On positive maps in quantum information.

Transcription

1 On positive maps in quantum information. Wladyslaw Adam Majewski Instytut Fizyki Teoretycznej i Astrofizyki, UG ul. Wita Stwosza 57, Gdańsk, Poland fizwam@univ.gda.pl IFTiA Gdańsk University Poland

2 Preliminaries: In operator theory, some of the most important families of linear maps are positive, k-positive, completely positive, and decomposable maps. In quantum physics dynamical maps should be positive, continuous and unital - to have well defined probabilistic scheme. A characterization of the structure of the set of positive maps is a necessary step in many problems in Quantum Information. Theory of positive maps constitute the essential ingredient in the study of non-commutative correlations (entanglements). Classification of entanglement; PPT states IFTiA Gdańsk University Poland 1

3 Definitions and notations let A and B be C -algebras (with unit), A h = {a A;a = a }, A + = {a A h ;a 0} - the set of all positive elements in A, S(A) the set of all states on A, i.e. the set of all linear functionals ϕ on A such that ϕ(1) = 1 and ϕ(a) 0 for any a A +. (A h, A + ) is an ordered Banach space. We say that a linear map α : A B is positive if α(a + ) B +. a linear map τ : A B is CP iff τ n : M n (A) M n (B); [a ij ] [τ(a ij )] (1) is positive for all n. Here, M n (A) stands for n n matrices with entries in A. IFTiA Gdańsk University Poland 2

4 The set of all (linear, bounded) positive maps α : A B will be denoted by L + (A, B). The set of all (linear, bounded) completely positive maps τ : A B will be denoted by CP(A, B). The structure of CP(A, B) is known. BUT this is not the case for L + (A, B). Stinespring theorem, 55 : any CP map T : B(H 1 ) B(H 2 ) is of the form T(A) = W π(a)w, where π : B(H 1 ) B(H 0 ) is a representation, and W : H 2 H 0 is a linear bounded operator. Problem: Structure of L + (A, B). IFTiA Gdańsk University Poland 3

5 Denote by B(X Y ) the Banach space of bounded bilinear mappings B from X Y into the field of scalars with the norm given by B = sup{ B(x, y) ; x 1, y 1}. There is an operator L B L(X, Y ) associated with each bounded bilinear form B B(X Y ). It is defined by y, L B (x) = B(x, y). The mapping B L B is an isometric isomorphism between the spaces B(X Y ) and L(X, Y ). Hence, there is an identification (X π Y ) = L(X, Y ), (2) such that the action of an operator S : X Y as a linear functional on X π Y is given by n x i y i, S = i=1 n y i, Sx i. (3) i=1 IFTiA Gdańsk University Poland 4

6 Note that the identification (2) and the relation (3) determines the linear duality. Moreover, we have used the classical Grothendieck theorem on tensor product of Banach spaces. B(H) equipped with the trace norm will be denoted by T (we have assumed that dim H = n <!). Finally, we denote by B(H) T the algebraic tensor product of B(H) and T and B(H) π T means its Banach space closure under the projective norm defined by n π(x) = inf{ a i b i 1 : x = i=1 n a i b i, a i B(H), b i T}, (4) i=1 where 1 stands for the trace norm. IFTiA Gdańsk University Poland 5

7 Lemma 1. There is an isometric isomorphism φ φ between L(B(H), B(H)) and (B(H) π T) given by n ( φ)( a i b i ) = i=1 n Tr(φ(a i )b t i), (5) i=1 where n i=1 a i b i A T. Furthermore, φ L + (B(H), B(H)) if and only if φ is positive on B(H) + π T +. (see e.g Størmer 86 ; Wickstead 73 ) One should give a credit to Grothendieck 55 for establishing the main idea of the above Lemma: he left the proof of B(X Y ) = (X π Y ) as an exercise! IFTiA Gdańsk University Poland 6

8 Remark There is not any restriction on the dimension of Hilbert space. In other words, this result can be applied to true quantum systems. 2. Størmer 87, showed that in the special case when A = M n (C) and H has dimension equal to n, the above Lemma is a reformulation of Choi result One should note that the positivity of a functional is defined by the projective cone A +ˆ T A generalization of the Choi result (mentioned in 2.2) was also treated by Belavkin and Staszewski 86 (also VPB 81 in Russian). Define (Choi matrix): C φ = ij e i >< e j φ( e i >< e j ), where {e i } n< 1 a CONS in H while φ L + (A, B). Where is the problem? IFTiA Gdańsk University Poland 7

9 Theorem 3. Let A be a Banach algebra. Then there are no two elements a,b in A such that ab ba = 1 for details - H.Wieland, Math. Ann. 121, 21 (1949) and A. Winter, The unboundness of quantum mechanical matrices, Phys. Rev (1947) Consequently, a Hilbert space H should be infinite dimensional one and the Choi matrices are not able to describe genuine quantum channels. On the other hand, quantum channels are described by CP maps, so by positive Choi matrices C φ ; therefore to compare quantum channels Slava has proposed to a version of Radon-Nikodym theorem for CP maps. Slava s result is: IFTiA Gdańsk University Poland 8

10 Definition 4. Let φ, ψ CP(A, B(H)). φ is called to be completely absolutely continuous with respect to ψ if for any n N inf m n (f i, ψ(a ima km f k ) = 0, ik=1 where f i H, for any increasing family of matrices {[a im a km]} implies inf m n i,k=1 (f i, φ(a ima km )f k ), for any f k H, j = 1,2,...n. Then IFTiA Gdańsk University Poland 9

11 Theorem 5. Let φ, ψ CP(A, B(H)), and let φ(a) = F π(a)f be the Stinespring representation for φ. Then ψ is completely absolutely continuous with respect to φ if and only if it has a spacial representation ψ(a) = K π(a)k with π(a)k = Wπ(a)F, where W is a densely defined operator in the minimal H 0, commuting with π(a) = {π(a), a A} on the linear manifold D = { j π(a j)ff j }. Consequently, one can compare certain quantum channels. This Radon-Nikodem type theorem is a version of Arveson result 69 (different assumptions). IFTiA Gdańsk University Poland 10

12 Some consequences of Lemma 1 To take into account that the isomorphism given in Lemma 1 is also isometric: Definition 6. α( ϕ) = sup 0 a B(H) π B(H) Tr ϕa π(a) (6) Definition 7. The set of bp normalized density matrices is defined as D = { φ : α( φ) = 1, φ = φ, φ bp 0, Tr φ = n} (7) Our first characterization of P 1 {linear, positive, unital maps} is IFTiA Gdańsk University Poland 11

13 Proposition Lemma 1 gives an isometric isomorphism between the convex set of unital positive maps P 1 and the set of bp normalized density matrices D. This map sends extreme (exposed) points of P 1 onto extreme (exposed) points of D. 2. D is a convex, compact set. 3. The family of exposed point of D is a dense subset of Ext{D}. Definition 9. (Straszewicz) Let C be a convex set in a Banach space X. A point x C is an exposed point of C (x Exp{C}) if there is f X (dual of X) such that f attains its maximum on C at x and only at x. IFTiA Gdańsk University Poland 12

14 Remark 10. Geometrically speaking, we are using the correspondence between two flat subsets of balls in L(B(H)) and in the set of all selfadjoint bp-density matrices on (B(H) π T) h, respectively. The interest of this remark follows from the fact that the considered balls are not so nicely shaped as the closed ball of real Euclidean 2 or 3 space. The structure and properties of D. Proposition 11. D is globally invariant with respect to the following operations: 1. local operations, LO for short, i.e. maps implemented by unitary operators U : H H H H of the form U = U 1 U 2 where U i : H H is unitary, i = 1,2; 2. partial transpositions τ p = id H τ : B(H) B(H) B(H) B(H) where τ stands for transposition. IFTiA Gdańsk University Poland 13

15 Let φ, ψ be unital in L + (A, B). We say φ dominates ψ if there exists a real number λ > 1 such that (λφ ψ) is a positive map on A. φ is said to be extremal if it can not be written as a convex combination of two others. φ is pure iff it dominates no others. in terms of D: σ φ is extremal if λ > 1 σ ψ σ φ f, g H (f g, λσ φ f g) < (f g, σ ψ f g). IFTiA Gdańsk University Poland 14

16 Thus σ ψ f 0 &g 0 : (f 0 f 0, σ ψ f 0 g 0 ) 0 (f 0 and g 0 depend on σ ψ ) but (f 0 f 0, σ φ f 0 g 0 ) = 0. Note similarity to a characterization of a genuine pure state! IFTiA Gdańsk University Poland 15

17 States: conv(s(a) S(B)) are called separable states. The subset of states S(A B) \ conv(s(a) S(B)) is called the set of entangled states. We will be interested in the special subset of states: S(A B) PPT S PPT = {ϕ S(A B);ϕ (t id) S(A B)} (8) where t stands for transposition. Such states are called PPT states. It is worth observing that the condition in the definition of S PPT is non-trivial; namely the partial transposition does not need to be a positive map! IFTiA Gdańsk University Poland 16

18 S S PPT S sep. A characterization of entanglement; Belavkin-Ohya 2002, M-Matsuoka-Ohya The basic concepts of this approach are the entangling operator H and entanglement mapping φ. Consider a composite system consisting of two subsystems 1, 2. Assume that 1 is defined by the pair (H, B(H)) while 2 by the pair (K, B(K)) respectively, where H and K are separable Hilbert spaces. Let ω be a normal compound state on, i.e. B (H K). ω is a normal state on IFTiA Gdańsk University Poland 17

19 Thus ω (a b) = Trρ ω (a b) with a B (H), b B (K). ρ ω ρ is a density matrix with the spectral resolution ρ = λ i e i e i. i Define a linear bounded operator T ζ : K H K by T ζ η = ζ η (9) where ζ H, η K. The adjoint operator T ζ : H K K is given by T ζ ζ η = (ζ, ζ )η. (10) IFTiA Gdańsk University Poland 18

20 Following B-O scheme, define the operator H : H H K K by the formula: Hζ = i λ 1 2 i ( JH K T J H ζ) ei e i (11) where J H K is a complex conjugation defined by J H K f J H K ( (e i, f) e i i ) = (e i, f)e i where {e i } is any CONS (complete i orthonormal system) extending (if necessary) the orthogonal system {e i } determined by the spectral resolution of ρ. J H is defined analogously with the spectral resolution given by H H; using the explicit form of H, easy calculations show that the spectrum of H H is discrete. IFTiA Gdańsk University Poland 19

21 Theorem 12. The normal state ω can be represented as ω (a b) = Tr H a t H (1 b) H (12) where a t = J H a J H. B [ B (H), B(K) ] stands for the set of all linear, bounded, normal maps from B(H) into B(H) There is the following modification of Lemma 1. IFTiA Gdańsk University Poland 20

22 Lemma 13. (1) An isomorphism ψ Ψ between B [ B (H),B(K) ] and (B(H) B(K)) ( ) Ψ a i b i = Tr K ψ (a i )b t i, a i B (H), b i B (K). i i The isomorphism is isometric if Ψ is considered on B(H) π B(K). Furthermore Ψ is positive on (B(H) B(K)) + iff ψ is complete positive. (2) An isomorphism φ Φ between B [ B (H),B(K) ] and (B(H) B(K)) ( ) Φ a i b i = Tr K φ(a i )b i, a i B (H), b i B (K). i i The isomorphism is isometric if Φ is considered on B(H) π B(K). Furthermore Φ is positive on (B(H) B(K)) + iff φ is complete co-positive. IFTiA Gdańsk University Poland 21

23 Now we are in position to define the entanglement mapping φ, which is the crucial ingredient of B-O approach. Namely: φ (b) = (H (1 b) H) t = J H H (1 b) HJ H. (13) Proposition 14. The entanglement mapping (i) φ : B (H) B (K) has the following explicit form φ (a) = Tr H K Ha t H (14) (ii) The state ω on B (H K) can be written as where φ was defined in (13). ω (a b) = Tr H aφ(b) = Tr K bφ (a) (15) IFTiA Gdańsk University Poland 22

24 Recall, that by definition, any PPT state composed with partial transposition is again a state. This observation and Proposition (14) lead to Corollary 15. PPT states are completely characterized by entanglement mappings φ which are both CP and co-cp. Consequently, for an arbitrary but fixed normal state, there exists the explicit construction of entanglement map. If this mapping, for given state, is both CP and co-cp (what is verifiable) the considered state is PPT. PPT states can be equivalently characterized in terms of Tomita-Takesaki theory. M-Matsuoka-Ohya The latter approach offers a possibility to define measures of entanglement. M-Matsuoka-Ohya. IFTiA Gdańsk University Poland 23