Unique decompositions, faces, and automorphisms of separable states
aa r X i v : . [ m a t h . OA ] S e p UNIQUE DECOMPOSITIONS, FACES, ANDAUTOMORPHISMS OF SEPARABLE STATES
ERIK ALFSEN AND FRED SHULTZ
Abstract.
Let S k be the set of separable states on B ( C m ⊗ C n )admitting a representation as a convex combination of k pure prod-uct states, or fewer. If m > , n >
1, and k ≤ max ( m, n ), we showthat S k admits a subset V k such that V k is dense and open in S k ,and such that each state in V k has a unique decomposition as aconvex combination of pure product states, and we describe allpossible convex decompositions for a set of separable states thatproperly contains V k . In both cases we describe the associatedfaces of the space of separable states, which in the first case aresimplexes, and in the second case are direct convex sums of facesthat are isomorphic to state spaces of full matrix algebras. Asan application of these results, we characterize all affine automor-phisms of the convex set of separable states, and all automorphismsof the state space of B ( C m ⊗ C n ) that preserve entanglement andseparability. Introduction
A state on the algebra B ( C m ⊗ C n ) of linear operators is separableif it is a convex combination of product states. States that are notseparable are said to be entangled, and are of substantial interest inquantum information theory. Easily applied conditions for separabilityare known only for special cases, e.g., if m = n = 2, then a state isseparable iff its associated density matrix has positive partial transpose,cf. [14, 6]. Other necessary and sufficient conditions are known, e.g.[6], but are not easily applied in practice. An open question of greatinterest is to find a simple necessary and sufficient condition for a stateto be separable.A product state ω ⊗ τ is a pure state iff ω and τ are pure states.Thus a separable state is precisely one that admits a representation as Date : September 17, 2009.2000
Mathematics Subject Classification.
Primary 46N50, 46L30; Secondary81P68, 94B27.
Key words and phrases. entanglement, separable state, face, affineautomorphism. a convex combination of pure product states. It is natural to ask theextent to which this decomposition is unique. That is the main topicof this article.For the full state space K of B ( C m ⊗ C n ) each non-extreme pointcan be decomposed into extreme points in many different ways. But forthe space S of separable states the situation is totally different. Whilenon-extreme points with many different decompositions exist (and areeasy to find) in S as well as in K , there are in S also plenty of pointsfor which the decomposition is unique.DiVincenzo, Terhal, and Thapliyal [4] defined the optimal ensemblecardinality of a separable state ρ to be k if k is the minimal number ofpure product states required for a convex decomposition of ρ . Lockhart[11] used the term “optimal ensemble length” for the same notion. Forbrevity, we will simply call this number the length of ρ , and we denotethe set of separable states of length at most k by S k . We show inTheorem 6 that for m > , n > k ≤ max( m, n ), the set S k has asubset V k which is dense and open in S k , with each σ ∈ V k admittinga unique decomposition into pure product states. Actually, we exhibitsuch a set V k consisting of states with the property that each generatesa face of S which is a simplex, from which the uniqueness follows.We remark that the sets V k are open and dense in the relative topol-ogy on S k , but are not open or dense in S or K if mn >
1. (See theremarks after Theorem 6). Indeed it would be surprising if a subset oflow rank separable states were open and dense in the set of all statesof that rank, since low rank states are almost surely entangled [17, 22],and in general S has measure which is a decreasingly small fraction ofthe measure of K as m, n increase, cf. [3, 20].While dimensions are too high to be able to accurately visualize theabove results, the reader may be curious about the relationship to thewell known tetrahedron/octahedron picture for m = n = 2, cf. [5]. Inthat picture, there is a subset T of states which is a tetrahedron, andwhich has the property that for every state ρ which restricts to thenormalized trace on B ( C ) ⊗ I and on I ⊗ B ( C ), there are unitaries U and V such that ( U ⊗ V ) ∗ ρ ( U ⊗ V ) ∈ T . The midpoints of the six edgesof this tetrahedron are the vertices of an octahedron that consists ofthe separable states in T . Each vertex of the octahedron is a convexcombination of two distinct pure product states (which of course arenot in T ), cf. [12, eqn. (63)]. In fact, the vertices are the only statesin the octahedron of length ≤ max( m, n ) = 2.It can be checked (e.g., by applying our Corollary 5) that the decom-position of each of these vertices into pure product states is unique.Each state in the interior of this octahedron has rank 4 = mn , so is an NIQUE DECOMPOSITIONS OF SEPARABLE STATES 3 interior point of the full state space K , hence has a non-unique convexdecomposition into pure product states (see the remarks after Theorem6.) The tetrahedron also arises as a parameterization for a set of unitalcompletely positive trace preserving maps from M ( C ) to M ( C ), withthe octahedron consisting of the entanglement breaking maps in thisset, cf. [16, Appendix B], [15, Thm. 4], and [17, Fig. 2].We also define a broader class of states that we show have a uniquedecomposition as a convex combination of product states ρ i ⊗ σ i that arenot necessarily pure, but with the property that each of them generatesa face of S which is also a face of K and is affinely isomorphic to thestate space of B ( C p i ) for a suitable p i . From this it follows that theambiguity in decompositions for a given state in this class is restrictedto the ambiguity in decompositions for points in the state space ofthe matrix algebras B ( C p i ). For a complete description of the possibledecompositions of a state on B ( C p ), see [10, 18, 24].We use our results on the facial structure of S to show that everyaffine automorphism of the space S of separable states on B ( C m ⊗ C n ) isgiven by a composition of the duals of the maps that are (i) conjugationby local unitaries (i.e., unitaries of the form U ⊗ U ) (ii) the two partialtranspose maps, or (iii) the swap automorphism that takes A ⊗ B to B ⊗ A (if m = n ). A consequence is a description of the affineautomorphisms Φ of the state space such that Φ preserves entanglementand separability.There is related work of Hulpke et al [7]. They say a linear map L on C m ⊗ C n preserves qualitative entanglement if L sends separable(i.e., product) vectors to product vectors, and entangled vectors toentangled vectors. They show that a linear map L preserves qualitativeentanglement of vectors on C m ⊗ C n iff L is a local operator (i.e. one ofthe form L ⊗ L ), or if L is a local operator composed with the swapmap that takes x ⊗ y to y ⊗ x . They then show that if L preserves acertain quantitative measure of entanglement, then L must be a localunitary.We thank Mary Beth Ruskai for helpful comments and references.2. Background: states on B ( C n )We review basic facts about states on B ( C n ), and develop some factsabout the relationship of independence of vectors x in C n and of thecorresponding vector states ω x . In the following sections we will spe-cialize to the case of interest: separable states. Notation. If x is a vector in any vector space, [ x ] denotes the subspacegenerated by x . C n denotes the set of n -tuples of complex numbers ERIK ALFSEN AND FRED SHULTZ viewed as an inner product space with the usual inner product (linearin the first factor). B ( C n ) denotes the linear transformations from C n into itself. For each unit vector x ∈ C n , we denote the associatedvector state by ω x , so that ω x ( A ) = ( Ax, x ). The convex set of stateson B ( C n ) will be denoted by K n .We recall that faces of the state space K n of B ( C n ) are in 1-1 corre-spondence with the projections in B ( C n ), and thus with the subspacesof C n that are the ranges of these projections. If Q is a projection in B ( C n ), then the associated face F Q of K n consists of all states takingthe value 1 on Q . The restriction map is an affine isomorphism from F Q onto the state space of Q B ( C n ) Q ∼ = B ( Q ( C n )). Thus F Q is affinelyisomorphic to the state space of B ( L ), where L = Q ( C n ). The setof extreme points of K n are the vector states, and it follows that theextreme points of F Q are the vector states ω x with x in the range of Q ,and F Q is the convex hull of these vector states. For background, see[2, Chapter 4] Definition.
Recall that a convex set C is said to be the direct convexsum of a collection of convex subsets C , . . . , C p if each point ω ∈ C can be uniquely expressed as a convex combination(1) ω = X i ∈ I λ i ω i where I ⊂ { , . . . , p } , λ i > i ∈ I , ω i ∈ C i for all i ∈ I , and P i ∈ I λ i = 1.If C is a convex subset of a real linear space and is located on anaffine hyperplane which does not contain the origin (as is the case forour state spaces), then it is easily seen that C is the direct convex sumof convex subsets C , . . . , C p iff the span of C is the direct sum of thereal subspaces spanned by C , . . . , C p .A finite dimensional convex set is a simplex if it is the direct convexsum of a finite set of points. If the affine span of the points does notcontain the origin, then their convex hull is a simplex iff the points arelinearly independent (over R ). Lemma 1.
Let L be a subspace of C n and suppose that L is the directsum of subspaces L , . . . , L p . Let F , . . . , F p be the corresponding facesof the state space of B ( C n ) . Then the convex hull of F , . . . , F p isthe direct convex sum of those faces. In particular, if x , . . . , x p arelinearly independent unit vectors, then the corresponding vector statesare linearly independent and the convex hull of the corresponding vectorstates is a simplex. NIQUE DECOMPOSITIONS OF SEPARABLE STATES 5
Proof.
Let I ⊂ { , . . . , p } , and suppose { ω i | i ∈ I } are nonzero func-tionals on B ( C n ) with ω i ∈ span R F i for each i . To prove independenceof { ω i | i ∈ I } , suppose that for scalars { γ i } i ∈ I we have(2) X i ∈ I γ i ω i = 0 . Let L be the orthogonal complement of L . Then C n as a linear spaceis the direct sum of L , L , . . . , L p .For each i ∈ I , let P i be the projection associated with F i . Then wecan find A i ∈ P i B ( C n ) P i such that ω i ( A i ) = 0. Let B i ∈ B ( C n ) be anoperator such that B i is zero on P j = i L j , and such that ω i ( B i ) = 0 (e.g.,set B i = A i on L i ). If x ∈ L j and j = i , then ω x ( B i ) = ( B i x, x ) = 0.Since every state in F j is a convex combination of vector states ω x with x ∈ L j , then ω j ( B i ) = 0 if j = i .Now apply both sides of (2) to B k to conclude that γ k ω k ( B k ) = 0for all k ∈ I , so γ k = 0 for all k ∈ I . Thus the set of vectors ω , . . . , ω p is independent. We conclude that co( F , . . . , F p ) is the direct convexsum of F , . . . , F p .If x , . . . , x p are linearly independent unit vectors, applying the resultabove with F i = { ω x i } shows that the convex hull of the vector states ω x i is a simplex. Hence the set { ω x , . . . , ω x p } is linearly independent. (cid:3) Note that the converses of the statements above are not true. Forexample, while no set of more than two vectors in C is independent,it is easy to find a set of three linearly independent vector states on B ( C ).3. Uniqueness of decompositions of separable states
We now turn to faces of the set of separable states on B ( C m ⊗ C n ), andto the question of uniqueness of convex decompositions of such states.We identify B ( C m ⊗ C n ) with B ( C m ) ⊗ B ( C n ) by ( A ⊗ B )( x ⊗ y ) = Ax ⊗ By . We denote the convex set of all states on B ( C m ⊗ C n ) by K ,and the convex set of all separable states by S . Lemma 2.
Let e , e , . . . , e p and f , f , . . . , f p be unit vectors in C m and C n respectively. We assume that f , f , . . . , f p are linearly independent.If e ∈ C m and f ∈ C n are unit vectors such that e ⊗ f is in the linearspan of { e i ⊗ f i | ≤ i ≤ p } , then there is an index j such that [ e ] = [ e j ] and such that f is in the span of those f i such that [ e i ] = [ e j ] . In thespecial case where [ e ] , . . . , [ e p ] are distinct, then [ e ] = [ e j ] and [ f ] = [ f j ] for some index j , and { e i ⊗ f i | ≤ i ≤ p } is independent. ERIK ALFSEN AND FRED SHULTZ
Proof.
Extend f , . . . , f p to a basis f , . . . , f n of C n , and let b f , . . . , b f n be the dual basis. For 1 ≤ k ≤ n , let T k : C m ⊗ C n → C m be the linearmap such that T k ( x ⊗ y ) = b f k ( y ) x for x ∈ C m , y ∈ C n .Suppose that the product vector e ⊗ f is a linear combination(3) e ⊗ f = p X i =1 α i e i ⊗ f i . For j > p , applying T j to both sides of (3) gives b f j ( f ) e = 0, so b f j ( f ) = 0for all such j . Now if 1 ≤ j ≤ p , applying T j to both sides of (3) gives(4) b f j ( f ) e = α j e j . Since b f j ( f ) can’t be zero for all j , then e is a multiple of some e j . Fixsuch an index j . If 1 ≤ i ≤ p and [ e i ] = [ e j ], then e i can’t be a multipleof e , so b f i ( f ) e = α i e i implies α i = 0, and then also b f i ( f ) = 0. We haveshown that b f i ( f ) = 0 if i > p , or if i ≤ p and [ e i ] = [ e j ]. It follows that f is in the linear span of those f i such that [ e i ] = [ e j ].If it also happens that [ e ] , . . . , [ e p ] are distinct, and [ e ] = [ e j ], then[ f ] = [ f j ]. Suppose now that P i α i e i ⊗ f i = 0. If α k = 0, then e k ⊗ f k is a linear combination of { e i ⊗ f i | i = k } . Thus by the conclusionjust reached, we must have [ e k ] = [ e i ] for some i = k , contrary to thehypothesis that [ e ] , . . . , [ e p ] are distinct. We conclude that α k = 0 forall k , and we have shown that { e i ⊗ f i | ≤ i ≤ p } is independent. (cid:3) Lemma 3.
Let e , . . . , e p ∈ C m and f , . . . , f p ∈ C n be unit vectors. If [ e ] = [ e ] = . . . = [ e p ] , then the face F of S generated by the states { ω e i ⊗ f i | ≤ i ≤ p } is also a face of K , and this face of K is associatedwith the subspace L = e ⊗ span { f , . . . , f p } of C m ⊗ C n , and F isaffinely isomorphic to the state space of B ( L ) .Proof. Let G be the face of K which is associated with the subspace L of C m ⊗ C n . By assumption each e i is a multiple of e , so that L = span { e ⊗ f i | ≤ i ≤ p } = span { e i ⊗ f i | ≤ i ≤ p } . Hence G is the face of K generated by { ω e i ⊗ f i | ≤ i ≤ p } .We would like to show G = F . For brevity we denote the convexhull of the set { ω e i ⊗ f i | ≤ i ≤ p } by C , and observe that G and F are the faces of K and S respectively generated by C . It follows easilyfrom the definition of a face that the face generated by the convex set C in either one of the two convex sets S or K consists of all points ρ in S or K respectively which satisfy an equation(5) ω = λρ + (1 − λ ) σ NIQUE DECOMPOSITIONS OF SEPARABLE STATES 7 where 0 < λ < ω ∈ C , and where σ is in S or K respectively. Itfollows that F = face S ( C ) ⊂ face K ( C ) = G .Since each vector in L is a product vector, the extreme points of G are pure product states, so G ⊂ S . If ρ is in the face G of K generatedby C , then we can find σ ∈ K and ω ∈ C such that (5) holds. Then σ is also in G ⊂ S , so both ρ and σ are in S . Hence ρ is in the face F of S generated by C . Thus G ⊂ F , and so F = G follows. (cid:3) So far we have considered collections of product vectors { e i ⊗ f i } with { f , . . . , f p } linearly independent. In Lemma 3 we have describedthe face F of S generated these states in the special case where all ofthe e i are multiples of each other. In this case F is also a face of K .We now remove the restriction that all of the one dimensional sub-spaces [ e i ] coincide. We are going to partition the set of vectors e i ⊗ f i into subsets for which these subspaces coincide, and apply Lemma 3 toeach such subset. For simplicity of notation, we renumber the vectorsin the fashion we now describe. Theorem 4.
Let e , e , . . . , e p and f , f , . . . , f p be unit vectors in C m and C n respectively, and with f , . . . , f p linearly independent. We as-sume that the vectors are ordered so that [ e ] , . . . , [ e q ] are distinct, andso that for i > q each [ e i ] equals one of [ e ] , . . . , [ e q ] . For ≤ i ≤ q , let F i be the face of S generated by the states { ω e j ⊗ f j | [ e j ] =[ e i ] } and ≤ j ≤ p } . Then each F i is also a face of K , and the face F of S generated by { ω e i ⊗ f i | ≤ i ≤ p } is the direct convex sumof F , . . . , F q . Moreover, each F i is affinely isomorphic to the statespace of B ( L i ) , where L i = e i ⊗ span { f j | [ e i ] = [ e j ] } . In the spe-cial case when [ e ] , . . . , [ e p ] are distinct, then F is the convex hull of { ω e i ⊗ f i | ≤ i ≤ p } , and F is a simplex.Proof. By Lemma 3, the face F i of S is equal to the face of K generatedby { ω e j ⊗ f j | [ e j ] = [ e i ] } , and is affinely isomorphic to the state space of B ( L i ).We will show L , . . . , L q are independent (i.e., that L + L + · · · L q is a vector space direct sum). For 1 ≤ i ≤ q let e i ⊗ g i be a nonzerovector in L i . For i = j , g i and g j are linear combinations of disjointsubsets of f , f , . . . , f p , so by independence of f , f , . . . , f p , the subset { g , . . . , g q } is independent. Thus by Lemma 2, { e ⊗ g , . . . , e p ⊗ g p } is independent, and hence the subspaces L , . . . , L q are independent.Hence by Lemma 1, the convex hull of the faces F i is a direct convexsum of those faces. ERIK ALFSEN AND FRED SHULTZ
Finally, we need to show that this convex hull coincides with theface F of S . Extreme points of F are extreme points of S , so arepure product states. Suppose that ω x ⊗ y is a pure product state in F .Then ω x ⊗ y is in the face of K generated by { ω e i ⊗ f i | ≤ i ≤ p } , so x ⊗ y is in span { e i ⊗ f i | ≤ i ≤ p } . By Lemma 2, [ x ] = [ e j ] forsome j , and y ∈ span { y i | [ e i ] = [ e j ] } . Hence ω x ⊗ y ∈ F j . Thus eachextreme point of F is in some F j , so F is contained in the convex hullof { F i | ≤ i ≤ q } . Evidently F contains every F j , so this convex hullequals F . (cid:3) In Theorem 4 we showed that the face F is the direct convex sum offaces that are affinely isomorphic to state spaces of full matrix algebras.Convex sets of this type were studied by Vershik (in both finite andinfinite dimensions), who called them block simplexes [21]. Other exam-ples are provided by state spaces of any finite dimensional C*-algebra.Our Theorem 4 provides new examples of such block simplexes. Corollary 5.
Let e , e , . . . , e p and f , f , . . . , f p be unit vectors in C m and C n respectively. We assume that [ e i ] = [ e j ] for i = j , and that f , f , . . . , f p are linearly independent. If λ , . . . , λ k are nonnegativenumbers with sum 1, then the separable state ω = P i λ i ω e i ⊗ f i has aunique representation as a convex combination of pure product states.Proof. Suppose ω equals the convex combination P j γ j τ j where each τ j is a pure product state. Then each τ j is in the face F of S generatedby ω . By Theorem 4, F is a simplex, and the extreme points of F are all of the form ω e i ⊗ f i . Since each τ j is a vector state, it is a purestate as well, so each state τ j must be an extreme point of F , and thusmust equal some ω e i ⊗ f i . Uniqueness of the representation of ω followsfrom the uniqueness of convex decompositions into extreme points of a(finite dimensional) simplex. (cid:3) Definition.
A separable state ω has length k if ω can be expressed as aconvex combination of k pure product states and admits no decompo-sition into fewer than k pure product states. We denote by S k the setof separable states of length at most k . Definition.
A separable state ω has a unique decomposition if it can bewritten as a convex combination of pure product states in just one wayBy the above result, roughly speaking decompositions of separablestates on B ( C m ⊗ C n ) of length ≤ max( m, n ) generically are unique.Here’s a more precise statement. NIQUE DECOMPOSITIONS OF SEPARABLE STATES 9
Let k ≤ max( m, n ), and let V k be the set of states ω admitting a con-vex decomposition ω = P ki =1 λ i ω e i ⊗ f i , where e , . . . , e k and f , . . . , f k are unit vectors in C m and C n respectively, 0 < λ i for 1 ≤ i ≤ k ,[ e ] , . . . , [ e k ] are distinct, and { f , . . . , f k } is linearly independent. Theorem 6.
Let m, n > . For a given k ≤ max( m, n ) , the states in V k have length k , and have unique decompositions. The set V k is openand dense in the set S k of separable states of length at most k .Proof. Without loss of generality, we may assume m ≤ n . By Corollary5, each ω ∈ V k admits a unique representation as a convex combinationof pure product states, and each state in V k has length k . We will showthat V k is open and dense in S k .To prove density, let ω ∈ S k have a convex decomposition ω = P ki =1 λ i ω x i ⊗ y i . By slightly perturbing the coefficients λ i if necessary,we may assume that λ i > i .Given ǫ >
0, by perturbing each x i and y i if necessary, we can find asecond convex combination of pure product states ω ′ = P ki =1 λ i ω e i ⊗ f i with k ω − ω ′ k < ǫ , with [ e ] , . . . , [ e k ] distinct, and with { f , . . . , f k } inde-pendent. (Indeed, to achieve independence we may append unit vectors y k +1 , . . . , y n to the vectors y , . . . , y k to give the subset { y , y , . . . , y n } of C n , and by small perturbations arrange that the determinant of thematrix with columns y , . . . , y n is nonzero.) Thus V k is dense in S k .Let I = { ( λ , λ , . . . , λ k ) ∈ [0 , k | P i λ i = 1 } . Let U m be the unitsphere of C m and U n the unit sphere of C n . Let X = I × ( U m ) k × ( U n ) k .Define ψ : X → S by ψ (( λ , . . . , λ k ) , ( x , . . . , x k ) , ( y , . . . , y k )) = X i λ i ω x i ⊗ y i . Note that ψ is continuous, that X is compact with respect to theproduct topology, and that ψ ( X ) = S k .Now let X be the set { (( λ , . . . , λ k ) , ( x , . . . , x k ) , ( y , . . . , y k )) } ofmembers of X of such that [ x ] , . . . , [ x k ] are distinct, such that { y , . . . , y k } is linearly independent, and such that λ i > ≤ i ≤ k . By lowersemicontinuity of the rank of a matrix whose columns are y , . . . , y k ,the set of elements (( λ , . . . , λ k ) , ( x , . . . , x m ) , ( y , . . . , y k )) of X with { y , . . . , y k } linearly independent is open in X , so it is clear that X isan open subset of X . By construction, ψ ( X ) = V k . Since X is openin X , then X \ X is closed and hence compact. Since ψ maps X \ X onto S k \ ψ ( X ), then the latter is closed, so V k = ψ ( X ) is open in S k . (cid:3) As remarked in the introduction, the sets V k are open and dense inthe relative topology on S k , but are not open or dense in S or K if mn >
1. To see this recall that a point σ in a convex set C is analgebraic interior point if for every point ρ in C there is a point τ in C such that σ lies on the open line segment from ρ to τ . Clearly for everyalgebraic interior point σ of S and every pure product state ρ , thereis a convex decomposition of σ that includes ρ with positive weight.Since there are infinitely many pure product states, there are infinitelymany convex decompositions for every algebraic interior point of S .Every nonempty subset which is open in S contains an algebraicinterior point of S ([19, pp. 88-91]), so contains points with nonuniquedecompositions. Thus V k is not open in S or K . It is not dense in S or K , since for any m, n there exists r > σ withina distance r from the normalized tracial state are separable, cf. [25,Thm. 1]. Every such state σ is an algebraic interior point of S , and sofails to have a unique decomposition.Observe that Theorem 6 implies that V k is also open and dense inthe set of separable states of length equal to k .4. Description of convex decompositions
Let e , e , . . . , e p and f , f , . . . , f p be unit vectors in C m and C n re-spectively, with f , . . . , f p linearly independent. Suppose ω is a convexcombination of { ω e i ⊗ f i | ≤ i ≤ p } . In this section, we will describeall convex decompositions of ω into pure product states.Let ω = P i λ i ω i be any convex decomposition of ω into pure prod-uct states. Then following the notation of Theorem 4, each ω i is inface S ( ω ) ⊂ F . Since each ω i is an extreme point of S , and F is thedirect convex sum of the faces F i , then each ω i must be in some F k . Ifwe define γ k = P { i | ω i ∈ F k } λ i and σ k = γ − k P { i | ω i ∈ F k } λ i ω i , then ω hasthe convex decomposition(6) ω = X k γ k σ k with σ k ∈ F k for each k. Since F is the direct convex sum of the F k , the decomposition of ω in(6) is unique.All possible convex decompositions of ω into pure product states canbe found by starting with the unique decomposition ω = P k γ k σ k with σ k ∈ F k , and then decomposing each σ k into pure states. (Every statein F k is separable, so pure states are pure product states). Since F k isaffinely isomorphic to the state space of B ( L k ), unless each σ k is itselfa pure state, this can be done in many ways, as we discussed in theintroduction. The possibilities have been described in [24, 18, 10].A decomposition of a separable state ω as a convex combination ofpure product states can be interpreted as a representation of ω as the NIQUE DECOMPOSITIONS OF SEPARABLE STATES 11 barycenter of a probability measure on the extreme points of S . Withthis interpretation the statement above can be rephrased in terms ofthe concept of dilation of measures (as defined e.g. in [1, p. 25]). If ω is given as above, then the probability measures on pure product statesthat represent ω are precisely those which are dilations of the uniquelydetermined probability measure µ = P k γ k µ k obtained from (6) with µ k = δ σ k .5. Affine automorphisms of the space S of separablestates Notation.
Fix m, n . We denote the state space of B ( C m ) by K m , thestate space of B ( C n ) by K n , and the state space of B ( C m ⊗ C n ) by K or K m,n . The convex set of separable states in K is denoted by S or S m,n . We will sometimes deal with a second algebra B ( C m ′ ⊗ C n ′ ),whose state space and separable state spaces we will denote by K ′ or S ′ respectively.From Theorem 4, the face of S generated by two distinct pure prod-uct states ω ⊗ σ and ω ⊗ σ is a line segment (if ω = ω and σ = σ )or is isomorphic to the state space of B ( C ) and hence is a 3-ball (when ω = ω but σ = σ , or when σ = σ but ω = ω ). (By a 3-ballwe mean a convex set affinely isomorphic to the closed unit ball of R .The fact that the state space of B ( C ) is a 3-ball can be found in manyplaces, e.g., [2, Thm. 4.4].)We define a relation R on the pure product states of K by ρ R τ ifface S ( ρ, τ ) is a 3-ball. By the remarks above, ( ω ⊗ σ ) R ( ω ⊗ σ ) iff( ω = ω but σ = σ ) or ( σ = σ but ω = ω ). Note that an affineisomorphism Φ : S → S ′ will take faces of S to faces of S ′ , and willtake 3-balls to 3-balls, so for pure product states ρ, τ we have ρ R τ iffΦ( ρ ) R Φ( τ ).The idea of the following lemmas is to show that if Φ( ω ⊗ σ ) = φ ( ω, σ ) ⊗ ψ ( ω, σ ), then φ depends only on the first argument and ψ depends only on the second argument, or possibly vice versa. Althoughwe are interested in affine automorphisms of a single space of separablestates, it will be easier to establish the needed lemmas in the contextof affine isomorphisms from S to S ′ .We use the notation ∂ e C for the set of extreme points of a convexset C . For example, ∂ e K is the set of pure states on B ( C m ⊗ C n ). Lemma 7.
Let
Φ : S m,n → S m ′ ,n ′ be an affine isomorphism. Let ω , ω be distinct pure states in K m and σ , σ distinct pure states in K n . Then the following four equations cannot hold simultaneously. Φ( ω ⊗ σ ) = ρ ⊗ τ Φ( ω ⊗ σ ) = ρ ⊗ τ Φ( ω ⊗ σ ) = ρ ⊗ τ Φ( ω ⊗ σ ) = ρ ⊗ τ (7) for ρ , ρ , ρ ∈ ∂ e K m ′ and τ , τ , τ ∈ ∂ e K n ′ .Proof. We assume for contradiction that all four equations hold. Since( ω ⊗ σ ) R ( ω ⊗ σ ), then ( ρ ⊗ τ ) R ( ρ ⊗ τ ). Hence(8) ρ = ρ or τ = τ . Similarly ( ω ⊗ σ ) R ( ω ⊗ σ ), so ( ρ ⊗ τ ) R ( ρ ⊗ τ ). Hence(9) ρ = ρ or τ = τ . Since we are assuming that ω = ω and σ = σ , the four states { ω i ⊗ σ j | ≤ i, j ≤ } are distinct, so the four states on the right sideof (7) must be distinct. Combining (8) and (9) gives four possibilities,each contradicting the fact that the states on the right side of (7) aredistinct. Indeed:( ρ = ρ and ρ = ρ ) = ⇒ ρ ⊗ τ = ρ ⊗ τ ( ρ = ρ and τ = τ ) = ⇒ ρ ⊗ τ = ρ ⊗ τ ( τ = τ and ρ = ρ ) = ⇒ ρ ⊗ τ = ρ ⊗ τ ( τ = τ and τ = τ ) = ⇒ ρ ⊗ τ = ρ ⊗ τ . We conclude that the four equations in (7) cannot hold simultaneously. (cid:3)
Definition.
Recall that we identify B ( C m ⊗ C n ) with B ( C m ) ⊗ B ( C n ).The swap isomorphism ( α m,n ) ∗ : B ( C n ⊗ C m ) → B ( C m ⊗ C n ) is the*-isomorphism that satisfies ( α m,n ) ∗ ( A ⊗ B ) = B ⊗ A . If operators in B ( C m ⊗ C n ) are identified with matrices, the swap isomorphism is thesame as the “canonical shuffle” discussed in [13, Chapter 8]. The dualmap α m,n is an affine isomorphism from the state space of B ( C m ⊗ C n )to the state space of B ( C n ⊗ C m ), with α m,n ( ω ⊗ σ ) = σ ⊗ ω . Thisrestricts to an affine isomorphism from S m,n to S n,m , which we alsorefer to as the swap isomorphism. If m = n , then ( α m,m ) ∗ is a *-automorphism of B ( C m ⊗ C m ), α m,m is an affine automorphism of thestate space K , and restricts to an affine automorphism of the space S of separable states. NIQUE DECOMPOSITIONS OF SEPARABLE STATES 13
Lemma 8.
Let
Φ : S m,n → S m ′ ,n ′ be an affine isomorphism. At leastone of the following two possibilities occurs: (i) For every ω ∈ ∂ e K m there exists ρ ∈ ∂ e K m ′ such that Φ( ω ⊗ K n ) = ρ ⊗ K n ′ , and for every σ ∈ ∂ e K n there exists τ ∈ ∂ e K n ′ such that Φ( K m ⊗ σ ) = K m ′ ⊗ τ . (ii) For each ω ∈ ∂ e K m there exists τ ∈ ∂ e K n ′ such that Φ( ω ⊗ K n ) = K m ′ ⊗ τ , and for every σ ∈ ∂ e K n there exists ρ ∈ ∂ e K m ′ such that Φ( K m ⊗ σ ) = ρ ⊗ K n ′ .If (i) occurs, then m = m ′ and n = n ′ . If (ii) occurs, then m = n ′ and n = m ′ .Proof. For fixed ω ∈ ∂ e K m and distinct σ , σ ∈ ∂ e K n we have ( ω ⊗ σ ) R ( ω ⊗ σ ), so Φ( ω ⊗ σ ) R Φ( ω ⊗ σ ). Thus either there exist ρ ∈ ∂ e K m ′ and distinct τ , τ ∈ ∂ e K n ′ such that(10) Φ( ω ⊗ σ i ) = ρ ⊗ τ i for i = 1 , , or there exist distinct ρ , ρ ∈ ∂ e K m ′ and τ ∈ ∂K n ′ such that(11) Φ( ω ⊗ σ i ) = ρ i ⊗ τ for i = 1 , . We will show that (10) implies (i), and (11) implies (ii).Suppose that (10) holds. Let σ ∈ ∂ e K n with σ = σ and σ = σ ,and let Φ( ω ⊗ σ ) = ρ ⊗ τ . Since ( ω ⊗ σ ) R ( ω ⊗ σ i ) for i = 1 , ρ ⊗ τ ) R ( ρ ⊗ τ i ) for i = 1 ,
2. Hence ( ρ = ρ or τ = τ ) and( ρ = ρ or τ = τ ). Since τ = τ , then ρ = ρ . It follows thatΦ( ω ⊗ K n ) ⊂ ρ ⊗ K n ′ . Thus(12) Φ( ω ⊗ σ i ) = ρ ⊗ τ i for i = 1 , ⇒ Φ( ω ⊗ K n ) ⊂ ρ ⊗ K n ′ . Now (10) also implies(13) Φ − ( ρ ⊗ τ i ) = ω ⊗ σ i for i = 1 , . If (10) holds (and hence also (13), then applying the implication (12)to (13) with Φ − in place of Φ shows Φ − ( ρ ⊗ K n ′ ) ⊂ ω ⊗ K n , so by(12) equality holds. Hence we have shown(14) Φ( ω ⊗ σ i ) = ρ ⊗ τ i for i = 1 , ⇒ Φ( ω ⊗ K n ) = ρ ⊗ K n ′ . Now suppose instead that (11) holds. Let α m ′ ,n ′ be the swap affineisomorphism defined above, so that α m ′ ,n ′ : S m ′ ,n ′ → S n ′ ,m ′ . Then(15) ( α m ′ ,n ′ ◦ Φ)( ω ⊗ σ i ) = α m ′ ,n ′ ( ρ i ⊗ τ ) = τ ⊗ ρ i for i = 1 , . By the implication (14) applied to α m ′ ,n ′ ◦ Φ we conclude that( α m ′ ,n ′ ◦ Φ)( ω ⊗ K n ) = τ ⊗ K m ′ , so Φ( ω ⊗ K n ) = α − m ′ ,n ′ ( τ ⊗ K m ′ ) = K m ′ ⊗ τ . Thus we have proven the implication(16) Φ( ω ⊗ σ i ) = ρ i ⊗ τ for i = 1 , ⇒ Φ( ω ⊗ K n ) = K m ′ ⊗ τ . By Lemma 7 and the implications (14) and (16), either (10) musthold for all ω ∈ ∂ e K m or (11) must hold for all ω ∈ ∂ e K m . We concludethat either(17) ∀ ω ∈ ∂ e K m ∃ ρ ∈ ∂ e K m ′ such that Φ( ω ⊗ K n ) = ρ ⊗ K n ′ or(18) ∀ ω ∈ ∂ e K m ∃ τ ∈ ∂ e K n ′ such that Φ( ω ⊗ K n ) = K m ′ ⊗ τ. Similarly, either(19) ∀ σ ∈ ∂ e K n ∃ τ ′ ∈ ∂ e K n ′ such that Φ( K m ⊗ σ ) = K m ′ ⊗ τ ′ or(20) ∀ σ ∈ ∂ e K n ∃ ρ ′ ∈ ∂ e K m ′ such that Φ( K m ⊗ σ ) = ρ ′ ⊗ K n ′ . Suppose that (17) and (20) both held. For ω ∈ K m and σ ∈ K n notethat ω ⊗ σ is in both ω ⊗ K n and K m ⊗ σ , so ρ ⊗ K n ′ and ρ ′ ⊗ K n ′ are notdisjoint. This implies ρ = ρ ′ , so Φ( ω ⊗ K n ) = Φ( K m ⊗ σ ). Since Φ isbijective, ω ⊗ K n = K m ⊗ σ follows. This is possible only if m = n = 1.If m = n = 1, then all of (17), (18), (19), (20) hold. Similarly if(18) and (19) both held then m = n = 1 is again forced. Thus thepossibilities are that (17) and (19) both hold (which is the same asstatement (i) of the lemma), or that (18) and (20) hold (equivalent to(ii)), or that m = n = 1, in which case both (i) and (ii) hold.Finally, since the affine dimensions of K p and K q are different when p = q , the statement in the last sentence of the lemma follows. (cid:3) If ψ : K m → K m and ψ : K n → K n are affine automorphisms,then we can extend each to linear maps on the linear span, and formthe tensor product ψ ⊗ ψ . This will be bijective, but not necessarilypositive. (A well known example of this phenomenon occurs when ψ is the identity map and ψ is the transpose map.) However, ψ and ψ will map pure states to pure states, and hence ψ ⊗ ψ will mappure product states to pure product states. Thus ψ ⊗ ψ will map S onto S , and hence will be an affine automorphism of S . We will nowsee that all affine automorphisms of S are either such a tensor productof automorphisms or such a tensor product composed with the swapautomorphism. NIQUE DECOMPOSITIONS OF SEPARABLE STATES 15
Theorem 9. If m = n , and Φ : S → S is an affine automorphism,then there exist unique affine automorphisms ψ : K m → K m and ψ : K n → K n such that Φ = ψ ⊗ ψ . If m = n then either wecan write Φ = ( ψ ⊗ ψ ) or Φ = α m,m ◦ ( ψ ⊗ ψ ) , where ψ , ψ areagain unique affine automorphisms of K m and K n respectively, and α m,m : S → S is the swap automorphism.Proof. We apply Lemma 8. For each ω ∈ ∂ e K m and σ ∈ ∂ e K n , define φ σ : K m → K m and ψ ω : K n → K n byΦ( ω ⊗ σ ) = φ σ ( ω ) ⊗ ψ ω ( σ ) . Suppose first that case (i) of Lemma 8 occurs. Then ψ σ ( ω ) is indepen-dent of σ and ψ ω ( σ ) is independent of ω . Therefore there are functions ψ : K m → K m and ψ : K n → K n such thatΦ( ω ⊗ σ ) = ψ ( ω ) ⊗ ψ ( σ ) . Since Φ is bijective and affine, so are ψ and ψ .Suppose instead that case (ii) of Lemma 8 occurs. Then m = n . Ifwe define Φ ′ = α m,m ◦ Φ, then Φ ′ : S → S satisfies case (i) of Lemma8. Then from the first paragraph we can choose affine automorphisms ψ : K m → K m and ψ : K n → K n such that Φ ′ = ψ ⊗ ψ . Since α m,m is the identity map, then Φ = α m,m ◦ ( ψ ⊗ ψ ). (cid:3) We review some well known facts about affine automorphisms ofstate spaces and maps on the underlying algebra. Let ψ be an affineautomorphism of K m . Then ψ extends uniquely to a linear map on thelinear span of K m , which we also denote by ψ , and this map is the dualof a unique linear map ψ ∗ on B ( C m ). By a result of Kadison [9] ψ ∗ willbe a *-isomorphism or a *-anti-isomorphism. (Since the restriction ofan affine automorphism to pure states preserves transition probabili-ties, this also follows from Wigner’s theorem [23]). The map ψ ∗ willbe a *-isomorphism iff ψ ∗ is completely positive, which is equivalentto ψ being completely positive. If ψ ∗ is a *-isomorphism, then ψ ∗ isimplemented by a unitary, i.e., there is a unitary U ∈ B ( C m ) such that ψ ∗ ( A ) = U AU ∗ .If ψ ∗ is a *-anti-isomorphism, then the composition of ψ ∗ with thetranspose map (in either order) gives a *-isomorphism, and the map ψ ∗ is completely copositive. It follows that an affine automorphism ψ of K m is either completely positive or completely copositive, and ψ is completely positive iff ψ − is completely positive. If t denotes thetranspose map on B ( C m ) or B ( C n ), then t is positive but t ⊗ id and id ⊗ t are not positive on B ( C m ) ⊗ B ( C n ) if m, n >
1. Background canbe found in [2, Chapters 4, 5].
Recall that a local unitary in B ( C m ⊗ C n ) is a tensor product U ⊗ U of unitaries. Theorem 10.
Every affine automorphism of the space S of separablestates on B ( C m ⊗ C n ) is the dual of conjugation by local unitaries, oneof the two partial transpose maps, the swap map (if m = n ), or acomposition of these maps. An affine automorphism Φ of S extendsuniquely to an affine automorphism of the full state space K iff it canbe expressed as one of the compositions just mentioned with both orneither of the partial transpose maps involved.Proof. We note first that if m = 1 or n = 1, the result is clear, so weassume hereafter that m ≥ n ≥ ψ : K m → K m and ψ : K n → K n are affineautomorphisms, then Φ = ψ ⊗ ψ is an affine automorphism of K iff ψ and ψ are both completely positive or both completely copositive.If ψ and ψ are completely positive, then Φ = ψ ⊗ ψ = ( id ⊗ ψ ) ◦ ( ψ ⊗ id ) is positive; hence Φ( K ) ⊂ K . Furthermore, ψ − and ψ − willbe completely positive, so Φ − is positive, and hence Φ( K ) = K . If ψ and ψ are completely copositive, then ( t ◦ ψ ) ⊗ ( t ◦ ψ ) is positive.Composing with t ⊗ t shows ψ ⊗ ψ is positive and as above we concludethat Φ( K ) = K . On the other hand, if ψ is completely positive and ψ is completely copositive, then ψ ⊗ ( t ◦ ψ ) is positive, so ( id ⊗ t ) ◦ ( ψ ⊗ ψ )is positive. If ( ψ ⊗ ψ )( K ) = K , then id ⊗ t would be positive, acontradiction since m, n ≥
2. Thus in this case ψ ⊗ ψ is not an affineautomorphism of K .If ψ and ψ are completely positive, then they are implemented byunitaries, so Φ = ψ ⊗ ψ is implemented by a local unitary. If bothare completely copositive, then t ◦ ψ and t ◦ ψ are implemented byunitaries, so ( t ⊗ t ) ◦ ( ψ ⊗ ψ ) is implemented by a local unitary. ThenΦ = ( t ⊗ t ) ◦ ( t ⊗ t ) ◦ ( ψ ⊗ ψ ) is the composition of the transpose mapon K and conjugation by local unitaries.The first statement of the theorem now follows from Theorem 9.Uniqueness follows from the fact that the linear span of S contains K . (cid:3) Definition.
Let Φ : K → K be an affine automorphism. We say Φ preserves separability if Φ takes separable states to separable states,i.e., if Φ( S ) ⊂ S . A state ω in K is entangled if ω is not separable. Φ preserves entanglement if Φ takes entangled states to entangled states. Corollary 11.
Let
Φ : K m,n → K m,n be an affine automorphism.Then Φ preserves entanglement and separability iff Φ is a composition NIQUE DECOMPOSITIONS OF SEPARABLE STATES 17 of maps of the types (i) conjugation by local unitaries, (ii) the transposemap, (iii) the swap automorphism (in the case that m = n ).Proof. If Φ preserves entanglement and separability, then Φ maps S into S and K \ S into K \ S , which is equivalent to Φ( S ) = S . (cid:3) Corollary 12. If Φ t : S → S is a one-parameter group of affine au-tomorphisms, then there are one-parameter groups of unitaries U t and V t such that Φ t ( ω ( A )) = ω (( U t ⊗ V t ) A ( U ∗ t ⊗ V ∗ t )) .Proof. For each t , factor Φ t = φ t ⊗ ψ t or Φ t = α ◦ ( φ t ⊗ ψ t ), where α isthe swap automorphism. In the latter case,Φ t = Φ t ◦ Φ t = α ◦ ( φ t ⊗ ψ t ) ◦ α ◦ ( φ t ⊗ ψ t )= ( φ t ⊗ ψ t ) ◦ ( φ t ⊗ ψ t ) = ( φ t ◦ φ t ) ⊗ ( ψ t ◦ ψ t ) . It follows that the swap automorphism is not needed for Φ t , and hencefor Φ t for any t . Uniqueness of the factorization Φ t = φ t ⊗ ψ t showsthat φ t and ψ t are also one parameter groups of affine automorphisms.By a result of Kadison [8], such automorphisms are implemented byone parameter groups of unitaries. (cid:3) Corollary 13. If Φ t : K → K is a one-parameter group of entangle-ment preserving affine automorphisms, then there are one-parametergroups of unitaries U t and V t such that Φ t ( ω ( A )) = ω (( U t ⊗ V t ) A ( U ∗ t ⊗ V ∗ t )) .Proof. Since Φ t and (Φ t ) − = Φ − t preserve entanglement, then Φ t maps S onto S , so this corollary follows from Corollary 12. (cid:3) References [1] E. M. Alfsen,
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