Factorizable completely positive maps and quantum information theory. Magdalena Musat. Sym Lecture Copenhagen, May 9, 2012

Transcription

1 Factorizable completely positive maps and quantum information theory Magdalena Musat Sym Lecture Copenhagen, May 9, 2012 Joint work with Uffe Haagerup based on: U. Haagerup, M. Musat: Factorization and dilation problems for completely positive maps on von Neumann algebras, Comm. Math. Phys., and work in progress (part of which in collaboration with M. B. Ruskai, Tufts University) 1

2 The classical Birkhoff theorem (Birkhoff, 1946): Every doubly stochastic matrix is a convex combination of permutation matrices. Let D = l ({1, 2,..., n}) with trace τ(χ {i} ) = 1/n, 1 i n. The unital positive trace-preserving maps on D are precisely the linear operators on D given by doubly stochastic n n matrices. Note that every automorphism of D is given by a permutation of {1, 2,..., n}. The quantum Birkhoff conjecture: Every unital completely positive τ n -preserving (for short, UCPT n ) map T : M n (C) M n (C), n 2, satisfies T conv(aut(m n (C))). (Here τ n is the normalized trace on M n (C).) Note: φ Aut(M n (C)) if and only if u M n (C) unitary, so that φ(x) = u xu, for all x M n (C). Choi (1975): a linear map T : M n (C) M n (C) is CP if and only if d T x = a i xa i, x M n (C), (1) i=1 where a 1,..., a d M n (C) can be chosen to be linearly independent, in which case d: = Choi-rank(T ). We refer to (1) as the (Choi) canonical form of T. Then T UCP n if and only if d i=1 a i a i = 1 n, while T CPT n if and only if d i=1 a ia i = 1 n. 2

3 Kümmerer (1983): UCPT 2 = conv(aut(m 2 (C))). However, UCPT n conv(aut(m n (C))), n 3, cf. Kümmerer (1986): n = 3, Kümmerer-Maasen (1987): n 4, Landau-Streater (1993): another counterexample for n = 3. Gregoratti-Werner (2003): Channels which are convex combinations of unitarily implemented ones allow for complete error correction, given suitable feedback of classical information from the environment. V. Paulsen brought to our attention the following asymptotic version of the quantum Birkhoff conjecture, listed as Problem 30 on R. Werner s web page of open problems in quantum information theory: The asymptotic quantum Birkhoff conjecture (J. A. Smolin, F. Verstraete, A. Winter, 2005): Let T UCPT n (unital quantum channel in dimension n), n 3. Then T satisfies the following asymptotic quantum Birkhoff property: ( k ) lim d cb T, conv(aut( k M n (C))) = 0. (AQBP) k i=1 C. Mendl and M. Wolf (2009): Significant evidence in support of the AQBP, namely, they provided examples of unital quantum channels T such that T / conv(aut(m n (C))), but i=1 T T conv(aut(m n (C))). 3

4 Theorem (Haagerup-M, 2011): For every n 3, there exist UCPT n maps which fail the AQBP. Key tool: The existence of non-factorizable maps, a notion introduced by C. Anantharaman-Delaroche, Let (M, φ) and (N, ψ) be von Neumann algebras with normal, faithful, tracial states. Definition (Anantharaman-Delaroche, 2005): A UCP map T : M N which is state-preserving (i.e., ψ T = φ), is called factorizable if a vn algebra P with a normal, faithful tracial state χ and injective state-preserving unital -homomorphisms α: M P and β : N P such that T = β α. M T α P N β =β 1 E β(n) Remark: The set of factorizable maps F(M, N) is convex, and it is closed under composition and by taking adjoints. In particular, conv(aut(m n (C))) F(M n (C)) UCPT n. By Kümmerer s 1983 result, we have equality throughout for n = 2. We will later see that the first inclusion is strict, for all n 3. Problem (C. Delaroche, 2005): Is every UCP state-preserving map factorizable? 4

5 Factorizability of unital quantum channels Theorem 1 (Haagerup-M., 2011): Let T UCPT n, n 3, written in Choi canonical form T x = d a i xa i, x M n (C). i=1 The following are equivalent: 1) T is factorizable 2) There exist a vn algebra N with a normal faithful tracial state τ N and u U(M n (N)) such that T x = (id Mn (C) τ N )(u (x 1 N )u), x M n (C). We say that T has an exact factorization through M n (C) N. 3) There exist a vn algebra N with a normal faithful tracial state τ N and v 1,..., v d N such that u: = d i=1 a i v i U(M n (N)) and, moreover, τ N (vi v j) = δ ij, 1 i, j d. Interpretation in QIT (Werner): Factorizable maps is the class of operations one can get by coupling the input system to a maximally mixed ancillary one, executing a unitary rotation on the combined system and tracing out the ancilla. 5

6 Proposition: On the inclusion conv(aut(m n (C))) F(M n (C)) Let T UCPT n, n 3. Then T conv(aut(m n (C)) if and only if T has an exact factorization through M n N, where N is abelian. Corollary 1: Let T UCPT n be of the form T x = d i=1 a ixa i, x M n (C), where a 1,..., a d M n (C) are self-adjoint and satisfy a i a j = a j a i, 1 i, j d. Then T is factorizable. Furthermore, if d 3 and {a i a j : 1 i j d} is a linearly independent set, then T / conv(aut(m n (C))). Example: Let β = 1/ 5 and set 1 β β β β β β 1 β β β β β β 1 β β β B : = β β β 1 β β β β β β 1 β β β β β β 1. Then the associated Schur multiplier T B satisfies the hypotheses of Corollary 1, hence T B is factorizable, but T B / conv(aut(m 6 (C))). 6

7 Corollary 2: Examples of non-factorizable maps Let T UCPT n, with canonical form T = d i=1 a i xa i. If d 2 and {a i a j : 1 i, j d} is a linearly independent set, then T is not factorizable. Example: As an application of Corollary 2, by letting a 1 = , a 2 = , a 3 = we obtained a (first example of a) non-factorizable unital channel. Turned out to be the Holevo-Werner channel, W3, in dimension n = 3. Remark: Choi proved that T e (UCP n ) if and only if the set {a i a j : 1 i, j d} is linearly independent. (So, if T e (UCP n ), then Choi-rank(T ) n.) By Corollary 2, if then T is not factorizable. T e (UCP n ) UCPT n Starting from this remark, in very recent work (in progress) with M.B. Ruskai, we constructed large classes of non-factorizable unital quantum channels in all dimensions. 7

8 Landau and Streater (1993): T e (UCPT n ) if and only if the set {a i a j a j a i : 1 i, j d} is lin. indep. (Note: T e (UCPT n ) Choi-rank(T ) 2n 2 1.) Hence e (UCPT n ) ( e (UCP n ) e (CPT n )) UCPT n. It was pointed out by Mendl and Wolf (2009) that the above inclusion is strict for n = 3. Later, Ohno (2010) provided concrete examples (satisfying the strict inclusion) in dimensions n = 3 and n = 4. Further examples (Haagerup-M.-Ruskai): Motivated by a question of D. Farenick (Mittag-Leffler, 2010), and B. Ruskai s own interest in the problem, we recently constructed a family (T t ) t [0,1] UCPT 3 with the property that if t / {0, 1/2, 1}, then T t e (UCPT 3 ), while T t / e (UCP 3 ) e (CPT 3 ). Moreover, for 0 t 1, each T t is factorizable (in fact, through M 3 (C) M 2 (C)). 8

9 We also constructed an example of a semigroup (T t ) t 0 of UCPT 4 maps for which there exists t 0 > 0 such that T (t) is not factorizable, for any 0 < t < t 0. Remarks: Kümmerer and Maassen (1987) proved that if (T t ) t 0 is a one-parameter semigroup of UCPT n maps satisfying T (t) = T (t), t 0, (2) then T (t) conv(aut(m n (C))), for all t 0. In particular, T (t) is factorizable, t 0. Our example shows that the self-adjointness condition (2) is necessary for the Kümmerer and Maassen result to hold. In very recent work, M. Junge, E. Ricard and D. Shlyakhtenko have generalized the result of Kümmerer and Maassen, by showing that if (T t ) t 0 is a strongly continuous one-parameter semigroup of UCPT maps on a finite von Neumann algebra M, satisfying T (t) = T (t), t 0, then T (t) is factorizable, for all t 0. This result has been obtained independently (by different methods) by Y. Dabrowski. 9

10 Non-factorizable maps and the AQBP Theorem (Haagerup-M, 2011): Let T UCPT n, where n 3. Then, for all k 1, ( k ( k )) d cb T, F M n (C) d cb (T, F(M n (C))). i=1 i=1 Note that if T is not factorizable, then d cb (T, F(M n (C))) > 0, since F(M n (C)) is norm-closed. Therefore, since conv(aut(m n (C))) F(M n (C)), we deduce that any non-factorizable unital channel T fails the AQBP. 10

11 Natural question: Does every factorizable unital quantum channel in dimension n satisfy the AQBP, for all n 3? This question (that arose during discussions at a 2010 Banff meeting) gained a lot of interest due to the following connection to the Connes embedding problem, whether every II 1 -factor (on a separable Hilbert space) embeds in an ultrapower R ω of the hyperfinite II 1 factor R. Theorem (Haagerup-M, 2011): If for any n 3, every factorizable UCPT n map satisfies the AQBP, then the Connes embedding problem has a positive answer. Earlier, we have constructed a factorizable unital Schur channel T B on M 6 (C), satisfying T B / conv(aut(m 6 (C)). Question: Does T B satisfy the AQBP? Theorem (Haagerup-M): Let T be a unital Schur channel on M n (C) and S be a unital Schur channel on M k (C), k, n 1. Then d cb (T S, conv(aut(m nk (C)))) 1 2 d cb(t, conv(aut(m n (C)))). In particular, if T / conv(aut(m n (C))), then T fails the AQBP. Conclusion: The factorizable map T B above fails the AQBP. 11

12 On the connection to the Connes embedding problem Theorem (Haagerup-M): Let T UCPT n be factorizable. The following are equivalent: (1) T has an exact factorization through M n (C) N, where (N, τ N ) embeds into R ω, i.e., u U(M n (N)) such that T x = (id Mn (C) τ N )(u (x 1 N )u), x M n (C). (2) ε > 0 there exist k N and T UCPT n having an exact factorization through M n (C) M k (C) such that T T cb < ε. (3) T satisfies the following asymptotic property: lim d cb(t S k, conv(aut(m nk (C)))) = 0, k where S k is the completely depolarizing channel S k (y) = τ k (y)1 k, y M k (C). Theorem (Haagerup-M): The Connes embedding problem has a positive answer if and only if every factorizable UCPT n map satisfies one of the equivalent conditions in above theorem, for all n 3. 12

13 Theorem (Haagerup-M): The Connes embedding problem has a positive answer if and only if every factorizable UCPT n map satisfies one of the equivalent conditions in above theorem, for all n 3. Idea of proof: Dykema and Jushenko (2009) considered for n 1: F n : = { B = (bij ) M n (C) : b ij = τ k (u i u j ), u 1,..., u n U(M k (C)) } k 1 { G n : = B = (b ij ) M n (C) : b ij = τ M (u i u j), u 1,..., u n U(M), for } some (M, τ M ) von Neumann algebra with n.f. tracial state τ M By deep results of Kirchberg (1993), the Connes embedding problem has a positive answer if and only if F n = G n, for all n 1. Assume that Connes embedding problem has a negative answer. Then G n \ F n, for some n 1. Choose B = (b ij ) n i,j=1 G n \ F n. It follows that the associated Schur multiplier T B is factorizable. This implies that u 1,... u n unitaries in R ω so that b ij = τ R ω(u i u j ), 1 i, j n. Then we approximate each b ij by τ R (v i v j), where v i s are unitaries in R, and further down with unitaries coming from matrix algebras (via Kaplansky s density theorem). Hence B can be approximated by a sequence of matrices B k for which the Schur multiplier T Bk admits an exact factorization through a matrix algebra. This will imply that B F n, a contradiction! 13

14 Some concrete examples Let n 2. Consider the Holevo-Werner channel in dimension n: It has an analogue W n (x) = 1 n 1 ( Tr(x)1 n x t ), x M n (C). W + n (x) = 1 n + 1 ( Tr(x)1 n + x t ), x M n (C). Note: W n and W + n are unital quantum channels. The completely depolarizing channel S n conv(w n, W + n ), since for x M n (C), S n (x) = 1 n Tr(x)1 n = n 1 2n W n (x) + n + 1 2n W + n (x). Furthermore, if we denote a ij = e ij e ji, b ij = e ij + e ji, 1 i, j n then for all x M n (C), we have (Choi canonical forms): W n (x) = W + n (x) = 1 a ij xa ij n 1 i<j 1 b ij xb ij + 2 n + 1 i<j i e ii xe ii. Remark: W 3 is not factorizable. 14

15 Theorem (Mendl-Wolf, 2009): (1) W n + conv(aut(m n (C))), for all n 2. (2) Wn conv(aut(m n (C))), for all n even. (3) For n odd and 0 λ 1, λ W n + + (1 λ) Wn conv(aut(m n (C))) if and only if 1 n λ 1. In particular, W n / conv(aut(m n (C))). Proposition (Haagerup-M): For n odd, d cb (Wn, conv(aut(m n (C)))) = 2/n. Theorem (Haagerup-M): (1) d cb (W 3, F(M 3(C)) = (2) W n is factorizable, for all n odd, n 3. Moreover, W n has an exact factorization through M n (C) M 4 (C), and hence W n S 4 conv(aut(m 4n (C))). (3) λ W (1 λ) W 3 is factorizable if and only if 2 27 λ 1. Moreover, for all such λ, the map λ W 3 + +(1 λ) W 3 factorization through M n (C) M 3 (C), and hence ( λ W (1 λ) W 3 ) S 3 conv(aut(m 9 (C))). has an exact 15

16 For 0 λ 1, set T λ : = λ W (1 λ) W 3. We have seen (cf. Mendl-Wolf, 2009) that T λ conv(aut(m n (C))) if and only if λ 1 3. Theorem (Mendl-Wolf, 2009): There exists λ 0 ( 0, 1 3) such that for all λ > λ0, T λ T λ conv(aut(m 3 (C) M 3 (C))). Theorem (Haagerup-M): There exists λ 1 (< 1 3 ) such that for all λ > λ 1, T λ T λ T λ conv(aut(m 3 (C) M 3 (C) M 3 (C))). 16

17 An averaging technique For T B(M n (C)) set F (T ): = U(n) ρ u (T )du, where du is the Haar measure on U(n) ρ u (T )(x) = ut (u t xū)u = (ad(u) T ad(u t ))(x), x M n (C). It is easily seen that if T UCPT n, then F (T ) UCPT n. Moreover, F (T S) cb T S cb, T, S UCPT n. Furthermore, F (conv(aut(m n (C)))) conv(aut(m n (C))) and F (F(M n (C))) F(M n (C)). Also, F (W n + ) = W n + and F (Wn ) = Wn. Theorem (Haagerup-M): If T UCPT n, then F (T ) conv(w + n, W n ). 17