, 052101 共2006兲
Quantum entanglement and geometry of determinantal
Definition 2: Let ␾ : CPm−1 ⫻ CPn−1 → CPmn−1 be the mapping defined by
␾共r11, . . . rm1 ,r21, . . . ,r2n兲 = 共r11r21, . . . ,r1i r2j , . . . rm1 r2n兲
共5兲
(i.e., rij = r1i r2j is introduced).
k
k
Then VA:B
共␳兲 is defined as the preimage ␾−1共VAB
共␳兲兲.
k
k
Similarly VB:C共␳兲 , VA:C共␳兲 can be defined. In the following statement we just state the result
k
共␳兲. The conclusion holds similarly for other V’s.
for VA:B
For the mixed state ␳ on the multipartite system H = HAm1 丢 ¯ 丢 HAmk, we want to study the
1
k
entanglement under the cut Ai1 : Ai2 : ¯ : Ail : 共A j1 ¯ A jk−l兲, where 兵i1 , . . . , il其 艛 兵j1 , . . . , jk−l其
= 兵1 , . . . , k其. We can define the set VAk :¯:A 共␳兲 similarly.
i1
il
It is obvious we have the following results about the “invariance” under local unitary transformations.
Theorem 1: Let T = UA 丢 UB, where UA and UB are the unitary transformations on HAm and
n
HB, respectively. Then VAk共T共␳兲兲 = UA−1共VAk共␳兲兲, that is VAk共␳兲 关respectively, VBk共␳兲兴 is an “invariant”
up to a linear transformation of CPm−1 of the mixed state ␳ under local unitary transformations.
Theorem 1⬘: Let T = UA 丢 UB 丢 UC, where UA, UB, and UC are unitary transformations on HAm,
n
k
k
k
共T共␳兲兲 = UA−1 ⫻ UB−1共VA:B
共␳兲兲, that is VA:B
共␳兲 is an “invariant”
HB, and HCl, respectively. Then VA:B
m−1
n−1
up to a linear transformation of CP ⫻ CP of the mixed state ␳ under local unitary transformations.
Theorem 1⬙: Let T = UAi 丢 ¯ 丢 UAi 丢 U j1¯jk−l, where UAi , . . . , UAi , U j1¯jk−l are unitary
1
m
l
m
m
m
1
l
transformations on HA i1 , . . . , HA il and 共HA j1 丢 ¯ 丢 HA jk−l兲, respectively. Then VAk
i1
il
⫻ ¯ ⫻ UA−1共VAk :¯:A 共␳兲兲.
i1
il
il
1
Remark 1: Since UA−1 ⫻ ¯ ⫻ UA−1
i
i
= UA−1
i
1
j1
jk−l
i1:¯:Ail
共T共␳兲兲
certainly preserves the standard Euclid metric of complex
l
linear space and hence the 共product兲 Fubini-Study metric of the product of projective complex
spaces, all metric properties of VAk :¯:A 共␳兲 are preserved when the local unitary transformations
i1
il
are applied to the mixed state ␳.
Moreover from the proof it is easy to see that all algebraic-geometric properties 关since VAk共␳兲,
k
VB共␳兲 are algebraic sets as proved in Theorem 2兴 of VAk共␳兲 关respectively, VBk共␳兲兴 are preserved even
under local linear inversible transformations 共i.e., T = TA 丢 TB where TA, TB are just linear inversible operators of HAm, HBn兲
We observe VAk共␳PT兲 = 共VAk共␳兲兲*, here * is the conjugate mapping of CPm−1 defined by
*
兲. It is clear that this property holds for other V’s invariants.
共r1 , . . . , rm兲* = 共r*1 , . . . , rm
k
Theorem 2: VA共␳兲 关respectively, VBk共␳兲兴 is an algebraic set in CPm−1 共respectively, CPn−1兲.
From Definition 2 and Theorem 2 we immediately have the following result.
k
共␳兲 is an algebraic set in CPm−1 ⫻ CPn−1.
Theorem 2⬘: VA:B
The general result can be stated as follows.
Theorem 2⬙: VAk :¯:A 共␳兲 is an algebraic set in CPmi1−1 ⫻ ¯ ⫻ CPmil−1.
i1
il
It is easy to see from Definitions that we just need to prove Theorem 2.
Theorem 3: If ␳ is a separable mixed state, VAk共␳兲 关respectively, VBk共␳兲兴 is a linear subset in
m−1
共respectively, CPn−1兲, i.e., it is a union of the linear subspaces.
CP
In the following statement we give the separability criterion of the mixed state ␳ under the cut
A : B : C. The “linear subspace of CPm−1 ⫻ CPn−1” means the product of a linear subspace in CPm−1
and a linear subspace in CPn−1.
Theorem 3⬘: If ␳ is a separable mixed state on H = HAm 丢 HBn 丢 HCl under the cut A : B : C,
k
VA:B共␳兲 is a linear subset in CPm−1 ⫻ CPn−1, i.e., it is a union of the linear subspaces.
The general result can be stated as follows.
Theorem 3⬙: If ␳ is a separable mixed state on H = HAm1 丢 ¯ 丢 HAmk under the cut
1
k
Ai1 : Ai2 : ¯ : Ail : 共A j1 ¯ A jk−l兲, VAk :¯:A 共␳兲 is a linear subset in CPmi1−1 ⫻ ¯ ⫻ CPmil−1, i.e., it is a
i1
il
union of the linear subspaces.
We just proved Theorem 3 and Theorem 3⬘. The proof of Theorem 3⬙ is similar.
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052101-6
J. Math. Phys. 47, 052101 共2006兲
Hao Chen
For the purpose to prove Theorems 2 and 3 we need the following lemmas.
Lemma 1: Let 兵e1 , . . . , eh其 be an orthogonal base of a h dimension Hilbert space H, ␳
t
h
= ⌺l=1
pl Pvl, where vl is unit vector in H for l = 1 , . . . , t, vl = ⌺k=1
aklek, and A = 共akl兲1艋k艋h,1艋l艋t is the
h ⫻ t matrix. Then the matrix of ␳ with the base 兵e1 , . . . , eh其 is APA†, where P is the diagonal
matrix with diagonal entries p1 , . . . , pt.
Proof: We note that the matrix of Pvl with the basis is ␣l␣†l where ␣l = 共a1l , . . . , ahl兲␶ is just the
expansion of vl with the basis. The conclusion follows immediately.
The following conclusion is a direct matrix computation from Lemma 1 or see Ref. 24.
Corollary 1: Suppose pi ⬎ 0, then the image of ␳ is the linear span of vectors v1 , . . . , vt.
From Corollary 1 it is clear that the ranges of the separable mixed states and its partial
transposition are the linear span of separable pure states. This is the so-called range criterion of
Horodecki 共see Refs. 24 and 28兲.
兵e1 , . . . , emn其
be
a
orthogonal
bases
Now
let
H
be
the
HAm 丢 HBn,
t
pl Pvl with positive pl’s be a mixed state on H. We
兵兩11典 , . . . , 兩1n典 , . . . , 兩m1典 , . . . , 兩mn典其 and ␳ = ⌺l=1
may consider the mn ⫻ t matrix A as a m ⫻ 1 blocked matrix with each block Aw, where w
= 1 , . . . , m, a n ⫻ t matrix corresponding to 兵兩w1典 , . . . , 兩wn典其. Then it is easy to see ␳ij = Ai PA†j ,
where i = 1 , . . . , m, j = 1 , . . . , m. Thus
⌺rir*j ␳ij = 共⌺riAi兲P共⌺riAi兲† .
共6兲
⌺rir*j ␳ij
is a (semi) positive definite n ⫻ n matrix. Its rank equals the rank of 共⌺riAi兲.
Lemma 2:
Proof: The first conclusion is clear. The matrix ⌺rir*j ␳ij is of rank k if and only if there exist
n − k linear independent vectors c j = 共c1j , . . . , cnj兲 with the property,
c j共⌺ijrir*j ␳ij兲共c j兲* = 共⌺iric jAi兲P共⌺iric jAi兲† = 0.
共7兲
Since P is a strictly positive definite matrix, our conclusion follows immediately.
Proof of Theorem 2: From Lemma 1, we know that VAk共␳兲 is the zero locus of all determinants
of 共k + 1兲 ⫻ 共k + 1兲 submatrices of 共⌺riAi兲. The conclusion is proved.
Because the determinants of all 共k + 1兲 ⫻ 共k + 1兲 submatrices of 共⌺riAi兲 are homogeneous polynomials of degree k + 1, thus VAk共␳兲 关respectively, VBk共␳兲兴 is an algebraic subset 共called determinantal varieties in algebraic geometry16,17兲 in CPm−1 关respectively, CPn−1兴.
The point here is for different representations of ␳ as ␳ = ⌺ j p j Pv j with p j’s positive real
numbers, the determinantal varieties from their corresponding ⌺iriAi’s are the same.
Now suppose that the mixed state ␳ is separable, i.e., there are unit product vectors a1
s
丢 b1 , ¼ . , as 丢 bs such that ␳ = ⌺l=1ql Pa 丢 b , where q1 , . . . , qs are positive real numbers. Suppose
l
l
1
1
m
au = au兩1典 + ¯ + au 兩m典, bu = bu兩1典 + ¯ + bnu兩n典 for u = 1 , . . . , s. Hence the vector representation of
au 丢 bu with the standard basis is au 丢 bu = ⌺ijaiubuj兩ij典. Consider the corresponding mn ⫻ s matrix C
of a1 丢 b1 , . . . , as 丢 bs as in Lemma 1, we have ␳ = CQC†, where Q is diagonal matrix with diagonal entries q1 , . . . , qs. As before we consider C as m ⫻ 1 blocked matrix with blocks Cw, w
= 1 , . . . , m. Here Cw is a n ⫻ s matrix of the form Cw = 共awj bij兲1艋i艋n,1艋j艋s = BTw, where B
= 共bij兲1艋i艋n,1艋j艋s is a n ⫻ s matrix and Tw is a diagonal matrix with diagonal entries aw1 , . . . , asw.
Thus from Lemma 1, we have ␳ij = CiQC†j = B共TiQT†j 兲B† = BTijB†, where Tij is a diagonal matrix
with diagonal entries q1ai1共a1j 兲† , . . . , qsasi共asj兲†.
Proof of Theorem 3: As in the proof of Theorem 2, we have
⌺rir†j ␳ij = ⌺rir†j BTijB† = B共⌺rir†j Tij兲B† .
共8兲
Here we note ⌺rir†j Tij is a diagonal matrix with diagonal entries q1共⌺riai1兲
⫻共⌺riai1兲† , . . . , qs共⌺riasi兲共⌺riasi兲†. Thus ⌺rir*j ␳ij = BGQG†B†, where G is a diagonal matrix with
diagonal entries ⌺riai1 , . . . , ⌺riasi. Because Q is a strictly positive definite matrix, from Lemma 1
we know that ⌺rir†j ␳ij is of rank smaller than k + 1 if and only if the rank of BG is strictly smaller
than k + 1. Note that BG is just the multiplication of s diagonal entries of G 共which are linear forms
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052101-7
Quantum entanglement and geometry of determinantal
J. Math. Phys. 47, 052101 共2006兲
of r1 , . . . , rm兲 on the s columns of B, thus the determinants of all 共k + 1兲 ⫻ 共k + 1兲 submatrices of BG
共in the case s 艌 k + 1, otherwise automatically linear兲 are the multiplications of a constant 共possibly
zero兲 and k + 1 linear forms of r1 , . . . , rm. Thus the conclusion is proved.
From Lemma 1 and the proof of Theorems 2 and 3, ⌺iriAi play a key role. If we take the
standard ␳ = ⌺rj=1 p j P␸ j, where p j , ␸ j , j = 1 , . . . , r are eigenvalues and eigenvectors, the corresponding ⌺iriAi measures the geometric positions of eigenvectors in HAm 丢 HBn. It is obvious from the
proof of Theorem 2, the invariants defined in Definition 1 are independent of p1 , . . . , pr, the global
eigenvalue spectra of the mixed states.
Proof of Theorem 3⬘: We first consider the separability of ␳ under the cut AB : C, i.e., ␳
= ⌺gf=1 p f Pa f 丢 c f , where a f 苸 HAm 丢 HBn and c f 苸 HCl for f = 1 , . . . , g. Consider the separability of ␳
under the cut A : B : C, we have a f = a⬘f 丢 a⬙f , a⬘f 苸 CAm , a⬙f 苸 CBm. Let a f = 共a1f , . . . , amn
f 兲, a⬘f
= 共a⬘f 1 , . . . , a⬘f m兲, and a⬙f = 共a⬙f 1 , . . . , a⬙f n兲 be the coordinate forms with the standard orthogonal basis
兵兩ij典其, 兵兩i典其, and 兵兩j典其, respectively, we have that aijf = a⬘f ia⬙f j. Recall the proof of Theorem 3, the
diagonal entries of G in the proof of Theorem 3 are
⌺ijrijaijf = ⌺ijr1i a⬘f ir2j a⬙f j = 共⌺ir1i a⬘f i兲共⌺ jr1j a⬙f j兲.
共9兲
k
共␳兲 must be the zero locus of the multiplicaThus as argued in the proof of Theorem 3, VA:B
tions of the linear forms in 共9兲. The conclusion is proved.
III. NUMERICAL INVARIANTS
It is a standard fact in algebraic geometry that V’s defined in Sec. II are the sum of irreducible
algebraic varieties 共components兲. Suppose VAk共␳兲 = V1 艛 ¯ 艛 Vt. From Theorem 1 and Remark 1,
we know that t is a numerical invariant of ␳ when local linear inversible transformations are
applied to ␳. Actually, since there are a lot of numerical algebraic-geometric invariants of these
components, e.g., dimensions, cohomology classes 关represented by Vi’s in H*共CPm−1兲兴, cohomology rings of Vi’s, etc. We can get many numerical invariants of the mixed state when local linear
inversible transformations are applied to them. In this way, we get a very powerful tool of
numerical invariants to distinguish the entangled classes of the mixed states in composite quantum
systems.
On the other hand, if local unitary transformations are applied to the mixed states, it is known
that even the metric properties of V’s 共the metric on V is from the standard Fubini-Study metric of
projective spaces兲 are invariant. Thus any complex differential geometric quantity, such as the
volumes of Vi’s, the integrations 共over the whole component兲 of some curvature functions of Vi’s,
are the invariants of the mixed states under local unitary transformations.
For any given pure state 兩v典 on a bipartite quantum system, 兩v典 苸 HAm 丢 HBn, there exist orthogonal basis 兩␾1典 , . . . , 兩␾m典 of HAm and orthogonal basis 兩␺1典 , . . . , 兩␺n典 of HBn, such that 兩v典 = ␭1兩␾1典
丢 兩␺1典 + ¯ + ␭d兩␾d典 丢 兩␺d典, where d 艋 min兵m , n其. This is Schmidt decomposition 共see Ref. 39兲. It is
clear that d is an invariant under local unitary transformations. This number is called the Schmidt
rank of the pure state 兩v典. It is clear that 兩v典 is separable if and only if its Schmidt rank is 1.
Schmidt rank of pure states on a bipartite quantum system is a classical concept in the theory of
quantum entanglement, it is actually the codimension of the invariant VA0 共␳兲 for the pure state ␳
= P 兩v典.
Let ␳ = P兩v典 be a pure state on HAm 丢 HBn with m 艋 n. From Theorem 1 about the invariance of
0
d
␭i兩␾i典 丢 兩␺i典. It is clear that
VA共␳兲, we can compute it from its Schmidt decomposition 兩v典 = ⌺i=1
0
m−1
␶
VA共␳兲 = 兵共r1 , . . . , rm兲 苸 CP : 共␭1r1 , . . . , ␭drd , 0 , . . . , 0兲 = 0其.
Proposition 1: For the pure state ␳ = P兩v典, d = m if and only if VA0 共␳兲 = 쏗 and d = m − 1
− dim共VA0 共␳兲兲 = n − 1 − dim共VB0 共␳兲兲 if d 艋 m − 1.
In this way we show that the Schmidt rank of a pure state is just the codimension of the
algebraic set, and thus it seems interesting to study the quantity m − 1 − dim共VAk共␳兲兲 for mixed
states, since it is nonlocal invariant and the generalization of the classical concept of Schmidt rank
of pure states.14,42
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052101-8
J. Math. Phys. 47, 052101 共2006兲
Hao Chen
IV. EXAMPLES
Now we give some examples to show how to use Theorems 1, 2, and 3 to construct and
distinguish the entangled classes of the mixed states.
n
Pai 丢 兩i典兲, where ai, i = 1 , . . . , n, are unit
Example 3: Let H = HAm 丢 HBn and ␳a1,. . .,an = 共1 / n兲共⌺i=1
m
vectors in HA . This is a rank n separable mixed state. Suppose ai = 共a1i , . . . , am
i 兲, i = 1 , . . . , n, the
expansion with respect to the standard basis 兩1典 , . . . , 兩m典 of HAm. Let li共r1 , . . . , rm兲 = a1i r1 + ¯
n−1
+ am
i rm for i = 1 , . . . , n be n linear forms. It is easy to check that VA 共␳兲 = 兵共r1 , . . . , rm兲 : l1 ¯ ln = 0其.
Proposition 2: The mixed states ␳a1,. . .,an and ␳b1,. . .,bn are equivalent under the local unitary
transformations if and only if there exists a unitary transformation UA on HAm such that the n
vectors b1 , . . . , bn are exactly UA共a1兲 , . . . , UA共an兲, i.e., bi = UA共ai j兲, where 兵i1 , . . . , in其 = 兵1 , . . . , n其.
Proof: The “if” part is clear. Let li⬘共r1 , . . . , rm兲 = b1i r1 + ¯ + bm
i rm for i = 1 , . . . , n. Then
n−1
VA 共␳a1,. . .,an兲 关respectively, VAn−1共␳b1,. . .,bn兲兴 are the union of n hyperplanes defined by li = 0 共respectively li⬘ = 0兲 for i = 1 , . . . , n. It should be noted here that these hyperplanes are counted with
multiplicities. From Theorem 1 we get the conclusion.
Segre variety ⌺n,m, which is the image of the following map, ␴ : CPn ⫻ CPm → CP共n+1兲共m+1兲−1
where ␴共关X0 , . . . , Xn兴 , 关Y 0 , . . . , Y m兴兲 = 关. . . , XiY j , . . . 兴, is a famous determinantal variety 共see Ref. 23,
pp. 25 and 26兲. It is clear that Segre variety is irreducible and not a linear subvariety. We consider
the Segre variety ⌺1,m in the case n = 1, actually ⌺1,m = 兵共r1 , . . . , r2n兲 : rank共M兲 艋 1其 where M is the
following matrix:
冉
r1
r2
¯
rn
rn+1 rn+2 ¯ r2n
冊
共10兲
.
Example 4 (entangled mixed state from Segre variety): Let H = HA2m 丢 HB2 , 兩␾i典 = 共1 / 冑2兲共兩i1典
+ 兩共m + i兲2典兲 for i = 1 , . . . , m and ␳ = 共1 / m兲共P兩␾1典 + ¯ + P兩␾m典兲. This is a mixed state of rank m. By
computing ⌺iriAi as in the proof of Theorem 2, we get VA1 共␳兲 = ⌺1,m. Thus ␳ is an entangled mixed
state.
The rank 1 locus Xl,n−l−1 = 兵rank共R兲 艋 1其 of the following 2 ⫻ 共n − 1兲 matrix R:
冉
r0 . . . rl−1 rl+1 . . . rn−1
r1 . . .
rl
rl+2 . . .
rn
冊
is the rational normal scroll 共see p. 106 of Ref. 23兲. The mixed states corresponding to them are
as follows.
Example 5 (entangled mixed state from rational normal scroll): Let H = HAn+1 丢 HB2 and 兩␾1典
= 共1 / 冑2兲共兩01典 + 兩12典兲 , . . . , 兩␾i典 = 共1 / 冑2兲共兩共i − 1兲1典 + 兩i2典兲 , . . . , 兩␾l典 = 共1 / 冑2兲共兩共l − 1兲1典 + 兩l2典兲 , 兩␾l+1典
= 共1 / 冑2兲共兩共l + 1兲1典 + 兩共l + 2兲2典兲 , . . . , 兩␾ j典 = 共1 / 冑2兲共兩j1典 + 兩共j + 1兲2典兲 , . . . , 兩␾n−1典 = 共1 / 冑2兲共兩共n − 1兲1典
+ 兩n2典兲. We consider the mixed state ␳l = 关1 / 共n − 1兲兴共P兩␾1典 + ¯ + P兩␾n−1典兲 of rank n − 1 on H. It is
clear from Sec. II that VA1 共␳兲 = Xl,n−l−1 傺 CPn. From the well-known fact in algebraic geometry 共see
pp. 92 and 93 of Ref. 23兲 we have the following result.
Proposition 3. The mixed states ␳l , l = 1 , . . . , 关共n − 1兲 / 2兴 are entangled and ␳l and ␳l⬘ for l
⫽ l⬘ are not equivalent under local unitary transformations.
We need to recall a well-known result in the theory of determinantal varieties 共see Proposition
on p. 67 of Ref. 8兲. Let M共m , n兲 = 兵共xij兲 : 1 艋 i 艋 m , 1 艋 j 艋 n其 共isomorphic to CPmn−1兲 be the projective space of all m ⫻ n matrices. For an integer 0 艋 k 艋 min兵m , n其, M共m , n兲k is defined as the
locus 兵A = 共xij兲 苸 M共m , n兲 : rank共A兲 艋 k其. M共m , n兲k is called generic determinantal varieties.
Proposition 4: M共m , n兲k is an irreducible algebriac subvariety of M of codimension 共m − k兲
⫻共n − k兲.
Suppose m 艋 n, we now construct a mixed state ␳ with VAm−1共␳兲 = M共m , n兲共m−1兲.
Example 6 (generic entangled mixed state): Let H = HAmn 丢 HBm, where m 艋 n, and Aij, i
= 1 , . . . , m, j = 1 , . . . n be m ⫻ n matrix with only nonzero entry at ij position equal to 1. Let A be a
blocked mn ⫻ 1 matrix with ij block Aij. Here the kth row of Aij in A corresponds to the vector
兩共ij兲k典 in the standard basis of H. Hence A is a size m2n ⫻ n matrix. Let ␳ = 共1 / D兲AA† be a mixed
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052101-9
J. Math. Phys. 47, 052101 共2006兲
Quantum entanglement and geometry of determinantal
state on H. 共Here D is a normalizing constant.兲 It is a rank n mixed state.
It is easy to compute ⌺ijrijAij = 共rij兲1艋i艋m,1艋j艋n 共up to a constant兲. Thus we have VAm−1共␳兲
= M共m , n兲共m−1兲. From Proposition 4 and Theorem 3, ␳ is an entangled mixed state.
m
Example 7: Let H = HAm 丢 HBn 丢 HCm, 兩␾l典 = 共1 / m兲⌺i=1
兩ili典 for l = 1 , . . . , n and ␳ = 共1 / n兲共P兩␾1典
+ ¯ + P兩␾n典兲 be a rank n mixed state on H. It is clear that under the cut B : AC, ␳ is separable.
However, under the cut AB : C, we can check that ␳ is just the mixed state in Example 6 and thus
entangled. Similarly under the cut A : BC, ␳ is also the mixed state in Example 6 and thus entangled. Hence this is a mixed state on tripartite quantum system with the property that it is
separable under B : AC cut and entangled under AB : C and A : BC cuts.
Example 8 (entangled mixed states from Eisenbud theorem (Ref. 17): Let H = HAh 丢 HBm, where
h 艌 nm − m + 2 and n 艌 m are positive integers, Ai, i = 1 , . . . , h be m ⫻ n matrix with the property:
the space M of linear forms 共of r1 , . . . , rh兲 span by the entries of T共r1 , . . . , rh兲 = ⌺iriAi is of dimension h. Let A be the hm ⫻ n matrix with ith block Ai. Here the kth row of Ai in A corresponds to
the vector 兩ik典 in the standard basis of H. Let ␳ = 共1 / D兲AA† 共D is a normalizing constant兲 be a
共rank n兲 mixed state on H. From the proof of Theorem 2 we have VAm−1共␳兲
= 兵共r1 , . . . , rh兲 : rank共T共r1 , . . . , rh兲兲 艋 m − 1其. We observe that when h = mn it is just the mixed state in
Example 6.
Theorem 6: ␳ is an entangled mixed state.
Proof: It is clear that M共m , n兲 is 1-generic 共see Ref. 17, p. 548兲. We can see that the space M
has codimension 关in M共m , n兲兴 smaller or equal to m − 1 共here v = n, w = m, k = m − 1 as referred to
Theorem 2.1 on p. 552 of Ref. 17兲. From definition it is clear that VA共␳兲 is just M m−1, which is
reduced and irreducible and of codimension n − m + 1 in M共m , n兲 from Theorem 2.1 of Ref. 17. The
conclusion is proved.
Eisenbud theorem 共Theorem 2.1 in Ref. 17 and thus Theorem 6 here兲 gives us a general
method to construct many entangled states of low ranks, since the condition about M is not a very
strong restriction.
Example 9 (Bennett-DiVincenzo-Mor-Shor-Smolin-Terhal mixed state from UPB (Ref. 7)): Let
H = HA2 丢 HB2 丢 HC2 , 兩␾+典 = 共1 / 冑2兲共兩1典 + 兩2典兲, 兩␾−典 = 共1 / 冑2兲共兩1典 − 兩2典兲. Consider the linear subspace T
spaned by the following four vectors 兩1典 丢 兩2典 丢 兩␾+典, 兩2典 丢 兩␾+典 丢 兩1典, 兩␾+典 丢 兩1典 丢 兩2典, 兩␾−典 丢 兩␾−典
丢 兩␾−典. Now P is the projection to the complementary space T⬜ of T and ␳ = 共1 / D兲P is a rank 4
PPT mixed state on H for any bipartite cut 共see Ref. 7兲. It is proved in Ref. 7 that ␳ is entangled
under the cut A : B : C 共thus bound entanglement兲, however, it is separable under the cuts
1
1
共␳兲 and VA:B
共␳兲. It is easy to see from
A : BC , B : AC , C : AB. Now we can compute its invariants VAB
1
1
Theorem 3 that VAB共␳兲 should be linear, however we can see that VA:B共␳兲 is also linear from our
computation below, though it is entangled under the cut A : B : C.
It is easy to check that the following four vectors 兩010典–兩011典, 兩100典–兩110典, 兩001典–兩101典, 兩000典–
兩111典 are the base of T⬜. Thus the matrix A is of the following form 共with rows corresponding to
兩000典, 兩001典, 兩010典, 兩011典, 兩100典, 兩101典, 兩110典, 兩111典兲:
冢 冣
0
0
0
1
0
0
1
0
1
0
0
0
−1
0
0
0
0
1
0
0
0
0
−1
0
0
−1
0
0
0
0
0
−1
and ⌺ijrijAij is of the following form:
冉
r01
r10 − r11
− r01
0
0
r00
r00 − r10 − r11
共11兲
冊
.
共12兲
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052101-10
J. Math. Phys. 47, 052101 共2006兲
Hao Chen
1
Thus VAB
共␳兲 is the union of the following three points 共1:1:0:0兲, 共0:1:0:1兲, 共0:0:1:0兲 in CP3
1
and VA:B共␳兲 is union of CP1 ⫻ 共1 : 0兲, 共0 : 1兲 ⫻ CP1, and 共1 : 0兲 ⫻ 共0 : 1兲 in CP1 ⫻ CP1.
V. GENERALIZED SMOLIN STATE
Smolin40 introduced a rank 4 mixed state ␳ = 41 共P兩v1典 + P兩v2典 + P兩v3典 + P兩v4典兲 on 4-party quantum
2
, where
system HA2 丢 HB2 丢 HC3 丢 HD
兩v1典 = 21 共兩0000典 + 兩0011典 + 兩1100典 + 兩1111典兲,
兩v2典 = 21 共兩0000典 − 兩0011典 − 兩1100典 + 兩1111典兲,
兩v3典 = 21 共兩0101典 + 兩0110典 + 兩1001典 + 兩1010典兲,
兩v4典 = 21 共兩0101典 − 兩0110典 − 兩1001典 + 兩1010典兲.
This mixed state ␳ is a bound entangled state when four parties A , B , C , D are isolated.
The following example, which is a continuous family 共depending on eight parameters兲 of
2
separable for any 2:2 cut but
mixed state on the four-party quantum system HA2 丢 HB2 丢 HC2 丢 HD
entangled for any 1:3 cut 共thus bound entangled mixed state when A , B , C , D are isolated兲, can be
thought of as a generalization of Smolin’s mixed state in Ref. 40. We prove that the generic
members in this family of mixed states are not equivalent under local unitary transformations
共Theorem 7 below兲.
2
and h1 , h2 , h3 , h4 共understood as row vectors兲 be four
Example 10: Let H = HA2 丢 HB2 丢 HC2 丢 HD
4
mutually orthogonal unit vectors in C and a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 be eight nonzero real parameters satisfying a1a8 = a2a7 = a3a6 = a5a4. Consider the 16⫻ 4 matrix T with 16 rows as T
= 共a1h1␶ , 0 , 0 , a2h2␶ , 0 , a3h3␶ , a4h4␶ , 0 , 0 , a5h4␶ , a6h3␶ , 0 , a7h2␶ , 0 , 0 , a8h1␶ 兲␶. Let 兩␾1⬘典, 兩␾2⬘典, 兩␾3⬘典, 兩␾4⬘典 be
four vectors in H whose expansions with the basis 兩0000典, 兩0001典, 兩0010典, 兩0011典, 兩0100典, 兩0101典,
兩0110典, 兩0111典, 兩1000典, 兩1001典, 兩1010典, 兩1011典, 兩1100典, 兩1101典, 兩1110典, 兩1111典 are exactly the four
columns of the matrix T. Let ␳ = 共1 / D兲TT† 共D is a normalizing constant兲 be the mixed states
共matrix with respect to the standard base of H as above兲.
It is easy to check that when h1 = 共1 / 冑2兲共1 , 1 , 0 , 0兲, h2 = 共1 / 冑2兲共1 , −1 , 0 , 0兲, h3 = 共1 / 冑2兲
⫻共0 , 0 , 1 , 1兲, h4 = 共1 / 冑2兲共0 , 0 , 1 , −1兲 and all ai’s are 1, it is just the Smolin’s mixed state in Ref.
40.
Now we prove that ␳ is invariant under the partial transposes of the cuts
AB : CD , AC : BD , AD : BC.
Let the “representation” matrix T = 共btijkl兲i=0,1,j=0,1,k=0,1,l=0,1,t=1,2,3,4 be the matrix with columns
corresponding the expansions of ␾1 , ␾2 , ␾3 , ␾4. Then we can consider that T = 共T1 , T2 , T3 , T4兲␶ as a
blocked matrix of size 4 ⫻ 1 with each block Tij = 共btkl兲k=0,1,l=0,1,=1,2,3,4 a 4 ⫻ 4 matrix, where ij
= 00, 01, 10, 11. Because h1 , h2 , h3 , h4 are mutually orthogonal unit vectors we can easily check
that TijT†i j = Ti⬘ j⬘T†ij from the condition a1a8 = a2a7 = a3a6 = a5a4. Thus it is invariant when the
⬘⬘
partial transpose of the cut AB : CD is applied.
With the same methods we can check that ␳ is invariant when the partial transposes of the cuts
AC : BD, AD : BC are applied. Hence ␳ is PPT under the cuts AB : CD, AC : BD , AD : BC. Thus from
a result in Ref. 31 which claims that the PPT mixed states on HTm 丢 HnS with their ranks not bigger
than max兵m , n其 are separable, we know ␳ is separable under these cuts AB : CD, AC : BD , AD : BC.
1
Now we want to prove ␳ is entangled under the cut A : BCD by computing VBCD
共␳兲. From the
1
previous arguments, we can check that VBCD共␳兲 is the locus of the condition: a1h1r000 + a2h2r011
+ a3h3r101 + a4h4r110 and a7h1r100 + a8h2r111 + a5h3r001 + a6h4r010 are linear dependent. This is
equivalent to the condition that the matrix 共8兲 is of rank 1,
冉
冊
a7r100 a8r111 a5r001 a6r010
.
a1r000 a2r011 a3r101 a4r110
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052101-11
Quantum entanglement and geometry of determinantal
J. Math. Phys. 47, 052101 共2006兲
1
From Ref. 23, pp. 25 and 26 we can check that VBCD
共␳兲 is exactly the famous Segre variety
in algebraic geometry. It is irreducible and thus cannot be linear. From Theorem 3, ␳ is entangled
under the cut A : BCD. Similarly we can prove that ␳ is entangled under the cuts B : ACD, C : ABD,
D : ABC.
3
共␳兲. From the previous arguments and Definition 2, it is just the locus
Now we compute VA:B
of the condition that the vectors h1共a1r10r20 + a7r11r21兲, h3共a3r10r21 + a5r11r20兲, h4共a4r10r21 + a6r11r20兲,
h2共a2r10r20 + a8r11r21兲 are linear dependent. Since h1 , h2 , h3 , h4 are mutually orthogonal unit vectors,
we have
3
共␳兲 = 兵共r10,r11,r20,r21兲 苸 CP1 ⫻ CP1:
VA:B
共a1r10r20 + a7r11r21兲共a3r10r21 + a5r11r20兲共a4r10r21 + a6r11r20兲共a2r10r20 + a8r11r21兲 = 0其.
3
Set ␭1 = a1 / a7 = a2 / a8 and ␭2 = a3 / a5 = a4 / a6, we know that VA:B
共␳兲 is the union of V1 and V2
with multiplicity 2, where
V1 = 兵共r10,r11,r20,r21兲 苸 CP1 ⫻ CP1:r10r20 − ␭1r11r21 = 0其,
V2 = 兵共r10,r11,r20,r21兲 苸 CP1 ⫻ CP1:r10r21 − ␭2r11r20 = 0其.
From Theorem 3 we know that ␳ is entangled for the cut A : B : CD, A : C : BD, and A : D : BC for
generic parameters, since r10r20 + ␭1r11r21, etc., cannot be factorized to two linear forms for generic
␭1. This provides another proof that the mixed state is entangled if the four parties are isolated.
Theorem 7: The generic members in this continuous family of mixed states are inequivalent
2
.
under the local unitary transformations on H = HA2 丢 HB2 丢 HC2 丢 HD
3
Proof: From the above computation, VA:B共␳␭1,2兲 is the union of V1 and V2 with multiplicity 2.
From Theorem 1⬘, if ␳␭1,2 and ␳␭1,2
⬘ are equivalent by a local operation, there must exist two
fractional linear transformations T1 , T2 of CP1 such that T = T1 ⫻ T2 共acting on CP1 ⫻ CP1兲 transforms the two varieties V1 , V2 of ␳␭1,2 to the two varieties V1⬘ , V2⬘ of ␳␭1,2,
⬘ , i.e., T共Vi兲 = V⬘j .
Introduce the inhomogeneous coordinates x1 = r10 / r11, x2 = r20 / r21. Suppose T共Vi兲 = Vi⬘ i = 1 , 2
without loss of generality. Then we have ␭1␭2 = 1. This means that there are some algebraic
relations of parameters if the T exists. This implies that there are some algebraic relations on
parameters if ␳␭1,2 and ␳␭1,2
⬘ are equivalent by local unitary transformations. Hence our conclusion
follows immediately.
VI. NONEMPTY THEOREM
In this section, we prove that the algebraic set invariants introduced in Sec. II are not empty
for low rank mixed states.
Theorem 8: Let H = HAm 丢 HBn be a bipartite quantum system and ␳ is a rank r mixed state on
H with r 艋 m + n − 2. Then VAn−1共␳兲 and VBm−1共␳兲 are not empty.
r
pi Pvi, where p1 , . . . , pr, v1 , . . . , vr are
Proof: We take “the standard representation” ␳ = ⌺i=1
eigenvalues and eigenvectors of ␳ and r = rank共␳兲. Recall the proof of Theorem 2, VAn−1共␳兲 is the
locus of the condition that ⌺iriAi 共a n ⫻ r matrix兲 has its rank smaller than n. From Proposition 4
the variety of generic n ⫻ r matrices of rank less than or equal to n − 1 has its codimension smaller
or equal to 共n − 共n − 1兲兲共r − 共n − 1兲兲 = r − n + 1, we know that the codimension of VAn−1共␳兲 in CPm−1 is
smaller or equal to r − n + 1. Hence dim共VA共␳兲兲 艌 m − 1 − r + n − 1 艌 0 and VAn−1共␳兲 is not empty. The
conclusion for VBm−1共␳兲 can be proved similarly.
VII. A RELATION OF DETERMINANTS
As indicated in Sec. II, we can have the following statement from Lemma 2.
t
s
Theorem 9: Let H = HAm 丢 HBn be a bipartite quantum system and ␳ = ⌺l=1
pl Pvl = ⌺l=1
ql Pv⬘ be a
l
mixed state with two “representations” as convex combinations of projections with
p1 , . . . , pt , q1 , . . . , qs ⬎ 0. Let A (respectively, A⬘) be the mn ⫻ t (respectively, mn ⫻ s) matrix of
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052101-12
J. Math. Phys. 47, 052101 共2006兲
Hao Chen
vectors v1 , . . . , vt (respectively, v1⬘, . . . , vs⬘) in the standard basis 兩11典 , . . . , 兩1n典 , . . . , 兩m1典 , . . . , 兩mn典
as in Lemma 1. We represent A (respectively, A⬘) as m ⫻ 1 blocked matrix with blocks A1 , . . . , Am
(respectively, A1⬘ , . . . , Am
⬘ ). Then the determinantal varieties defined by the conditions that the
ranks of R = ⌺iriAi and R⬘ = ⌺iriAi⬘ are smaller than k , k = 0 , 1 , . . . , n − 1, are the same.
Actually we can get more information about the determinants of n ⫻ n submatrices of ⌺iriAi
and ⌺iriAi⬘ from the proof of Theorems 2 and 3. This relation seems to be helpful to extract
information of ␳’s one unknown “representation” from its another known “representation,” as in
the proof of Theorem 3.
Theorem 9⬘: Let H , ␳ , p1 , . . . pt , q1 , . . . , qs , A , A⬘ , R , R⬘ be as above and Ri1,. . .in (respectively,
Ri⬘⬘ , . . . , in⬘) be the n ⫻ n submatrix of R (respectively, R⬘) consisting of i1 ⬍ ¯ ⬍ in (respectively,
1
i1⬘ ⬍ ¯ ⬍ in⬘兲th columns, where i1 , . . . , in 苸 兵1 , . . . t其 and i1⬘ , . . . , in⬘ 苸 兵1 , . . . , s其 are distinct indices.
Then we have
⌺i1⬍¯⬍in pi1 ¯ pin兩Ri1,. . .,in兩2 = ⌺i⬘⬍¯⬍i⬘qi⬘ ¯ qi⬘兩Ri⬘⬘,. . .,i⬘兩2 .
1
n
1
n
1
n
共13兲
The above result follows from the following lemma immediately, since both sides of the
equality are just det共⌺ijrir†j ␳ij兲.
Lemma 3 (Binet-Cauchy formula): Let B be a n ⫻ t matrix with t ⬎ n and Bi1,. . .,in be the n
⫻ n submatrix of B consisting of i1 ⬍ ¯ ⬍ in-th columns. Then det共BB†兲 = ⌺i1⬍¯⬍in兩det Bi1,. . .,in兩2.
It is clear that Theorems 2 and 3 follow from Theorem 8⬘ here immediately.
The following result was previously known in Refs. 29 and 31.
n
Proposition 5: Let H = HAn 丢 HBn, ␳ = 共1 / n兲⌺i=1
pi Pai 丢 bi where a1 , . . . , an (respectively, b1 , . . . , bn)
n
t
qi Pci 丢 di is
are linearly independent unit vectors in HA (respectively, HBn). Suppose that ␳ = 共1 / t兲⌺i=1
another representation of ␳ as a convex combination with qi’s positive, then actually we have t
= n and 兵a1 丢 b1 , . . . , an 丢 bn其 = 兵c1 丢 d1 , . . . , cn 丢 dn其.
Proof: We apply Theorem 9⬘ to the 2 “representations” here. First of all, we know that
det共⌺ijrir†j ␳ij兲 is 共up to a nonzero constant兲 the square of the absolute value of a multiplication of
linear forms bi共r1 , . . . , rn兲 = ⌺ jb1j r j, where bi = ⌺ jbij兩j典 is the coordinate form of bi for i = 1 , . . . , n,
from one known “representation.” Thus we know from Theorem 8⬘ that there are at least n vectors
in 兵d1 , . . . , dt其, without loss of generality, suppose they are d1 , . . . , dn, are just b1 , . . . , bn. Using
Theorem 9⬘ for the second factor and consider the first through nth columns of R⬘, this implies
that the multiplication of the linear forms c1共r1 , . . . , rn兲 , . . . , cn共r1 , . . . , rn兲 are just the multiplication of the linear forms a1共r1 , . . . , rn兲 , . . . , an共r1 , . . . , rn兲. Hence we know that the set 兵c1 , . . . , cn其
are just the set 兵a1 , . . . , an其.
On the other hand, it is easy to see that ai 丢 b j with i ⫽ j is not in the linear span of a1
丢 b1 , . . . , an 丢 bn, since a1 , . . . , an 共respectively, b1 , . . . , bn兲 are linear independent. Thus ci = ai from
Corollary 1.
Applying Theorem 9⬘ to other columns of R and R⬘ by a similar argument, we have c j
苸 兵a1 , . . . , an其 and d j 苸 兵b1 , . . . , bn其. Since ai 丢 b j with i ⫽ j cannot be in the image of ␳, c j 丢 d j must
be the form ai j 丢 bi j. The conclusion is proved.
Remark 2: If we compute VAn−1共␳兲 from the representation of ␳’s standard form, i.e., linear sum
of projections to its eigenvectors, it can be seen that our invariants defined in Sec. II are independent of eigenvalues 共p1 , . . . , pt in Sec. II兲. However the information of p1 , . . . , pt or eigenvalues is
certainly reflected in Theorem 9⬘ here. Thus Theorem 9⬘ might be more useful in determining
whether a given mixed state is entangled or not, provided that we know how to extract sufficient
information from Theorem 9⬘.
Remark 3: As shown in Example 1, our invariants might be an empty set for high rank mixed
states, however it seems that Theorem 9⬘ is still useful in determining whether a given high rank
mixed state is entangled or not in this case, provided that we know how to extract information
from Theorem 9⬘.
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052101-13
Quantum entanglement and geometry of determinantal
J. Math. Phys. 47, 052101 共2006兲
VIII. SIMULATION OF HAMILTONIANS
Historically, the idea of simulating Hamiltonian 共self-adjoint operators on the Hilbert space
corresponding to the quantum system, see Ref. 39兲 time evolutions was the first motivation for
quantum computation because of the famous paper of Feynman. Recently the ability of nonlocal
Hamiltonians to simulate one another is a popular topic, which has applications in quantum
control theory, quantum computation, and the task of generating entanglement. For the general
treatment of this topic and the references, we refer to Ref. 5.
We say, for two bipartite Hamiltonians H and H⬘ on HAm 丢 HBn, H⬘ can be efficiently simulated
by H with local unitary operations, write as H⬘ⱮLUH, if H⬘ can be written as a convex combination of conjugates of H by local unitary operations, H⬘ = p1共U1 丢 V1兲H共U1 丢 V1兲† + ¯ + ps共Us
丢 Vs兲H共Us 丢 Vs兲†, where p1 , . . . , ps are positive real numbers such that p1 + ¯ + ps = 1, U1 , . . . , Us,
and V1 , . . . , Vs are unitary operations on HAm and HBn, respectively. Here we use † for the adjoint.
This is equivalent to the notion “infinitesimal simulation” in Ref. 5. In Ref. 5 it is shown that
“local terms” like I 丢 KB and KA 丢 I are irrelevant to the simulation problem up to the second
order, thus they considered the simulation problem for Hamiltonians without local terms’ effect.
Our definition here is more restricted without neglecting the local terms.
We can have the following necessary conditions about the simulation of semipositive Hamiltonians based on the algebraic set invariants introduced in Sec. II.
Theorem 10: Let H and H⬘ be two semipositive Hamiltonians on the bipartite quantum
system HAm 丢 HBn. Suppose H⬘ⱮLUH, that is, H⬘ can be simulated by H efficiently by using local
unitary transformations. Then there are projective isomorphisms U1 of CPm−1 and V1 of CPn−1
such that U1共VAk共H⬘兲兲 傺 VAk共H兲 for k = 0 , . . . , n − 1 and V1共VBk共H⬘兲兲 傺 VBk共H兲 for k = 0 , . . . , m − 1.
The following observation about the computation of VAk共␳兲 is the key point of the proof of
Theorem 10 and Corollary 3. From Corollary 1 if ␳ = ⌺ti pi Pvi with pi’s positive real numbers, the
range of ␳ is the linear span of vectors v1 , . . . , vt. We take some vectors in the set 兵v1 , . . . , vt其, say
they are v1 , . . . , vs. Let B be the mn ⫻ s matrix with columns corresponding to the s vectors
v1 , . . . , vs’s coordinates in the standard basis of HAm 丢 HBn. We consider B as m ⫻ 1 blocked matrix
with blocks B1 , . . . , Bm n ⫻ s matrix as in Sec. II. It is clear that VAk共␳兲 is an algebraic subset of the
m
zero locus of the determinants of all 共k + 1兲 ⫻ 共k + 1兲 submatrices of ⌺m
i riBi, since ⌺i riBi is a
m
submatrix of ⌺i riAi. On the other hand, if v1 , . . . , vs are linear independent and s
= dim共range共␳兲兲, VAk共␳兲 is just the zero locus of the determinants of all 共k + 1兲 ⫻ 共k + 1兲 submatrices
of ⌺m
i riBi, since any column in ⌺iriAi is a linear combination of columns in ⌺iriBi 关rank共⌺iriAi兲
艋 k is equivalent to rank共⌺iriBi兲 艋 k兴.
Proof of Theorem 10: Suppose H⬘ⱮLUH, then there exist positive numbers p1 , . . . , ps and local
unitary transformations U1 丢 V1 , . . . , Us 丢 Vs, such that ⌺i piUi 丢 ViH共Ui 丢 Vi兲† = H⬘. Let H
= ⌺si qi P␺i, where s = dim共range共H兲兲, q1 , . . . , qs are eigenvalues of H and ␺1 , . . . , ␺s are eigenvectors
of H. Then it is clear that 共Ui 丢 Vi兲H共Ui 丢 Vi兲† = ⌺sj q j P共Ui 丢 Vi兲␺ j and thus H⬘ = ⌺i,j piq j P共Ui 丢 Vi兲␺ j. This
is a representation of H⬘ as a convex combination of projections. From our above observation
VAk共H⬘兲 is an algebraic subset in VAk共共U1 ⫻ V1兲H兲 关which can be computed from vectors 共U1
丢 V1兲␺1 , . . . , 共U1 丢 V1兲␺s兴. Thus the conclusion follows from Theorem 1.
Corollary 3 (see Ref. 13): Let H and H⬘ be two semipositive Hamiltonians on the bipartite
quantum system HAm 丢 HBn of the same rank, i.e., dim共range共H兲兲 = dim共range共H⬘兲兲. Suppose H⬘
ⱮLUH, that is, H⬘ can be simulated by H efficiently by using local unitary transformations. Then
VAk共H⬘兲 = VAk共H兲 for k = 0 , . . . , n − 1 and VBk共H⬘兲 = VBk共H兲 for k = 0 , . . . , m − 1. Here the equality of the
algebraic sets means they are isomorphic via projective linear transformations of complex projective spaces.
Proof: Suppose H⬘ⱮLUH, then there exist positive numbers p1 , . . . , ps and local unitary transformations U1 丢 V1 , . . . , Us 丢 Vs, such that, ⌺i piUi 丢 ViH共Ui 丢 Vi兲† = H⬘. Let H = ⌺si qi P␺i, where s
= dim共range共H兲兲, q1 , . . . , qs are eigenvalues of H and ␺1 , . . . , ␺s are eigenvectors of H. Then it is
clear that 共Ui 丢 Vi兲H共Ui 丢 Vi兲† = ⌺sj q j P共Ui 丢 Vi兲␺ j and thus H⬘ = ⌺i,j piq j P共Ui 丢 Vi兲␺ j. This is a representation of H⬘ as a convex combination of projections. From our above observation VAk共H⬘兲 can be
computed from vectors 共U1 丢 V1兲␺1 , . . . , 共U1 丢 V1兲␺s, since they are linear independent and s
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052101-14
J. Math. Phys. 47, 052101 共2006兲
Hao Chen
= dim共range共H⬘兲兲. Hence VAk共H⬘兲 = VAk共共U1 丢 V1兲H兲 from the definition. Thus the conclusion follows from Theorem 1.
Let S be the swap operator on the bipartite system HAn 丢 HBn defined by S兩ij典 = 兩ji典. For any
Hamiltonian H, S共H兲 = SHS† corresponds to the Hamiltonian evolution of H with A and B interchanged. It is very interesting to consider the problem if H can be simulated by S共H兲 efficiently.
This led to some important consequences in the discussion VII of Ref. 5. For example it was
shown there are examples that H and S共H兲 cannot be simulated efficiently with one another in
higher dimensions 共n 艌 3兲. Thus in higher dimensions nonlocal degrees of freedom of Hamiltonians cannot be characterized by quantities that are symmetric with respect to A and B, such as
eigenvalues. This conclusion is also obtained from our example and Corollary 5 in the next
section. From Corollary 3 we have the following necessary condition about HⱮLUS共H兲.
Corollary 4: Let H be a semipositive Hamiltonian on HAn 丢 HBn. Suppose HⱮLUS共H兲. Then
k
VA共H兲 = VBk共H兲 for k = 0 , . . . , n − 1.
The following is a Hamiltonian H on 3 ⫻ 3 system for which H cannot be simulated efficiently
by S共H兲.
Example 11: H = P兩␾1典 + P兩␾2典 + P兩␾3典, where
兩 ␾ 1典 =
兩 ␾ 2典 =
兩 ␾ 3典 =
1
冑3 共兩11典 + 兩21典 + 兩32典兲,
1
冑1 + 兩v兩2 共兩12典 + v兩22典兲,
共14兲
1
冑1 + 兩␭兩2 共兩13典 + ␭兩23典兲.
Then it is easy to compute that VA2 共H兲 is the sum of three lines in CP2 defined by r1 + r2 = 0,
r1 + vr2 = 0, and r1 + ␭r2 = 0 for v ⫽ ␭ and both v , ␭ are not 1, and VB2 共H兲 is the sum of 2 lines in CP2
defined by r2 = 0 and r3 = 0. Thus we cannot have HⱮLUS共H兲.
IX. A CONTINUOUS FAMILY OF STATES AND HAMILTONIANS RELATED TO ELLIPTIC
CURVES
From a physical point of view, it is very interesting to have isospectral 关i.e., eigenvalues of
␳ , trA共␳兲 , trB共␳兲 are the same兴 mixed states, but they are not equivalent under local unitary transformations. This phenomenon indicates that we cannot obtain a complete understanding of a
bipartite quantum system by just studying the local and global properties of the spectra of the
system. Some examples of such mixed states have been found by several authors 共see Nielsen and
Kempe37 and references therein兲. Their result would imply the existence of continuously many
isospectral no-local-equivalent mixed states. Here we give a continuous family of such mixed
states by the theory of elliptic curves.9
Let H = HA3 丢 HB3 and ␳␩1,␩2,␩3 = 31 共P兩v1典 + P兩v2典 + P兩v3典兲 共␩1 , ␩2 , ␩3 are real parameters兲, a continuous family of mixed states on H, where
1
i␩1
兩11典 + 兩22典 + 兩33典兲,
1
i␩2
兩12典 + 兩23典 + 兩31典兲,
1
i␩3
兩13典 + 兩21典 + 兩32典兲.
兩 v 1典 =
冑3 共e
兩 v 2典 =
冑3 共e
兩 v 3典 =
冑3 共e
共15兲
It is easy to calculate that ⌺riAi 共up to a constant兲 is the following 3 ⫻ 3 matrix:
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052101-15
Quantum entanglement and geometry of determinantal
冢
J. Math. Phys. 47, 052101 共2006兲
冣
e i␩1r 1
r3
r2
i␩2
r2
e r1
r3
.
i␩3
r3
r2
e r1
Thus VA2 共␳␩1,␩2,␩3兲 is defined by ei共␩1+␩2+␩3兲r31 + r32 + r33 − 共ei␩1 + ei␩2 + ei␩3兲r1r2r3 = 0 in CP2. With
ei共␩1+␩2+␩3兲/3r1 = r1⬘ we have r1⬘3 + r32 + r33 − 关共ei␩1 + ei␩2 + ei␩3兲 / 共ei共␩1+␩2+␩3兲/3兲兴r1⬘r2r3 = 0. This is a family
of elliptic curves.
It is easy to check that three nonzero eigenvalues of ␳␩1,␩2,␩3 , trA共␳␩1,␩2,␩3兲 , trB共␳␩1,␩2,␩3兲 are all
the same value 31 for different parameters. In this case we have a family of isospectral 共both global
and local兲 mixed states ␳␩1,␩2,␩3. Set g共␩1 , ␩2 , ␩3兲 = 共ei␩1 + ei␩2 + ei␩3兲 / 共ei共␩1+␩2+␩3兲/3兲.
Theorem 11: ␳␩1,␩2,␩3 is entangled mixed state when 共g共␩1 , ␩2 , ␩3兲兲3 ⫽ 27. Moreover ␳␩1,␩2,␩3
and ␳␩⬘,␩⬘,␩⬘ are not equivalent under local unitary transformations if k共g共␩1 , ␩2 , ␩3兲兲
1 2 3
⫽ k共g共␩1⬘ , ␩2⬘ , ␩3⬘兲兲, where k共x兲 = x3共x3 + 216兲3 / 共−x3 + 27兲3 is the moduli function of elliptic curves.
Proof: The conclusion follows from Theorem 3, Theorem 1 and the well-known fact about
elliptic curves 共see Ref. 9兲.
In Ref. 36 Nielsen gave a beautiful necessary and sufficient condition for the bipartite pure
state 兩␺典 that can be transformed to the pure state 兩␾典 by local operations and classical communication 共LOCC兲 based on the majorization between the eigenvalue vectors of the partial traces of
兩␺典 and 兩␾典. In Ref. 7 an example was given, from which we know that Nielsen’s criterion cannot
be generalized to multipartite case, 3EPR and 2GHZ are understood as pure states in a 4 ⫻ 4
⫻ 4 quantum system, they have the same eigenvalue vectors when traced over any subsystem.
However it is proved that they are LOCC incomparable in Ref. 7.
In the following example, a continuous family 兩兵␾其␩1,␩2,␩3典 of pure states in tripartite quantum
system HA3 丢 HA3 丢 HA3 is given, the eigenvalue vectors of trAi共P兩␾␩ ,␩ ,␩ 典兲 , trAiA j共P兩␾␩ ,␩ ,␩ 典兲 are
1
2
3
1
2
3
1
2
3
independent of parameters ␩1 , ␩2 , ␩3. However, the generic pure states in this family are entangled
and LOCC incomparable. This gives stronger evidence that it is hopeless to characterize the
entanglement properties of multipartite pure states by only using the eigenvalue spectra of their
partial traces.
Let H = HA3 丢 HA3 丢 HA3 be a tripartite quantum system and 兩␾␩1,␩2,␩3典 = 共1 / 冑3兲共兩v1典 丢 兩1典
1
2
3
+ 兩v2典 丢 兩2典 + 兩v3典 丢 兩3典兲, where 兩v1典 , 兩v2典 , 兩v3典 are as in 共15兲. This is a continuous family of pure
states in H parametrized by three real parameters. We can check that the eigenvalue vector of any
partial trace of P兩␾␩ ,␩ ,␩ 典 is a constant vector. On the other hand, it is clear that trA3共P兩␾␩ ,␩ ,␩ 典兲
1
2
3
1
2
3
= 31 共P兩v1典 + P兩v2典 + P兩v3典兲 is a rank 3 mixed state in HA3 丢 HA3 . 兩␾␩1,␩2,␩3典 and 兩␾␩⬘,␩⬘,␩⬘典 are not
1
2
1 2 3
equivalent under local unitary transformations if k共g共␩1 , ␩2 , ␩3兲兲 ⫽ k共g共␩1⬘ , ␩2⬘ , ␩3⬘兲兲, since their
corresponding traces over A3 are not equivalent under local unitary transformations of HA3
1
3
丢 HA from Theorem 11. Hence the generic members of this family of pure states in tripartite
2
quantum system H are entangled and LOCC incomparable from Theorem 1 in Ref. 7.
We can also consider the following continuous family of semipositive Hamiltonians depending on three real parameters, H␩1,␩2,␩3 = P兩v1典 + P兩v2典 + P兩v3典, where v1 , v2 , v3 are as in 共15兲. As calculated above, VA2 共H␩1,␩2,␩3兲 is just the elliptic curve in CP2 defined by r31 + r32 + r33 − 关共ei␩1 + ei␩2
+ ei␩3兲 / 共ei共␩1+␩2+␩3兲/3兲兴r1r2r3 = 0. The elliptic curve VA2 共H␩1,␩2,␩3兲 is not isomorphic to the elliptic
curve VA2 共H␩⬘,␩⬘,␩⬘兲 if k共g共␩1 , ␩2 , ␩3兲兲 ⫽ k共g共␩1⬘ , ␩2⬘ , ␩3⬘兲兲. Thus we have the following Corollary of
1 2 3
Theorem 9.
Corollary 5: H␩⬘,␩⬘,␩⬘ cannot be simulated by H␩1,␩2,␩3 efficiently by using local unitary
1 2 3
transformations, i.e., we cannot have H␩⬘,␩⬘,␩⬘ⱮLUH␩1,␩2,␩3, if k共g共␩1 , ␩2 , ␩3兲兲 ⫽ k共g共␩1⬘ , ␩2⬘ , ␩3⬘兲兲,
1 2 3
though the three nonzero eigenvalues of H␩1,␩2,␩3, H␩⬘,␩⬘,␩⬘ and their partial traces are all 1.
1
2
3
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052101-16
J. Math. Phys. 47, 052101 共2006兲
Hao Chen
X. CONSTRUCTING ENTANGLED PPT MIXED STATES
As mentioned in the Introduction, the first several entangled PPT mixed states were constructed in Ref. 24 based on Horodecki’s range criterion of separable states, which asserts that a
separable mixed state must include sufficiently many separable pure states in its own range 共see
Refs. 24 and 28兲. This range criterion of separable mixed states was also the base to construct PPT
entangled mixed states in the context of unextendible product base 共UPB兲 studied by Bennett,
DiVincenzo, Mor, Shor, Smolin, and Terhal in Ref. 6. 共We should mention that unextendible
product bases also have other physical significance nonlocality without entanglement, see Refs. 6
and 28.兲 It is always interesting and important to have more methods to construct entangled PPT
mixed states. In this section, we give an example to show how our separability criterion Theorem
3 can be used to construct entangled mixed states which are invariant under partial transposition
共thus PPT and bound entanglement兲 systematically.
In the following example we construct a family of rank 7 mixed states 兵␳e1,e2,e3其 共e1 , e2 , e3 are
real parameters兲 with ␳e1,e2,e3 = ␳ePT,e ,e 共hence PPT automatically兲 on H = HA4 丢 HB6 . We prove that
1 2 3
they are entangled by Theorem 3 共thus bound entanglement兲 for generic parameters e1 , e2 , e3 and
parameters 共e1 , e2 , e3兲 = 共0 , 0 , 1兲. This family and the method used here can be easily generalized to
construct entangled mixed states with ␳ = ␳PT systematically.
Consider the following four 6 ⫻ 7 matrices:
冢
冢
A1 =
A2 =
A3 =
冢
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 2 0 0 1
0 0 0 0 2 0 0
0 0 0 0 0 2 0
冣
冣
0
1 1 −1 0 0 1
1
0 1
0
0 0 0
1
1 0
0
0 0 0
−1 0 0
0
1 1 0
0
0 0
1
0 1 0
0
0 0
1
1 0 0
e2 + e3 e1
0
0
,
,
0
0
0
e2
e3
0
0
0
0
0
e3
e1 + e2
0
0
0
0
0
0
0
e2 + e3 e1
0
0
0
0
0
e1
e2
e3
0
0
0
0
0
e3 e1 + e2 0
e1
冣
,
where e1 , e2 , e3 are real parameters, and A4 = 共I6 , 0兲, where I6 is the 6 ⫻ 6 unit matrix.
Let A be a 24⫻ 7 matrix with four blocks A1 , A2 , A3 , A4 where the 24 rows correspond to the
standard basis 兵兩11典 , . . . , 兩16典 , . . . , 兩41典 , . . . , 兩46典其. Let ␳e1,e2,e3 be 共1 / D兲AA† 共where D is a normalizing constant兲, a mixed state on H. It is easy to check that AiA†j = A jA†i , hence ␳e1,e2,e3 is invariant
under partial transposition and thus PPT.
Let 兩␺1典 , . . . , 兩␺7典 苸 HA4 丢 HB6 be seven vectors corresponding to seven columns of the matrix A.
It is clear that the range of ␳e1,e2,e3 is the linear span of 兩␺1典 , . . . , 兩␺7典. When e1 = e2 = 0, e3 = 1,
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052101-17
J. Math. Phys. 47, 052101 共2006兲
Quantum entanglement and geometry of determinantal
兩␺2典 − 兩␺3典 = 共兩1典 + 兩4典 − 兩2典兲 丢 共兩2典 − 兩3典兲. Thus there are some separable pure states in the range of
␳0,0,1. We will show that ␳0,0,1 and ␳e1,e2,e3 for generic parameters e1 , e2 , e3 are entangled by our
separability criterion Theorem 3.
As in the proof of Theorem 2 it is easy to compute F = r1A1 + r2A2 + r3A3 + r4A4,
冢
u1
r 2 + e 1r 3
r2
− r2
0
0
r2
r 2 + e 1r 3
u1⬘
r 2 + e 3r 3
0
0
0
0
r2
r 2 + e 3r 3
u1⬙
0
0
0
0
− r2
0
0
u2
r 2 + e 1r 3
r2
r1
0
0
0
r 2 + e 1r 3
u2⬘
0
0
0
r2
r 2 + e 3r 3
r 2 + e 3r 3 0
u2⬙
0
冣
共16兲
冣
共17兲
where u1 = r1 + r4 + 共e2 + e3兲r3, u1⬘ = r1 + r4 + e2r3, u1⬙ = r1 + r4 + 共e1 + e2兲r3 and u2 = 2r1 + r4 + 共e2 + e3兲r3,
u2⬘ = 2r1 + r4 + e2r3, u2⬙ = 2r1 + r4 + 共e1 + e2兲r3.
We consider the following matrix F⬘ which is obtained by adding the seventh column of F to
the fourth column of F and adding r2 / r1 of the seventh column to the first column,
冢
v1
r 2 + e 1r 3
r2
0
0
0
r2
r 2 + e 1r 3
u1⬘
r 2 + e 3r 3
0
0
0
0
r2
r 2 + e 3r 3
u1⬙
0
0
0
0
0
0
0
v2
r 2 + e 1r 3
r2
r1
0
0
0
r 2 + e 1r 3
u2⬘
0
0
0
r2
r 2 + e 3r 3
r 2 + e 3r 3 0
u2⬙
0
,
where v1 = r1 + r4 + 共e2 + e3兲r3 + 共r22 / r1兲, v2 = 3r1 + r4 + 共e2 + e3兲r3.
It is clear that the determinantal varieties defined by F and F⬘ are the same in the affine chart
C3 defined by r1 ⫽ 0. Consider the zero locus Z1 defined by the condition that the determinants of
the two diagonal 3 ⫻ 3 submatrices of the first 6 ⫻ 6 submatrix in 共17兲 are zero, locus Z2 defined
by the condition that the first three rows in 共17兲 are linear dependent and the locus Z3 defined by
the condition that the last three rows in 共17兲 are linear dependent, it is clear that VA5 共␳e1,e2,e3兲 艚 C3
is the sum of Z1 , Z2 , Z3. We can use the affine coordinates r2⬘ = r2 / r1, r3⬘ = r3 / r1, r4⬘ = r4 / r1 on the
affine chart C3 of CP3 defined by r1 ⫽ 0. In this affine coordinate system all r2 , r3 , r4 should be
replaced by r2⬘ , r3⬘ , r4⬘ and r1 should be replaced by 1 in 共17兲. Now we analyze VA5 共␳0,0,1兲. It is clear
that the following two planes H1 = 兵共r2⬘ , r3⬘ , r4⬘兲 : r2⬘ = r4⬘ + 1其, H2 = 兵共r2⬘ , r3⬘ , r4⬘兲 : r2⬘ = r4⬘ + 2其 are in
VA5 共␳0,0,1兲 艚 C3, since in the case r2⬘ = r4⬘ + 1 the second and the third rows of 共17兲 are linearly
dependent and in the case r2⬘ = r4⬘ + 2 the fifth and sixth rows of 共17兲 are linearly dependent. The
determinants of two 3 ⫻ 3 diagonal submatrices of the first 6 ⫻ 6 submatrix of 共17兲 are
共r2⬘ − r4⬘ − 1兲共共r2⬘兲3 + 共r2⬘兲2r4⬘ − 共r2⬘兲2 + 共r4⬘兲2 + r2⬘r3⬘ + r2⬘r4⬘ + r3⬘r4⬘ + r2⬘ + r3⬘ + 2r4⬘ + 1兲
and
共r2⬘ − r4⬘ − 2兲共共r4⬘兲2 − 2共r2⬘兲2 + r2⬘r3⬘ + r2⬘r4⬘ + r3⬘r4⬘ + 3r2⬘ + 2r3⬘ + 5r4⬘ + 6兲.
Let X1 and X2 be the zero locus of the second factors of the above two determinants. It is
obvious that X1 艚 X2 is in VA5 共␳0,0,1兲 艚 C3, we want to show that X1 艚 X2 \ H1 艛 H2 is a curve, not a
line. Take the point P = 共0 , 2 , −1兲 苸 X1 艚 X2 艚 H1, the tangent plane H3 of X2 at P is defined by
4r2⬘ + r3⬘ + 5r4⬘ = −3. If X1 艚 X2 is a line around the point P, this line is contained in H3 艚 X2. However we can easily find that H3 艚 X2 is defined by 3共r2⬘兲2 + 2共r4⬘兲2 + 4r2⬘r4⬘ + 4r4⬘ = 0. This polynomial
is irreducible and thus H3 艚 X2 is a curve around the point P. Thus X1 艚 X2 is a curve around the
point P. It is easy to check that X1 艚 X2 is not contained in H1 around the point P. This implies that
VA5 共␳e1,e2,e3兲 艚 C3 共actually the locus Z1兲 contains a curve 共not a line兲 for generic parameters
e1 , e2 , e3 共including parameters 0,0,1兲 from algebraic geometry. Thus if VA5 共␳e1,e2,e3兲 艚 C3 is the sum
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052101-18
Hao Chen
J. Math. Phys. 47, 052101 共2006兲
of 共affine兲 linear subspaces, it must contain a dimension 2 affine linear subspace H4 other than H1
and H2 of the affine chart C3. Thus the determinants of all 6 ⫻ 6 submatrices of 共17兲 must contain
a 共fixed兲 affine linear form 共i.e., a degree one polynomial of r2 , r3 , r4 which may contain a constant
term兲 other than r2 − r4 − 1 and r2 − r4 − 1 as one of their factors. This affine linear form defines that
dimension 2 linear affine subspace H4 of C3. However it is easy to check that this is impossible for
generic parameters e1 , e2 , e3 共including parameters 0,0,1兲. We know that VA5 共␳e1,e2,e3兲 艚 C3 cannot
be the sum of 共affine兲 linear subspaces of C3 for generic e1 , e2 , e3 共including parameters 0,0,1兲.
Thus from Theorem 3, ␳e1,e2,e3 is entangled for generic parameters e1 , e2 , e3 共including parameters
0,0,1兲.
Theorem 12: The mixed states ␳e1,e2,e3’s, which are invariant under partial transposition, are
entangled for generic parameters and 共e1 , e2 , e3兲 = 共0 , 0 , 1兲.
Remark 4: ␳0,0,1 is the first example of PPT entangled mixed state 共thus bound entanglement兲
with some separable pure states in its range.
From the construction in this example we can see if A1 , . . . , Am are mn ⫻ t matrices satisfying
AiA†j = A jA†i , A is the m ⫻ 1 matrix with ith block Ai and the rows of A correspond to the basis
兩11典 , . . . , 兩1n典 , . . . , 兩m1典 , . . . , 兩mn典 of HAm 丢 HBn, then the mixed state ␳ = 共1 / D兲AA†, where D is a
normalized constant, is invariant under partial transpose. It is not very difficult to find such
matrices. For the purpose that the constructed mixed state ␳ is entangled 共thus a bound entangled
mixed state兲, we just need that the determinantal variety 兵共r1 , . . . , rm兲 : rank共⌺riAi兲 艋 n − 1其 is not
linear. We know from algebraic geometry, it is not very hard to find such matrices A1 , . . . , Am.
However, as illustrated in this Example we do need some explicit calculation to prove this point.
Thus our separability criterion and the method used in this Example offer a new systematic way to
construct PPT bound entangled mixed states.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China Grant Nos.
60542006, 60433050 and Distinguished Young Scholar Grant No. 10225106.
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