Partial separability/entanglement violates distributive rules
aa r X i v : . [ qu a n t - ph ] J un PARTIAL SEPARABILITY/ENTANGLEMENT VIOLATESDISTRIBUTIVE RULES
KYUNG HOON HAN, SEUNG-HYEOK KYE AND SZIL ´ARD SZALAY
Abstract.
We found three qubit Greenberger-Horne-Zeilinger diagonal states whichtell us that the partial separability of three qubit states violates the distributive ruleswith respect to the two operations of convex sum and intersection. The gaps betweenthe convex sets involving the distributive rules are of nonzero volume. Introduction
Pure states in classical probability theory are uncorrelated, which is not the case inquantum probability theory, where this nonclassical form of correlation is called entan-glement [1]. Beyond the conceptual questions it raises [2, 3], entanglement plays a keyrole in the physics of strongly correlated many-body systems [4], and also finds directapplications in quantum information theory [5]. Mixed states in classical probabilitytheory arise as statistical mixtures (convex combinations) of pure, hence uncorrelatedstates. Again, this is not the case in quantum probability theory, and states which aremixtures of uncorrelated states are called separable, while the others are entangled [6].In the case of multipartite systems, the partitions of the total system into sub-systems give rise to various notions of partial separability. In the tripartite case withthe elementary subsystems A , B and C , we have three nontrivial partitions A - BC , B - CA and C - AB , and the corresponding partial separability properties are called A - BC -separability, B - CA -separability and C - AB -separability, respectively. We call these basic biseparabilities.It is natural to consider the intersections (also called partial separability classes)and convex hulls of the three convex sets consisting of the above three kinds of basicbiseparable states. For example, the intersection of them [7], the intersections of two ofthem [9, 8] and the convex hull of them [11, 10] have been considered. More recently,the convex hulls of two of them have also been considered together with intersectionsand complements of convex sets arising in the way [12, 13, 14], leading to the descriptionof the hierarchy of the intersections [15]. See also [16, 17, 18] for further developments.Recall that the intersection and convex hull of the convex sets of three basic biseparablestates give rise to fully biseparable and biseparable states, respectively. Tripartite stateswhich are not biseparable are called genuinely multipartite entangled. Mathematics Subject Classification.
Key words and phrases. partially separable, partially entangled, lattice, distributive rule. n this context, we consider the lattice generated by three convex sets of threequbit basic biseparable states with respect to the above mentioned two operations,intersection and convex hull. In this way, we deal with convex hulls of intersections, aswell as intersections of convex hulls of convex sets arising from basic biseparability. Onemay go further to investigate the whole structures of partial separability and partialentanglement. Due to technical reasons, we will consider the three convex cones α , β and γ of all un-normalized A - BC , B - CA and C - AB biseparable three qubit states,respectively. We note that the convex hull σ ∨ τ of two convex cones σ and τ coincideswith the sum σ + τ of them. The intersection of two convex cones σ and τ will bedenoted by σ ∧ τ following the lattice notations. In general, the two operations ∧ and ∨ among convex sets obey associative rule and commutative rule. Furthermore, theyalso satisfy the following relations σ ∨ σ = σ, σ ∧ σ = σ, ( σ ∨ τ ) ∧ σ = σ, ( σ ∧ τ ) ∨ σ = σ, and so they give rise to a lattice.A lattice ( L , ∨ , ∧ ) is distributive if the identities x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( x ∧ z ) , x ∨ ( y ∧ z ) = ( x ∨ y ) ∧ ( x ∨ z )hold for all x, y, z ∈ L . In general, the inequalities x ∧ ( y ∨ z ) ≥ ( x ∧ y ) ∨ ( x ∧ z ) , x ∨ ( y ∧ z ) ≤ ( x ∨ y ) ∧ ( x ∨ z )always hold trivially. We denote by L the lattice generated by three convex cones α , β and γ . Therefore, L is the smallest lattice containing the convex cones α , β and γ in the 64-dimensional real vector space of all self-adjoint three qubit matrices. Thepurpose of this note is to show that the lattice L does not satisfy the distributive rules.More precisely, we show that both inequalities( α ∧ β ) ∨ ( α ∧ γ ) ≤ α ∧ ( β ∨ γ ) , (1) β ∨ ( γ ∧ α ) ≤ ( β ∨ γ ) ∧ ( β ∨ α )(2)are strict. Furthermore, the gaps between the two sets are of nonzero volume, in bothcases. We also show that the lattice L does not satisfy the modularity which is weakerthan distributivity.We note that a state ̺ in the gap α ∧ ( β ∨ γ ) \ ( α ∧ β ) ∨ ( α ∧ γ ) by the strictinequality in (1) has the following properties: • ̺ is A - BC biseparable, and it is a mixture of B - CA and C - AB biseparablestates, • but, it is not a mixture of a simultaneously A - BC and B - CA biseparable stateand a simultaneously A - BC and C - AB biseparable state.We will find such states among GHZ diagonal states. On the other hand, a state ̺ arising by the strict inequality in (2) has the following properties: ̺ is a mixture of B - CA and C - AB biseparable states, and it is also a mixtureof a B - CA and A - BC biseparable states, • but it is not a mixture of B - CA biseparable state and a simultaneously C - AB and A - BC biseparable state.Examples will be found among GHZ diagonal states. After we provide backgroundsfor this in the next section, we find analytic examples in Section 3. We also consider inSection 4 the lattices arising from partial separability in general multi-partite systemsto see that both distributivity and modularity are violated in them, too. We close thepaper to ask several questions.The authors are grateful to the referee for bringing our attention to the generalmulti-partite cases. 2. X -shaped states We will find required examples among so called X -shaped states whose entries arezeros except for diagonal and anti-diagonal entries by definition. A self-adjoint X -shaped three qubit matrix is of the form X ( a, b, z ) = a z a z a z a z ¯ z b ¯ z b ¯ z b ¯ z b , with a, b ∈ R and z ∈ C . Here, C ⊗ C ⊗ C is identified with the vector space C with respect to the lexicographic order of indices. Note that X ( a, b, z ) is a state if andonly if a i , b i ≥ √ a i b i ≥ | z i | for each i = 1 , , , X -state X ( a, b, z ) is GHZ diagonal if and only if a = b and z ∈ R . In this case, we use thenotation X (cid:18) az (cid:19) = X ( a, a, z ) . By a pair { i, j } , we will mean an unordered set with two distinct elements forsimplicity throughout this paper. For a given three qubit X -shaped state ̺ = X ( a, b, z ),we consider the inequalities S [ i, j ] : min {√ a i b i , p a j b j } ≥ max {| z i | , | z j |} ,S [ i, j ] : min (cid:8) √ a i b i + p a j b j , √ a k b k + √ a ℓ b ℓ (cid:9) ≥ max {| z i | + | z j | , | z k | + | z ℓ |} ,S : P j = i p a j b j ≥ | z i | , i = 1 , , , { i, j } , where { k, ℓ } is chosen so that { i, j, k, l } = { , , , } . By [22, Propo-sition 5.2], we have the following: ̺ ∈ α if and only if S [1 ,
4] and S [2 ,
3] hold, • ̺ ∈ β if and only if S [1 ,
3] and S [2 ,
4] hold, • ̺ ∈ γ if and only if S [1 ,
2] and S [3 ,
4] hold.We also have the following • ̺ ∈ β ∨ γ if and only if S [1 ,
4] (equivalently S [2 , • ̺ ∈ γ ∨ α if and only if S [1 ,
3] (equivalently S [2 , • ̺ ∈ α ∨ β if and only if S [1 ,
2] (equivalently S [3 , S will not be used in this paper, but it is thecharacteristic inequality for the convex cone α ∨ β ∨ γ [20, 22, 14, 21].We will consider the above inequality S for arbitrary two pairs { i, j } and { k, ℓ } asfollows: S [ i, j | k, ℓ ] : min (cid:8) √ a i b i + p a j b j , √ a k b k + √ a ℓ b ℓ (cid:9) ≥ max {| z i | + | z j | , | z k | + | z ℓ |} .If { i, j } = { k, ℓ } then the inequality S [ i, j | k, ℓ ] holds automatically for any X -states ̺ = X ( a, b, z ). If { i, j } ∩ { k, ℓ } = ∅ then the three inequalities S [ i, j ], S [ k, ℓ ] and S [ i, j | k, ℓ ] are identical. In the other cases, the resulting inequalities are new ones.In order to get required examples in the gaps between convex cones in the inequal-ities (1) and (2), we proceed to characterize the following convex cones(3) α ∨ ( β ∧ γ ) , β ∨ ( γ ∧ α ) , γ ∨ ( α ∧ β ) . For this purpose, we will use the duality among convex cones in a real vector spacewith a bi-linear pairing h , i . For a convex cone C , the dual cone C ◦ is defined by C ◦ = { x ∈ V : h x, y i ≥ y ∈ C } . We are now working in the real vector spaces of all three qubit self-adjoint X -shapedmatrices, where the bi-linear pairing is defined by h x, y i = Tr ( yx t ), as usual. See [14].Every closed convex cone C satisfies the relation C = ( C ◦ ) ◦ , which tells us that x ∈ C if and only if h x, y i ≥ y ∈ C ◦ .The dual cones of the cones in (3) have also been characterized in [14]. For agiven X -shaped self-adjoint matrix W = X ( s, t, u ) with s i , t i ≥ u ∈ C , we haveconsidered the inequalities W , W , W given by W [ i, j ] : √ s i t i + √ s j t j ≥ | u i | + | u j | ,W [ i, j ] : P k = j √ s k t k ≥ | u i | , P k = i √ s k t k ≥ | u j | ,W : P i =1 √ s i t i ≥ P i =1 | u i | , for a pair { i, j } . Then we have the following by [14, Proposition 3.3]: • W ∈ α ◦ if and only if W [1 ,
4] and W [2 ,
3] hold, • W ∈ β ◦ if and only if W [1 ,
3] and W [2 ,
4] hold, • W ∈ γ ◦ if and only if W [1 ,
2] and W [3 ,
4] hold. n the other hand, we also have the following by [14, Theorem 5.2]: • W ∈ β ◦ ∨ γ ◦ if and only if W [1 , W [2 ,
3] and W hold, • W ∈ γ ◦ ∨ α ◦ if and only if W [1 , W [2 ,
4] and W hold, • W ∈ α ◦ ∨ β ◦ if and only if W [1 , W [3 ,
4] and W hold.For given a, b ∈ R with nonzero entries, and z ∈ C with arg z i = θ i , we considerthe following self-adjoint matrices: W [ i,j | k,ℓ ] = X r b i a i E i + s b j a j E j , r a i b i E i + r a j b j E j , − e − i θ k E k − e − i θ ℓ E l ! for pairs { i, j } and { k, ℓ } , where { E i } is the usual orthonormal basis of C . Then theinequality h W [ i,j | k,ℓ ] , X ( a, b, z ) i ≥ √ a i b i + p a j b j ≥ | z k | + | z ℓ | . Now, we present the main result.By [14, Proposition 2.2], the conditions give rise to necessary criteria for general statesto belong to classes in (3) in terms of diagonal and anti-diagonal entries. Theorem 2.1.
For a given three qubit X -state ̺ = X ( a, b, z ) , we have the following: (i) ̺ ∈ α ∨ ( β ∧ γ ) if and only if S [ i, j | k, ℓ ] holds whenever { i, j } , { k, ℓ } are twoof { , } , { , } , { , } , { , } ; (ii) ̺ ∈ β ∨ ( γ ∧ α ) if and only if S [ i, j | k, ℓ ] holds whenever { i, j } , { k, ℓ } are twoof { , } , { , } , { , } , { , } ; (iii) ̺ ∈ γ ∨ ( α ∧ β ) if and only if S [ i, j | k, ℓ ] holds whenever { i, j } , { k, ℓ } are twoof { , } , { , } , { , } , { , } .Proof. We will prove (i). For the ‘only if’ part, we first consider the case a i , b i > i = 1 , , ,
4. Then we see that W [ i,j | k,ℓ ] belongs to α ◦ ∧ ( β ◦ ∨ γ ◦ ) whenever { i, j } , { k, ℓ } are two of { , } , { , } , { , } , { , } , by checking the required inequalities W [1 , W [2 , W [1 , W [2 ,
3] and W . Now, the inequality h W [ i,j | k,ℓ ] , ̺ i ≥ S [ i, j | k, ℓ ] for such { i, j } , { k, ℓ } . If a or b has a zero entry, then weconsider ̺ + ε X ( , , ) with arbitrary ε > ε → = (1 , , ,
1) and = (0 , , , • if ̺ = X ( a, b, z ) satisfies S [ i, j | k, ℓ ] whenever { i, j } , { k, l } are two of { , } , { , } , { , } , { , } , and W = X ( s, t, u ) satisfies W [1 , W [2 , W [1 , W [2 ,
3] and W , then h W, ̺ i ≥ ̺ ∈ α then there is nothing to prove,and so we may assume that ̺ / ∈ α . Without loss of generality, we may also assumethat | z | > √ a b . Putting p = max {| z | , | z |} , we havemin { p a b , p a b } ≥ | z | + (cid:16) p − p a b (cid:17) , y the inequalities S [ i, j | k, ℓ ] for { i, j } = { , } , { , } and { k, ℓ } = { , } , { , } .Therefore, we have X i =2 √ s i t i p a i b i − | u || z | ≥ ( √ s t + √ s t ) min { p a b , p a b } + √ s t | z | − | u || z |≥ X i =2 √ s i t i − | u | ! | z | + ( √ s t + √ s t ) (cid:16) p − p a b (cid:17) ≥ X i =2 √ s i t i − | u | ! p a b + ( √ s t + √ s t ) (cid:16) p − p a b (cid:17) = ( √ s t − | u | ) p a b + ( √ s t + √ s t ) p, by the inequality W [4 ,
1] and the assumption | z | > √ a b . We also have √ s t p a b − X i =1 | u i || z i | ≥ √ s t p a b − | u | p a b − | u || z | − | u || z |≥ √ s t p a b − | u | p a b − ( | u | + | u | ) p. Summing up the above two inequalities, we have X i =1 ( √ s i t i p a i b i − | u i || z i | ) ≥ ( √ s t + √ s t − | u | − | u | ) p a b + ( √ s t + √ s t − | u | − | u | ) p, which is nonnegative by the inequalities W [1 ,
4] and W [2 , h X ( s, t, u ) , X ( a, b, z ) i = 12 X i =1 ( s i a i + t i b i + 2Re ( u i z i )) ≥ X i =1 ( √ s i t i p a i b i − | u i || z i | ) ≥ , which completes the proof. (cid:3) Examples
In order to get analytic examples distinguishing the convex cones in the inequalities(1) and (2), we consider GHZ diagonal states ̺ , = 18 X (cid:18) (cid:19) , ̺ , = 18 X (cid:18) (cid:19) , ̺ , = 112 X (cid:18) (cid:19) , and define ̺ s,t = (1 − s − t ) ̺ , + s̺ , + t̺ , = 18 X (cid:18) t s − t − s − t ts + t t t s (cid:19) , for real numbers s and t . We consider the convex set D of all three qubit states,which is a 63 affine dimensional convex body. We slice D by the 2-dimensional plane ❅❅❅❅❅❅❅❅❅❅❅❅❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜✂✂✂✂✂✂✂✂✂✂✂✂✂✂✂ ❛❛❛❛❛❛❛❛❛(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)▲▲▲▲▲▲▲▲❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜❜✂✂✂✂✂✂✂✂✂▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ ❡❡❡❡❡❡❡❡❡✆✆✆✆✆✆✆✆✆❆❆❆❆❆❆❆❜❜❜❜❜❜❜❜❜❜❜❜❜❜❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ ❝ (1 , ❝ (0 , ❝ ( − , s ( , )( − , )( − , ) − − ( − , − ) s ( − , ) s ( − , ) s ( − , − ) ❝ ( , − s ( , − ) γ γββ β Figure 1.
The difference between the whole quadrilateral R and thebigger hexagon H shows us that the distributive rules do not hold. Thestates ̺ s,t in the region labeled by β and γ belong to β \ γ and γ \ β ,respectively. The smaller hexagon represents the convex set α ∧ β ∧ γ which consists of PPT states.Π determined by ̺ , , ̺ , and ̺ , to get the pictures for various convex sets. We seethat ̺ s,t is a state if and only if | s + t | ≤ t, | t | ≤ s − t, | t | ≤ − s − t, | s | ≤ t if and only if ( s, t ) belongs to the region R = { ( s, t ) : s + t ≤ , − s + t ≤ , − s − t ≤ , s − t ≤ } which is a quadrilateral on the st -plane with the four vertices (1 , , − − , , D ∩ Π is also a quadrilateral withthe vertices(4) ̺ , , ̺ , , ̺ , − = 112 X (cid:18) − − − (cid:19) , ̺ − , = 18 X (cid:18) − − (cid:19) . See Figure 1.It is easily checked by S [1 ,
4] and S [2 ,
3] that the four states in (4) belong to theconvex set α , and so the convex set α on the plane Π is represented by the quadrilateral R itself. Using S [1 , S [2 ,
4] and S [1 , S [3 , s, t )such that ̺ s,t belongs to to β and γ , respectively. One may check that ̺ s,t ∈ β ifand only if it is a state and satisfies both inequalities 2 s + t ≤ − s + t ≤ herefore, the region for β on the plane Π is a pentagon with vertices( − , ) , ( − , ) , ( , − ) , ( , − , ( − , − ) . We also see that the region for γ is determined by t ≤ − s − t ≤ s + t ≤ − , ) , ( , ) , ( , − , ( − , − ) , ( − , ) . It is clear that the region for α ∨ β or γ ∨ α on the plane Π occupies all of thequadrilateral R . One may also easily check by S [1 ,
4] that the four states in (4)belong to β ∨ γ , and so the region for β ∨ γ coincides with the quadrilateral R . Moreprecisely, the convex sets( α ∨ β ) ∩ Π = ( β ∨ γ ) ∩ Π = ( γ ∨ α ) ∩ Π = α ∩ Πare represented by the quadrilateral R . The whole quadrilateral R in Figure 1 thusrepresents the regions for the following convex sets α, α ∨ β, β ∨ γ, γ ∨ α, α ∨ β ∨ γ, α ∧ ( β ∨ γ ) , α ∨ ( β ∧ γ )on the st -plane. It should be noted that they are strictly bigger than the convex hullgenerated by β ∩ Π and γ ∩ Π. For example, the state ̺ , ∈ Π in Figure 1 belongs tothe convex hull β ∨ γ , but it is not a mixture of states in β ∩ Π and γ ∩ Π. In fact, if ̺ , = ̺ + ̺ with ̺ ∈ β and ̺ ∈ γ then one can easily see that the X -parts of ̺ and ̺ should be of the form18 X (cid:18) ∗ (cid:19) and 18 X (cid:18) ∗ (cid:19) , respectively. Therefore, they never belong to the plane Π.Now, we use Theorem 2.1 to find the region for the convex set β ∨ ( γ ∧ α ). Forpairs { i, j } = { , } , { , } , { , } , { , } , { , } , { , } , the form 8 √ a i b i + 8 p a j b j forthe state ̺ s,t has the values2 + s, − s, t, − t, s, − s, respectively. On the other hands, 8 | z i | + 8 | z j | becomes | s + t | + | t | , | s + t | + | t | , | s + t | + | s | , | t | , | s | + | t | , | s | + | t | . Therefore, we see that a state ̺ s,t belongs to β ∨ ( γ ∧ α ) if and only if it belongs to γ ∨ ( α ∧ β ) if and only if the inequalitymin { s, − s, t, − t }≥ max {| s + t | + | t | , | s + t | + | s | , | t | , | s | + | t |} holds. One may check a point ( s, t ) ∈ R satisfies this inequality if and only if s + t ≤ , s + t ≤ , − s − t ≤ , This region is represented by the bigger hexagon H with the vertices( − , ) , ( , ) , ( , − ) , ( , − , ( − , − ) , ( − , ) n Figure 1. Therefore, the difference R \ H consisting of three triangles gives usexamples in(5) ( β ∨ γ ) ∧ ( β ∨ α ) \ β ∨ ( γ ∧ α ) , which shows that the strict inequality holds in (2).In order to consider the inequality (1), we first note the inequality(6) ( α ∧ β ) ∨ ( α ∧ γ ) ≤ α ∧ ( β ∨ ( γ ∧ α )) ≤ α ∧ ( β ∨ γ ) , which holds in general. By the first inequality, the region for ( α ∧ β ) ∨ ( α ∧ γ ) is alsocontained in H . In fact, it fills up all of H . Indeed, it is clear that five vertices of H belong to ( α ∧ β ) ∨ ( α ∧ γ ) except for ( − , ) by S [ i, j ]. We also see that52 ̺ − , = X (cid:18) − − (cid:19) = X (cid:18) − − (cid:19) + X (cid:18) − (cid:19) belongs to ( α ∧ β ) ∨ ( α ∧ γ ). Therefore, this coincides with α ∧ ( β ∨ ( γ ∧ α )) for states ̺ s,t , that is, the first inequality in (6) becomes an identity on the plane Π. Therefore,the difference R \ H again gives rise to examples in(7) α ∧ ( β ∨ γ ) \ ( α ∧ β ) ∨ ( α ∧ γ ) , for the strict inequality in (1).Now, we proceed to show that the gaps (5) and (7) arising in the distributiveinequalities have nonzero volume. We first note that the convex set S of all fullyseparable three qubit states generates the same affine manifold as the convex set D ofall three qubit states. See the discussion at the end of Section 7 in [23]. Therefore, allthe convex sets in (5) and (7) generate the same affine manifold. We also recall thata point x of a convex set C is called an interior point of C if it is an interior pointof C with respect to the affine space generated by C . We note that the state ̺ , is acommon interior point of the convex sets appearing in (5) and (7) as well as S and D .If we consider the line segment x t = (1 − t ) x + tx between an interior point x of aconvex set C and an arbitrary point x ∈ C then x t is also an interior point of C for0 < t <
1. See [24, Lemma 2.3]. Therefore, every interior point of [ α ∧ ( β ∨ γ )] ∩ Πis actually an interior point of α ∧ ( β ∨ γ ). For example, ̺ , is an interior point of α ∧ ( β ∨ γ ) which is a boundary point of ( α ∧ β ) ∨ ( α ∧ γ ). From this, we may concludethat the difference (7) has the nonempty interior by [23, Proposition 7.4]. The exactlysame argument also shows that the difference (5) also has the nonempty interior.We also consider the smaller hexagon H with vertices( − , ) , (0 , ) , ( , , ( , − , ( − , − ) , ( − , ̺ s,t in β ∧ γ . This represents also α ∧ β ∧ γ . Note thatan X -shaped state belongs to α ∧ β ∧ γ if and only if it is of positive partial transposeby [22, Theorem 5.3]. Therefore, the region H represents the region for PPT statesfor ̺ s,t . ecall that a lattice is called modular if x ≤ z implies x ∨ ( y ∧ z ) = ( x ∨ y ) ∧ z .This is the case if and only if the modular identity( x ∧ z ) ∨ ( y ∧ z ) = (( x ∧ z ) ∨ y ) ∧ z holds for every x, y and z . Every distributive lattice is modular. See [25, 26] forelementary properties of modular lattices. We exhibit examples of states showing thatthe first inequality in (6) is also strict, to conclude that the lattice L is not modular.To do this, we consider ̺ = 112 X (cid:18) (cid:19) , and put ̺ t = (1 − t ) ̺ , + t̺ = 124 X (cid:18) t − t − t t t t (cid:19) . We also consider W = X (cid:18) − − (cid:19) , which satisfies the inequality W [ i, j ] for { i, j } = { , } , { , } , { , } , { , } and W . Therefore, we see that W belongs to theconvex cone ( α ◦ ∨ β ◦ ) ∧ ( α ◦ ∨ γ ◦ ), which is the dual of the convex cone ( α ∧ β ) ∨ ( α ∧ γ ).Now, we see that h W, ̺ t i = (6 − t ) ≥ t ≤ . Therefore, we concludethat ̺ t does not belong to ( α ∧ β ) ∨ ( α ∧ γ ) for < t ≤
1. On the other hand, one caneasily check that ̺ ∈ α ∧ ( β ∨ ( γ ∧ α )) by Theorem 2.1. Therefore, we see that ̺ t alsobelongs to the same cone. In fact, ̺ t is an interior point of the cone α ∧ ( β ∨ ( γ ∧ α ))for 0 ≤ t <
1, because ̺ , is an interior point. Hence, we also see that the gap for thefirst inequality in (6) has also nonzero volume.Finally, we also characterize the full separability for the states we are considering.To do this, we summarize the results in [28, 29, 27]. See also [30, 23]. For a given GHZdiagonal state ̺ = X ( a, a, c ) with a, c ∈ R , we consider the following: λ = 2(+ c + c + c + c ) , λ = 2( − c − c + c + c ) ,λ = 2( − c + c − c + c ) , λ = 2( − c + c + c − c ) ,t = c ( − c + c + c + c ) − c c c , t = c (+ c − c + c + c ) − c c c ,t = c (+ c + c − c + c ) − c c c , t = c (+ c + c + c − c ) − c c c . When all the following inequalities(8) λ λ λ λ > , t t λ λ < , t t λ λ > ̺ = X ( a, a, c ) is fully separable if and only if the inequality(9) min { a , a , a , a } ≥ p ( λ λ + λ λ )( λ λ + λ λ )( λ λ + λ λ )8 √ λ λ λ λ is satisfied. In the other cases, ̺ is fully separable if and only if it is of PPT. For thestate ̺ s,t , the conditions (8) are given by s (3 s + 4 t ) < , (9 s + 18 st + 4 t )(9 s + 6 st − t ) > . − − ( − , − ) ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ r rr r ❡❡❡❡❡❡❡❡❡❊❊❊❊❊❊❊✆✆✆✆✆❆❆❆❆❆❆❆❜❜❜❜❜❜❜ ........................... ........................ ...................... ................... .................. ................... ..................... ...................... ....................... ........................................................................................................................................................................... ............................... ............................. .......................... ....................... ..................... ................... ................... ( − , )( − √ , ) ( , − ) ( −
11 + 5 √ , − √ Figure 2.
Two line segments through the origin are given by the con-ditions (8), and two curves surrounding the region of full separability aregiven by (9).We note that a point ( s, t ) on the line t = as satisfies this condition if and only if − (3 + √ ≤ a < − . On the other hands, the square of right side of (9) is given by t (9 s + 12 st − t )432 s (3 s + 4 t )respectively. See Figure 2.4. General multi-partite cases
Now, we turn our attention to general multi-partite system M A ⊗ M A ⊗ · · · ⊗ M A n .For a given partition Π of local systems { A , A , . . . , A n } , we denote by α Π the convexcone consisting of all partially separable (unnormalized) states with respect to thepartition Π. When M A i is the d i × d i matrices, we denote by L d ,d ,...,d n the latticegenerated by α Π through all nontrivial partitions Π with respect to two operations,intersection and convex hull. Therefore, elements of this lattice are convex cones sittingin the real vector space of all d d · · · d n × d d · · · d n Hermitian matrices. So far, wehave considered the lattice L , , . We close this section by showing that the lattice L d ,d ,...,d n also violates distributivity and modularity.For d ≥
2, we consider the canonical embedding ι d : ̺ ∈ M ̺ ⊕ tr( ̺ )1 d − ∈ M d nd the compression Q d : ̺ ∈ M d (cid:0) I (cid:1) ̺ (cid:18) I (cid:19) ∈ M . Here, the trace is a normalized one. Then, the composition Q d ◦ ι d is the identity map.In other words, M is unital completely positive (u.c.p.) complemented in M n . Thismeans that every local property of M p ⊗ M q ⊗ M r is hereditary to M ⊗ M ⊗ M . Wewrite ι = ι p ⊗ ι q ⊗ ι r : M ⊗ M ⊗ M → M p ⊗ M q ⊗ M r Q = Q p ⊗ Q q ⊗ Q r : M p ⊗ M q ⊗ M r → M ⊗ M ⊗ M for brevity.We retain the notations α , β and γ for the three qubit case, that is, α = α A - BC , β = α B - CA and γ = α C - AB generate the lattice L , , . To make clear, we will usenotations ˜ α , ˜ β and ˜ γ for generators of the lattice L p,q,r . It is easily seen that ̺ ∈ α implies ι ( ̺ ) ∈ ˜ α , and ω ∈ ˜ α implies Q ( ω ) ∈ α , and similarly for β and γ . In order toshow that L p,q,r violates the distributive rules, we first take ̺ ∈ ( α ∧ ( β ∨ γ )) \ ( α ∧ β ) ∨ ( α ∧ γ )in M ⊗ M ⊗ M . We can write ̺ = ̺ + ̺ for ̺ ∈ β and ̺ ∈ γ . We have ι ( ̺ ) ∈ ˜ α, ι ( ̺ ) ∈ ˜ β, ι ( ̺ ) ∈ ˜ γ in M p ⊗ M q ⊗ M r . Thus, ι ( ̺ ) belongs to ˜ α ∧ ( ˜ β ∨ ˜ γ ) in M p ⊗ M q ⊗ M r . Assume tothe contrary that the distributive rule holds in M p ⊗ M q ⊗ M r . Then, ι ( ̺ ) belongs to( ˜ α ∧ ˜ β ) ∨ ( ˜ α ∧ ˜ γ ), and so we can write ι ( ̺ ) = ω + ω with ω ∈ ˜ α ∧ ˜ β and ω ∈ ˜ α ∧ ˜ γ .Therefore, we have ̺ = Q ◦ ι ( ̺ ) = Q ( ω ) + Q ( ω ) ∈ ( α ∧ β ) ∨ ( α ∧ γ ) , which is a contradiction. The same argument can be applied to the other distributiverule. It should be noted that ω and ω themselves need not belong to the image of ι in the above argument. Non-modularity also follows from that of three qubit systemin the same fashion.For general multi-partite cases, we consider the canonical embedding ι : x ∈ M d ⊗ M d ⊗ M d x ⊗ ∈ M d ⊗ M d ⊗ M d ⊗ M d ⊗ · · · ⊗ M d n and the (normalized) partial traceid ⊗ tr : M d ⊗ M d ⊗ M d ⊗ M d ⊗ · · · ⊗ M d n → M d ⊗ M d ⊗ M d . Then, the composition (id ⊗ tr) ◦ ι is the identity map. In other words, M d ⊗ M d ⊗ M d is u.c.p. complemented in M d ⊗ M d ⊗ M d ⊗ M d ⊗ · · · ⊗ M d n . This means that everylocal property of the latter is hereditary to M p ⊗ M q ⊗ M r . By the exactly sameargument as before, one may check that the lattice L d ,...,d also violates distributivityand modularity. . Summary and further questions
In this paper, we have considered the lattice L generated by three basic convex sets α , β and γ consisting of all A - BC biseparable, B - CA biseparable and C - AB biseparablethree qubit states, respectively, with respect to the operations of convex hull ∨ andintersection ∧ . In this way, we may consider convex sets of partially separable statesobtaining by arbitrary convex hulls and intersections of α , β and γ , and the wholestructure of partial separability may be revealed by mathematical properties of thelattice L . For general theory for lattices, we refer to the monographs [25], [31] and [26].We gave the negative answer to the first natural question asking if this lattice isdistributive. The lattice L is not even modular. Another interesting question is to askif the lattice L has infinitely many elements. We conjecture this is the case. This meansthat there are infinitely many kinds of partial separability and partial entanglement. Itis known that a free lattice with three generators must have infinitely many elements.In this regard, it would be interesting to know if the lattice L is free or not.The next question is whether the lattice L is complemented or not. A lattice iscalled complemented if every element x has a complement y which satisfies x ∧ y = 0and x ∨ y = 1, where 0 and 1 denote the least and greatest elements, respectively. Theleast and the greatest elements of L are given by α ∧ β ∧ γ and α ∨ β ∨ γ , respectively.They represent the set of all fully biseparable and biseparable states, respectively.Especially, we would like to ask if α has a complement, that is, we ask if there exist σ ∈ L such that α ∧ σ = α ∧ β ∧ γ and α ∨ σ = α ∨ β ∨ γ . Recall that the set of all closedsubspaces of a Hilbert space makes a lattice, the subspace lattice , with respect to theclosed linear hull and intersection. This plays an important role in quantum logic andtheory of operator algebras. We note that the subspace lattice is non-distributive, butcomplemented.Finally, we ask how the lattice L d ,...,d n depends on the dimensions of the localsystems. In the case of tri-partite systems, we are asking if two lattices L , , and L p,q,r are isomorphic to each other.Acknowledgement: Both KHH and SHK were partially supported by the grantNRF-2017R1A2B4006655, Korea. SzSz was supported by the National Research, De-velopment and Innovation Fund of Hungary within the Researcher-initiated ResearchProgram (project Nr: NKFIH-K120569) and within the
Quantum Technology NationalExcellence Program (project Nr: 2017-1.2.1-NKP-2017-00001), by the Ministry for In-novation and Technology within the ´UNKP-19-4 New National Excellence Program ,and by the Hungarian Academy of Sciences within the
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Kyung Hoon Han, Department of Data Science, The University of Suwon, Gyeonggi-do 445-743, Korea
E-mail address : kyunghoon.han at gmail.com Seung-Hyeok Kye, Department of Mathematics and Institute of Mathematics, SeoulNational University, Seoul 151-742, Korea
E-mail address : kye at snu.ac.kr Szil´ard Szalay, Wigner Research Centre for Physics, 29-33, Konkoly-Thege Mikl´os,H-1121 Budapest, Hungary
E-mail address : szalay.szilard at wigner.mta.huszalay.szilard at wigner.mta.hu