Free probability and quantum information

Transcription

1 Free probability and quantum information Benoît Collins WPI-AIMR, Tohoku University & University of Ottawa Tokyo, Nov 8, 2013

2 Overview

3 Overview Plan: 1. Quantum Information theory: the additivity problem 2. A random matrix model 3. Free probability and norm convergence for random matrices 4. Violation of additivity

4 Quantum information A quantum system is a Hilbert space C n. Its set of states D(C n ) = D n is the collection of positive trace one matrices of M n (C).

5 Quantum information A quantum system is a Hilbert space C n. Its set of states D(C n ) = D n is the collection of positive trace one matrices of M n (C). A quantum channel is a linear completely positive trace preserving map Φ from M n (C) to M k (C).

6 Quantum information A quantum system is a Hilbert space C n. Its set of states D(C n ) = D n is the collection of positive trace one matrices of M n (C). A quantum channel is a linear completely positive trace preserving map Φ from M n (C) to M k (C). The trace preservation condition: Φ maps D n to D k.

7 Quantum information A quantum system is a Hilbert space C n. Its set of states D(C n ) = D n is the collection of positive trace one matrices of M n (C). A quantum channel is a linear completely positive trace preserving map Φ from M n (C) to M k (C). The trace preservation condition: Φ maps D n to D k. Complete positivity: d 1, Φ I d : M nd (C) M kd (C) is a positive map.

8 Quantum systems Stinespring theorem: Theorem A linear map Φ : M n (C) M k (C) is a quantum channel if and only if there exists a finite dimensional Hilbert space K = C d, a matrix Y D d and an unitary operation U U(nd) such that Φ(X ) = (id Tr) [U(X Y )U ], X M n (C). (1)

9 Rényi & Shannon Entropy

10 Rényi & Shannon Entropy For a positive real number p > 0, the Rényi entropy of order p of a probability vector x = (x 1,..., x d ) R d (x i 0, i x i = 1) is H p (x) = 1 d 1 p log x p i. i=1

11 Rényi & Shannon Entropy For a positive real number p > 0, the Rényi entropy of order p of a probability vector x = (x 1,..., x d ) R d (x i 0, i x i = 1) is H p (x) = 1 d 1 p log x p i. Since lim p 1 H p (x) exists, we define the Shannon entropy of x to be this limit: d H(x) = H 1 (x) = x i log x i. i=1 i=1

12 von Neumann Entropy, Minimum output entropy Rényi & von Neumann entropy: Extension to density matrices by functional calculus: H p (ρ) = 1 1 p log Tr ρp ; H(ρ) = H 1 (ρ) = Tr ρ log ρ.

13 von Neumann Entropy, Minimum output entropy Rényi & von Neumann entropy: Extension to density matrices by functional calculus: H p (ρ) = 1 1 p log Tr ρp ; H(ρ) = H 1 (ρ) = Tr ρ log ρ. p-minimum output entropy of a quantum channel Φ : M n (C) M k (C): H p min (Φ) = min A D n H p (Φ(A)).

14 Additivity problem

15 Additivity problem Most important problem in QIT until the last decade: For all quantum channels Φ and Φ, does one have H min (Φ Φ) = H min (Φ) + H min ( Φ)?

16 Additivity problem Most important problem in QIT until the last decade: For all quantum channels Φ and Φ, does one have H min (Φ Φ) = H min (Φ) + H min ( Φ)? Loosely speaking: can we transmit more classical data than expected with quantum channels (and entanglement)?

17 Additivity problem, p-version A more general additivity problem (an approach to a positive solution of the original additivity problem via a p 1 limiting procedure):

18 Additivity problem, p-version A more general additivity problem (an approach to a positive solution of the original additivity problem via a p 1 limiting procedure): For all quantum channels Φ and Φ, are the minimum output p-rényi entropies are additive p > 1, H p min (Φ Φ) = H p min (Φ) + Hp min ( Φ)?

19 Additivity problem, p-version A more general additivity problem (an approach to a positive solution of the original additivity problem via a p 1 limiting procedure): For all quantum channels Φ and Φ, are the minimum output p-rényi entropies are additive p > 1, H p min (Φ Φ) = H p min (Φ) + Hp min ( Φ)? Both problems have been proved false by Hayden, Winter, Hastings, and others.

20 Additivity problem, p-version A more general additivity problem (an approach to a positive solution of the original additivity problem via a p 1 limiting procedure): For all quantum channels Φ and Φ, are the minimum output p-rényi entropies are additive p > 1, H p min (Φ Φ) = H p min (Φ) + Hp min ( Φ)? Both problems have been proved false by Hayden, Winter, Hastings, and others. All counterexamples are random. We still don t have an explicit counterexample.

21 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random.

22 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random. Take Φ = Φ (entry-wise conjugate).

23 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random. Take Φ = Φ (entry-wise conjugate). Get a good lower bound on H p min (Φ) = Hp min ( Φ)

24 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random. Take Φ = Φ (entry-wise conjugate). Get a good lower bound on H p min (Φ) = Hp min ( Φ) Get a good upper bound on H p min (Φ Φ).

25 Additivity problem: strategy for a solution Take Φ = Φ n a random sequence of quantum channels (input of increasing size, output of fixed size), at random. Take Φ = Φ (entry-wise conjugate). Get a good lower bound on H p min (Φ) = Hp min ( Φ) Get a good upper bound on H p min (Φ Φ). In practice, the bound is given by an estimation of H p (Φ Φ(ρ)) for a well chosen ρ (Bell state, i.e. maximally entangled state).

26 Digression: Random matrices

27 Digression: Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties, usually as dimension.

28 Digression: Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties, usually as dimension. Discrete mathematics?

29 Digression: Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties, usually as dimension. Discrete mathematics? Yes, from a non-commutative point of view. We will adopt the point of view that a random matrix is a discrete non-commutative random variable, and that a quantum channel is a discrete non-commutative Markov operator.

30 Random matrices Notation: let X n M n (C) be a (self-adjoint or normal) (random) matrix. Let λ (n) i be its eigenvalues, and µ n = n 1 δ (n) λ is the eigenvalue counting measure (this is a i random probability measure).

31 Random matrices Notation: let X n M n (C) be a (self-adjoint or normal) (random) matrix. Let λ (n) i be its eigenvalues, and µ n = n 1 δ (n) λ is the eigenvalue counting measure (this is a i random probability measure). Classical RMT question: When does the (random) probability measure µ n have an interesting behaviour at n?

32 Random matrices Notation: let X n M n (C) be a (self-adjoint or normal) (random) matrix. Let λ (n) i be its eigenvalues, and µ n = n 1 δ (n) λ is the eigenvalue counting measure (this is a i random probability measure). Classical RMT question: When does the (random) probability measure µ n have an interesting behaviour at n? More recent RMT question: how about the (random) set supp(µ n )? (largest eigenvalue)

33 Random matrices Notation: let X n M n (C) be a (self-adjoint or normal) (random) matrix. Let λ (n) i be its eigenvalues, and µ n = n 1 δ (n) λ is the eigenvalue counting measure (this is a i random probability measure). Classical RMT question: When does the (random) probability measure µ n have an interesting behaviour at n? More recent RMT question: how about the (random) set supp(µ n )? (largest eigenvalue) More recent RMT question: how about the eigenvectors?

34 Back to our problem: a Random matrix Model We fix an integer k and a real number t (0, 1).

35 Back to our problem: a Random matrix Model We fix an integer k and a real number t (0, 1). Let n, N be such that n tnk. Let U n be a random partial isometry in M kn,n (C) (think of it as a truncated random Haar unitary matrix in M Nk (C))

36 Back to our problem: a Random matrix Model We fix an integer k and a real number t (0, 1). Let n, N be such that n tnk. Let U n be a random partial isometry in M kn,n (C) (think of it as a truncated random Haar unitary matrix in M Nk (C)) Our counterexample will be Φ n : M n (C) M k (C) given by Φ n (X ) = (id k Tr)(U n XU n).

37 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate.

38 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i

39 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i Then we have a random matrix Z n = Φ n Φ n (X n ) M k 2(C).

40 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i Then we have a random matrix Z n = Φ n Φ n (X n ) M k 2(C). As often in probability theory, it is a sequence of random objects. Unlike in usual random matrix theory, the size of the matrix doesn t change.

41 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i Then we have a random matrix Z n = Φ n Φ n (X n ) M k 2(C). As often in probability theory, it is a sequence of random objects. Unlike in usual random matrix theory, the size of the matrix doesn t change. The study of this random matrix will give us a good upper bound on H p min (Φ Φ)

42 Random matrix Model Let Φ n be defined as Φ n where U n is replaced by its entry-wise conjugate. Let (E i ) n i=1 be the canonical basis of C n and X n is the rank 1 orthogonal projection onto E i E i Then we have a random matrix Z n = Φ n Φ n (X n ) M k 2(C). As often in probability theory, it is a sequence of random objects. Unlike in usual random matrix theory, the size of the matrix doesn t change. The study of this random matrix will give us a good upper bound on H p min (Φ Φ) (=H p min (Φ n Φ n ), for n large enough).

43 Weingarten calculus: integrating over compact groups The unitary Weingarten function Wg(n, σ): inverse of the function σ n #σ under the convolution for the symmetric group (#σ denotes the number of cycles of the permutation σ).

44 Weingarten calculus: integrating over compact groups The unitary Weingarten function Wg(n, σ): inverse of the function σ n #σ under the convolution for the symmetric group (#σ denotes the number of cycles of the permutation σ). Theorem Let n be a positive integer and i = (i 1,..., i p ), i = (i 1,..., i p), j = (j 1,..., j p ), j = (j 1,..., j p) be p-tuples of positive integers from {1, 2,..., n}. Then U(n) U i1 j 1 U ipj p U i 1 j 1 U i pj p du = σ,τ S p δ i1 i σ(1)... δ i pi σ(p) δ j 1 j τ(1)... δ j pj τ(p) Wg(n, τσ 1 ). (2)

45 Weingarten calculus: n limit Theorem For a permutation σ S p, let Cycles(σ) denote the set of cycles of σ. Then Wg(n, σ) = ( 1) n #σ Wg(n, c)(1 + O(n 2 )) (3) and c Cycles(σ) Wg(n, (1,..., d)) = ( 1) d 1 c d 1 where c i = (2i)! (i+1)! i! d+1 j d 1 is the i-th Catalan number. (n j) 1 (4)

46 Application of Weingarten calculus to QIT Theorem (Product channel) Almost surely, as n, the random matrix Φ n Φ n (X n ) = Z n M k 2(C) has non-zero eigenvalues converging towards γ (t) = t + 1 t, 1 t k 2 k 2,..., 1 t k 2 } {{ } k 2 1 times.

47 Application of Weingarten calculus to QIT Theorem (Product channel) Almost surely, as n, the random matrix Φ n Φ n (X n ) = Z n M k 2(C) has non-zero eigenvalues converging towards γ (t) = t + 1 t, 1 t k 2 k 2,..., 1 t k 2 } {{ } k 2 1 times. Consequence: get a good upper bound for H min (Φ Φ) for n large enough

48 Digression 2: Free probability

49 Digression 2: Free probability A non-commutative probability space : unital algebra A with tracial state ϕ. Elements: (non-commutative) random variables.

50 Digression 2: Free probability A non-commutative probability space : unital algebra A with tracial state ϕ. Elements: (non-commutative) random variables. E.g. random matrices (M n (L (Ω, P)), E[n 1 Tr( )]) (discrete non-commutative random variables)

51 Digression 2: Free probability A non-commutative probability space : unital algebra A with tracial state ϕ. Elements: (non-commutative) random variables. E.g. random matrices (M n (L (Ω, P)), E[n 1 Tr( )]) (discrete non-commutative random variables) Let A 1,..., A k be subalgebras of A. They are freely independent if for all a i A ji (i = 1,..., k) such that ϕ(a i ) = 0, one has ϕ(a 1 a p ) = 0 as soon as j 1 j 2, j 2 j 3,..., j p 1 j p.

52 Free probability & non-commutative probability

53 Free probability & non-commutative probability The joint distribution of a k-tuple (a 1,..., a k ) of selfadjoint random variables is the collection of its noncommutative moments.

54 Free probability & non-commutative probability The joint distribution of a k-tuple (a 1,..., a k ) of selfadjoint random variables is the collection of its noncommutative moments. Convergence in distribution = pointwise convergence of moments.

55 Free probability & non-commutative probability The joint distribution of a k-tuple (a 1,..., a k ) of selfadjoint random variables is the collection of its noncommutative moments. Convergence in distribution = pointwise convergence of moments. Sequences of random variables (a (n) 1 ) n,..., (a (n) k ) n are called asymptotically free as n iff the k-tuple (a (n) 1,..., a(n) k converges in distribution towards a family of free random variables. ) n

56 Free probability and random matrices Voiculescu provides the following crucial link between random matrix theory and free probability theory:

57 Free probability and random matrices Voiculescu provides the following crucial link between random matrix theory and free probability theory: Theorem Let U1 n,..., Un k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi n ) i I be a set of constant matrices of M n (C) admitting a joint limit distribution for large n with respect to the state n 1 Tr.

58 Free probability and random matrices Voiculescu provides the following crucial link between random matrix theory and free probability theory: Theorem Let U1 n,..., Un k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi n ) i I be a set of constant matrices of M n (C) admitting a joint limit distribution for large n with respect to the state n 1 Tr. Then, the family ((U1 n, Un 1 ),..., (Un k, Un k ),..., (W n i )) admits a limit distribution, and is asymptotically free with respect to n 1 Tr.

59 Free probability and random matrices: strong version Strong convergence in distribution = in addition to convergence in distribution, convergence of the set of singular values towards the limiting set in the Hausdorff distance sense.

60 Free probability and random matrices: strong version Strong convergence in distribution = in addition to convergence in distribution, convergence of the set of singular values towards the limiting set in the Hausdorff distance sense. Theorem Let U1 n,..., Un k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi n ) i I be a set of constant matrices of M n (C) admitting a STRONG joint limit distribution for large n with respect to the state n 1 Tr.

61 Free probability and random matrices: strong version Strong convergence in distribution = in addition to convergence in distribution, convergence of the set of singular values towards the limiting set in the Hausdorff distance sense. Theorem Let U1 n,..., Un k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi n ) i I be a set of constant matrices of M n (C) admitting a STRONG joint limit distribution for large n with respect to the state n 1 Tr. Then the family ((U1 n, Un 1 ),..., (Un k, Un k ),..., (W n i )) admits a STRONG limit distribution, and is STRONGLY asymptotically free with respect to n 1 Tr.

62 Example: Random projections Let P n, P n be iid selfadjoint random projection in M n (C) of rank n/2.

63 Example: Random projections Let P n, P n be iid selfadjoint random projection in M n (C) of rank n/2. The non-trivial eigenvalues of P n P np n arcsine on [0, 1], and

64 Example: Random projections Let P n, P n be iid selfadjoint random projection in M n (C) of rank n/2. The non-trivial eigenvalues of P n P np n arcsine on [0, 1], and The eigenvalues P n + P n arcsine on [0, 2]

65 Example: Random projections Let P n, P n be iid selfadjoint random projection in M n (C) of rank n/2. The non-trivial eigenvalues of P n P np n arcsine on [0, 1], and The eigenvalues P n + P n arcsine on [0, 2] Same distribution modulo a scaling and trivial eigenvectors... no direct matrix / probability explanation; the only conceptual explanation is through free probability.

66 The t-norm Let A M k (C), P be a random projection of rank n tnk in M kn (C). Then the previous theorem implies that the operator norm of the random matrix P(A 1 N )P converges with probability one to a quantity A (t).

67 The t-norm Let A M k (C), P be a random projection of rank n tnk in M kn (C). Then the previous theorem implies that the operator norm of the random matrix P(A 1 N )P converges with probability one to a quantity A (t). It turns out to be a Banach norm on M k (C) which we call the t-norm (in free probability jargon: free compression norm).

68 Output space for a single random channel Let K n = Φ n (D n ) be the output set of the quantum channel. This is a random convex body.

69 Output space for a single random channel Let K n = Φ n (D n ) be the output set of the quantum channel. This is a random convex body. Theorem With probability one, as n, K n K, where K = {B D k, A D k, Tr(AB) A (t) }

70 Bound for the single channel Theorem For all p > 1, the minimizer of H p on K is attained at a point that does not depend on p. This point x is characterized up to conjugation by the following properties

71 Bound for the single channel Theorem For all p > 1, the minimizer of H p on K is attained at a point that does not depend on p. This point x is characterized up to conjugation by the following properties x K The eigenvalues of x are as follows a > b =... = b.

72 Violation of additivity With the results we stated so far, we obtain by inspection the following results:

73 Violation of additivity With the results we stated so far, we obtain by inspection the following results: Additivity is typically violated with the Bell state iff the output dimension is k 183.

74 Violation of additivity With the results we stated so far, we obtain by inspection the following results: Additivity is typically violated with the Bell state iff the output dimension is k 183. For any p > 1 the violation is asymptotically as bad as it can get with the Bell state.

75 Violation of additivity With the results we stated so far, we obtain by inspection the following results: Additivity is typically violated with the Bell state iff the output dimension is k 183. For any p > 1 the violation is asymptotically as bad as it can get with the Bell state. For p = 1 the violation can be as bad as log 2 (one bit...)

76 Violation of additivity With the results we stated so far, we obtain by inspection the following results: Additivity is typically violated with the Bell state iff the output dimension is k 183. For any p > 1 the violation is asymptotically as bad as it can get with the Bell state. For p = 1 the violation can be as bad as log 2 (one bit...) Known results before: violation for k 10 4, of order 10 3

77 Selected References (1) The strong asymptotic freeness of Haar and deterministic matrices math/arxiv: With C. Male. To appear in Annales Scientifiques de l ENS (2) Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product math/arxiv: With S. Belinschi and I. Nechita. Inventiones Mathematicae December 2012, Volume 190, Issue 3, pp (3) Almost one bit violation for the additivity of the minimum output entropy, arxiv: with S. Belinschi and I. Nechita.