Free probability and quantum information

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

Overview

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

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).

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.

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)

Rényi & Shannon Entropy

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

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

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.

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)).

Additivity problem

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)?

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)?

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):

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.

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.

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).

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).

Digression: Random matrices

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

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

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.

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?

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

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))

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

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.

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)

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)

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.

Digression 2: Free probability

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

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.

Free probability & non-commutative probability

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.

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

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

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.

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.

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

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

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.

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).

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.

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) }

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

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

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.

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.

Selected References (1) The strong asymptotic freeness of Haar and deterministic matrices math/arxiv:1105.4345- 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:1008.3099- With S. Belinschi and I. Nechita. Inventiones Mathematicae December 2012, Volume 190, Issue 3, pp 647-697 (3) Almost one bit violation for the additivity of the minimum output entropy, arxiv:1305.1567 - with S. Belinschi and I. Nechita.