Various notions of positivity for bi-linear maps and applications to tri-partite entanglement

Transcription

1 Various notions of positivity for bi-linear maps and applications to tri-partite entanglement , Waterloo Seung-Hyeok Kye (Seoul National University) a joint work with Kyung Hoon Han [arxiv: ] 2015 tri-partite1 entangle / 37

2 1 Separability and entanglement 2 Duality between tensor products and mapping spaces 3 (p, q, r)-positive bi-linear maps 4 Schmidt numbers for tri-partite states 5 Relations with quantization 6 Examples , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite2 entangle / 37

3 1. Separability and entanglement A state on the Hilbert space H = C d 1 C d 2 C dn (genuinely) separable if it is the convex combination is said to be ϱ := i I p i z i z i of pure product states z i z i, with product vectors z i : z i = x 1i x 2i x ni C d 1 C d 2 C dn, i I. Therefore, ϱ is a d d density matrix, with the dimension d = n i=1 d i of the Hilbert space H. The set S consisting of all separable states is a convex set whose extreme points are pure product states. A state which is not separable is called entangled tri-partite3 entangle / 37

4 genuinely separable A-BC separable 2015 tri-partite4 entangle / 37

5 genuinely separable A-BC separable B-CA separable C-AB separable 2015 tri-partite5 entangle / 37

6 genuinely separable bi-separable genuinely entangled 2015 tri-partite6 entangle / 37

7 How to distinguish entanglement from separability? Use duality between tensor products and mapping spaces tri-partite7 entangle / 37

8 2. Duality between tensor products and mapping spaces For an multi-linear map φ from M d1 M dn 1 into M dn, we associate a matrix C φ M d = M d1 M dn 1 M dn by C φ = i 1,j 1,...,i n 1,j n 1 i 1 j 1 i n 1 j n 1 φ( i 1 j 1,, i n 1 j n 1 ). The correspondence φ C φ is nothing but the Choi-Jamiołkowski isomorphism for bi-partite case of n = tri-partite8 entangle / 37

9 It is easy to see that the following are equivalent for a multi-linear map φ from M d1 M dn 1 to M dn : ϱ, C φ 0 for each ϱ S, where A, B = Tr (A t B) = a ij b ij. φ(x 1,..., x n 1 ) M + d n whenever x i M + d i for i = 1, 2,..., d 1. We call that a multi-linear map φ : M d1 M dn 1 M dn x i M + d i, i = 1, 2,..., n 1 = φ(x 1,..., x n 1 ) M + d n. is positive if This is the case if and only if its linearization φ : M d1 M dn 1 M dn sends M + d 1 M + d n 1 into M + d n. We also define the bilinear pairing between ϱ in M d1 M dn 1 M dn and a multi-linear map φ : M d1 M dn 1 M dn by ϱ, φ = ϱ, C φ tri-partite9 entangle / 37

10 So far, we have seen that φ is positive if and only if ϱ, φ 0 for each separable ϱ. By the separation theorem for a point outside of a closed convex cone, we have the following: [K, J. Phys. A (2015), arxiv: ]: Theorem An n-partite state ϱ is separable if and only if ϱ, φ 0 for every positive (n 1)-linear map φ. bi-partite case of n = 2: M. Horodecki, P. Horodecki and R. Horodecki (1996) M. H. Eom + K (2000) B. M. Terhal (2000): introduce entanglement witnesses, Hermitian matrices which are the Choi matrices of positive, non-completely positive maps. One may go further when n = 2: , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite10 entangle / 37

11 [M. H. Eom + K, 2000] A linear map φ : M m M n is s-positive if and only if φ, ζ ζ 0 for every ζ C m C n with SR(ζ) s, where SR(ζ) denote the Schmidt rank of ζ C m C n. We denote by V s the convex cone in M m M n generated by ζ ζ with SR(ζ) s. φ is s-positive if and only if ϱ, φ 0 for each ϱ V s. ϱ V s if and only if ϱ, φ 0 for each s-positive map φ. The second statement follows from the first, by the separation theorem for a point outside of a closed convex set. B. M. Terhal and P. Horodecki (2000): introduce Schmidt numbers for states, which coincide with V s. A. Sanpera, D. Bruss, M. Lewenstein (2001): States with Schmidt number s are detected by s-positive maps; duality between P s and V s tri-partite11 entangle / 37

12 We may consider various other notions for positivity of multi-linear maps. We list up some candidates for bilinear maps φ : M A M B M C : φ is positive: φ(x, y) is positive whenever x and y are positive. y φ(x, y) : M B M C is completely positive for each positive x M A. x φ(x, y) : M A M C is completely positive for each positive y M B. The linearization M A M B M C is positive. The linearization M A M B M C is completely positive. The middle three properties are dual to A-BC, B-CA and C-AB separability, respectively. Explain the above properties in a single framework , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite12 entangle / 37

13 3. (p, q, r)-positive bi-linear maps (Unital) positive maps = morphisms for functions systems (AOU space) Function system = unital self-adjoint space of continuous functions Kadison (1951), Paulsen+Tomforde (2009): Abstract characterization Namioka+Phelps(1969), Effros(1972): Tensor product of function systems Han (2009) [arxiv: ] showed that the following are equivalent for a bilinear map φ : V 1 V 2 V 3 between function systems: v 1 V + 1 and v 2 V + 2 imply φ(v 1, v 2 ) V + 3. The linearization φ : V 1 max V 2 V 3 is positive. Note that (M m max M n ) + = M + m M + n as function systems So, our definition of positivity of multi-linear maps corresponds to the positivity of its linearization with respect to the maximal tensor products of function systems , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite13 entangle / 37

14 (Unital) Completely positive maps = morphisms for operator systems Operator system = unital self-adjoint space of bounded operators Choi + Effros (1977): Abstract characterizarion Kavruk+Paulsen+Todorov+Tomforde (2011) : define tensor products of operator systems, and show that the following are equivalent for a bi-linear map φ : V 1 V 2 V 3 between operator systems: The linearization φ : S T R is completely positive with respect to the operator system maximal tensor products [x i,j ] M p (S) +, [y k,l ] M q (T ) + = [φ(x i,j, y k,l )] M pq (R) + holds for every p, q = 1, 2,.... Note that (M m max M n ) + = (M m M n ) + as operator systems , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite14 entangle / 37

15 Definition Let S = (s 1, s 2,..., s n ) be an n-tuple of natural numbers. An (n 1)-linear map φ : S 1 S n 1 S n between operator spaces is said to be S-positive if the following condition holds: x k M sk (S k ) + for k = 1, 2,...,n 1 and α M sn,s 1 s n 1 (C) + = α[φ(x 1,..., x n 1 )]α M sn (S) + When n = 2, a linear map is (s, t)-positive if and only if it is (s t, s t)-positive if and only if it is s t-positive in the usual sense , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite entangle / 37

16 Some combinations of triplets are redundant: For examples, the following are equivalent for fixed p and q: φ satisfies the condition by Kavruk, V. I. Paulsen, I. G. Todorov and M. Tomforde. φ is (p, q, r)-positive for each r = 1, 2,... ; φ is (p, q, r)-positive for some r pq; φ is (p, q, pq)-positive. We also have: 1 φ is (1, q, r)-positive if and only if φ is (1, q r, q r)-positive. 2 φ is (p, 1, r)-positive if and only if φ is (p r, 1, p r)-positive. 3 φ is (p, q, 1)-positive if and only if φ is (p q, p q, 1)-positive, when the domains and the range are matrices tri-partite16 entangle / 37

17 For bi-linear maps in matrix algebras, we have the following: φ is positive if and only if φ is (1, 1, 1)-positive. y φ(x, y) : M B M C is completely positive for each positive x M A if and only if φ is (1, b, c)-positive. x φ(x, y) : M A M C is completely positive for each positive y M B if and only if φ is (a, 1, c)-positive. The linearization M A M B M C is positive if and only if φ is (a, b, 1)-positive. The linearization M A M B M C is completely positive if and only if φ is (a, b, c)-positive tri-partite17 entangle / 37

18 We also have the following multi-linear version of Choi isomorphism and Kraus decomposition: Theorem For a bi-linear map φ : M A M B M C, the following are equivalent: The linearization φ : M A M B M C is completely positive; φ is (p, q, r)-positive for each p, q, r = 1, 2,... ; φ is (a, b, ab)-positive; φ satisfies the condition [KPTT] for each p, q = 1, 2,... ; φ satisfies the condition [KPTT] with p = a and q = b; the Choi matrix C φ is positive; φ(x, y) = V i (x y)v i with c ab matrices V i s , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite18 entangle / 37

19 Recall that φ : M A M B is s-positive if and only if its dual φ : M B M A is s-positive. The dual map may be interpreted in terms of permutation: φ(x A ), x B = φ (x B ), x A = φ (x σa ), x σb. for the non-trivial permutation σ on {A, B}. For a permutation σ in the set {A, B, C} and a bi-linear map φ : M A M B M C, we define the dual bi-linear map with respect to the permutation σ by φ σ : M σa M σb M σc φ σ (x σa, x σb ), x σc = φ(x A, x B ), x C, x A M A, x B M B, x C M C. φ is (p, q, r)-positive if and only if φ σ is (p, q, r) σ -positive , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite19 entangle / 37

20 4. Schmidt numbers for tri-partite states We recall that the Schmidt rank of a vector η = n i=1 v i w i H B H C is equal to the rank of the associate map λ η : H B H C given by λ η (v) = n v i v w i. i=1 In fact, the correspondence η λ η follows from the natural isomorphisms H B H C (H B ) H C L(H B, H C ) tri-partite20 entangle / 37

21 Applying the above isomorphism twice, we consider the natural isomorphisms H A H B H C (H A ) (H B ) H C (H A ) L(H B, H C ) L(H A, L(H B, H C )), to get the analogous notion. We write ξ H A H B H C by ξ = n u i η i H A (H B H C ) i=1 with u i H A and η i H B H C, and consider the linear map Λ ξ : H A L(H B, H C ) given by Λ ξ (u) = n ū i u λ ηi. i= tri-partite21 entangle / 37

22 Now, we consider the following three numbers: α ξ = rankλ ξ, β ξ = dim {supp T : T ranλ ξ } γ ξ = dim {ran T : T ranλ ξ }. Definition We call the triplet (α ξ, β ξ, γ ξ ) the Schmidt rank of the vector ξ H A H B H C and write SR(ξ) = (α ξ, β ξ, γ ξ ) , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite22 entangle / 37

23 Theorem Suppose that ξ H A H B H C and 1 p a, 1 q b, 1 r c. Then the following are equivalent: α ξ p, β ξ q, γ ξ r; SR(ξ) (p, q, r) there exist vectors {u i } p i=1 H A, {v j } q j=1 H B, {w k } r k=1 H C and scalars c i,j,k such that ξ = p q r c i,j,k u i v j w k ; i=1 j=1 k=1 there exist orthonormal vectors {u i } p i=1 H A, {v j } q j=1 H B, {w k } r k=1 H C and scalars c i,j,k such that ξ = p q r c i,j,k u i v j w k. i=1 j=1 k= , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite23 entangle / 37

24 Possible combinations are given by Σ a,b,c = {(α, β, γ) N 3 : α βγ, β γα, γ αβ, 1 α a, 1 β b, 1 γ c}. The following are equivalent: There exists ξ H A H B H C such that SR(ξ) = (α, β, γ) (α, β, γ) Σ a,b,c , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite24 entangle / 37

25 Definition For a tri-partite unnormalized state ϱ M A M B M C, we write SN(ϱ) (p, q, r) if there exist ξ 1,, ξ n H A H B H C such that ϱ = n i=1 ξ i ξ i and SR(ξ i ) (p, q, r) for each i = 1, 2,..., n. We denote by S p,q,r the set of all tri-partite states ϱ in M A M B M C with the property SN(ϱ) (p, q, r). The set S p,q,r is compact convex. SN(ϱ) (1, 1, 1) if and only if ϱ is genuinely separable. SN(ϱ) (1, b, c) if and only if it is A-BC separable. SN(ϱ) (a, 1, c) if and only if it is B-CA separable. SN(ϱ) (a, b, 1) if and only if it is C-AB separable , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite25 entangle / 37

26 Theorem φ : M A M B M C is (p, q, r)-positive if and only if ϱ, φ 0 for each ϱ S p,q,r. A tri-partite state ϱ belongs to S p,q,r if and only if ϱ, φ 0 for each (p, q, r)-positive map φ. A state ϱ is bi-separable, that is, in the convex hull of three kinds of separability, if and only if ϱ, φ 0 for each φ P 1,b,c P a,1,c P a,b,1, where P p,q,r denotes the set of all (p, q, r)-positive. Therefore, bi-linear maps in these intersection play the role of witnesses for genuine entanglement tri-partite26 entangle / 37

27 5. Relations with quantization Theorem Suppose that φ : S T R is a bi-linear map for operator systems S, T and R. Then the following are equivalent: (i) φ is (p, q, r)-positive; (ii) φ : OMAX p (S) max OMAX q (T ) R is r-positive; (iii) φ : OMAX p (S) max OMAX q (T ) OMIN r (R) is completely positive. [B. Xhabli (2012)]: Defined OMAX k (S) for an operator system S, whose n-level positive cones are smallest among all possible cones under the condition that k-level cones coincide, and similarly for OMIN k (S) , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite27 entangle / 37

28 Theorem An (unnormalized) state ρ M A M B M C belongs to S p,q,r if and only if we have ϱ [OMAX p (M A ) max OMAX q (M B ) max OMAX r (M C )] , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite28 entangle / 37

29 [Kavruk, V. I. Paulsen, I. G. Todorov and M. Tomforde (2011)]: defined operator system maximal tensor products, and showed the isomorphism (S max T ) L(S, T ). [B. Xhabli (2012)]: showed the isomorphisms: OMAX k (M n ) OMIN k (M n ), OMIN k (M n ) OMAX k (M n ) tri-partite29 entangle / 37

30 Therefore, we have (OMAX p (M A ) max OMAX q (M B ) max OMAX r (M C ))) L(OMAX p (M A ) max OMAX q (M B ), OMAX r (M C ) ) L(OMAX p (M A ) max OMAX q (M B ), OMIN r (M C )). Under these isomorphisms, a functional on OMAX p (M A ) max OMAX q (M B ) max OMAX r (M C ) is positive (ϱ M A M B M C belongs to S p,q,r ) if and only if its associated linear map OMAX p (M A ) max OMAX q (M B ) OMIN r (M C ) is completely positive (φ is (p, q, r)-positive) tri-partite30 entangle / 37

31 6. examples Consider the following 8 8 matrix: s 1 u 1 s 2 u 2 s 3 u 3 s 4 u 4 ū 4 t 4 ū 3 t 3 ū 2 t 2 ū 1 t tri-partite31 entangle / 37

32 Theorem Suppose that φ : M 2 M 2 M 2 is a bilinear map given by its Choi matrix as above, and consider the inequalities si t i + s j t j u i + u j. (1) Then we have the following: φ is (1, 2, 2)-positive if and only if (1) hold for (i, j) = (1, 4) and (2, 3). φ is (2, 1, 2)-positive if and only if (1) hold for (i, j) = (1, 3) and (2, 4). φ is (2, 2, 1)-positive if and only if (1) hold for (i, j) = (1, 2) and (3, 4) , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite32 entangle / 37

33 Corollary Suppose that φ : M 2 M 2 M 2 is a bilinear map given by its Choi matrix as above. Then the following are equivalent: ϱ, φ 0 for each bi-separable three qubit state ϱ; inequality (1) holds for each possible choice of i, j with i j from {1, 2, 3, 4} , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite33 entangle / 37

34 Theorem Suppose that φ : M 2 M 2 M 2 is a bilinear map given by its Choi matrix as above. Then, φ is (1, 1, 1)-positive if and only if the inequality (s 1 + t 4 α 2 )(s 4 + t 1 α 2 )+ (s 2 + t 3 α 2 )(s 3 + t 2 α 2 ) holds for each α C. In particular, this holds when u 1 ᾱ + ū 4 α + u 2 ᾱ + ū 3 α 4 si t i i=1 4 u i. i= , Waterloo (Seung-Hyeok Kye Various (Seoul National notions of University) positivity afor joint bi-linear work maps with Kyung and applications Hoon Han Jun, to [arxiv: ]) 2015 tri-partite34 entangle / 37

35 Taking s 1 t 1 = 0, s 2 t 2 = 1, s 3 t 3 = 1, s 4 t 4 = 2, u i = 1 for i = 1, 2, 3, 4 to get a (1, 2, 2)-positive map which is neither (2, 1, 2) nor (2, 2, 1)-positive. If we take s 1 t 1 = 0, s 2 t 2 = 0, s 3 t 3 = 2, s 4 t 4 = 2, u i = 1 then the map is both (1, 2, 2) and (2, 1, 2)-positive but not (2, 2, 1)-positive. If we put s i t i = 0 for i = 1, 2, 3 and s 4 t 4 = 4, then we may find an example of (1, 1, 1)-positive map which is not (p, q, r)-positive for other (p, q, r). we may also get an example of (p, q, r)-positive map for each (p, q, r) in Σ 2,2,2 except for (2, 2, 2) by putting s 1 t 1 = 0 and s i t i = 2 for i = 2, 3, tri-partite35 entangle / 37

36 2015 tri-partite36 entangle / 37