Operator norm convergence for sequence of matrices and application to QIT

Transcription

1 Operator norm convergence for sequence of matrices and application to QIT Benoît Collins University of Ottawa & AIMR, Tohoku University Cambridge, INI, October 15, 2013

2 Overview

3 Overview Plan: 1. Norm convergence for simple random matrices. 2. Application: threshold for for the appt property. 3. Norm convergence for multiple random matrices. 4. Application: converse threshold for PPT. 5. Application: convergence of the collection of output states.

4 Random matrices

5 Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties as d.

6 Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties as d. Notation: let X d M d (C) be a (self-adjoint or normal) (random) matrix. Let λ (d) i be its eigenvalues, and µ d = d 1 λ (d) i is the eigenvalue counting measure (histogram of eigenvalues).

7 Random matrices Random matrix theory: probability theory with matrix valued random variables, study of its properties as d. Notation: let X d M d (C) be a (self-adjoint or normal) (random) matrix. Let λ (d) i be its eigenvalues, and µ d = d 1 λ (d) i is the eigenvalue counting measure (histogram of eigenvalues). Classical RMT question: When does the (random) probability measure µ d have an interesting behaviour at d? More recent RMT question: how about the (random) set supp(µ d )? (largest eigenvalue)

8 Single random matrix: Wishart matrices Let G M d s (C) be a Ginibre random matrix

9 Single random matrix: Wishart matrices Let G M d s (C) be a Ginibre random matrix ({G ij } are i.i.d. standard complex Gaussian random variables)

10 Single random matrix: Wishart matrices Let G M d s (C) be a Ginibre random matrix ({G ij } are i.i.d. standard complex Gaussian random variables) Let W = W d = GG be the corresponding Wishart matrix of parameters (d, s) (G denotes the Hermitian adjoint of G).

11 Single random matrix: Wishart matrices Let G M d s (C) be a Ginibre random matrix ({G ij } are i.i.d. standard complex Gaussian random variables) Let W = W d = GG be the corresponding Wishart matrix of parameters (d, s) (G denotes the Hermitian adjoint of G). One has E[W ij ] = sδ ij and E[Tr(W )] = ds.

12 Single random matrix: Wishart matrices The Marchenko-Pastur (or free Poisson) probability distributions, π c, (c > 0), is defined as follows (x a)(b x) π c = max(1 c, 0)δ πx [a,b] (x) dx, (1) (a = ( c 1) 2 and b = ( c + 1) 2 ).

13 Single random matrix: Wishart matrices The Marchenko-Pastur (or free Poisson) probability distributions, π c, (c > 0), is defined as follows (x a)(b x) π c = max(1 c, 0)δ πx [a,b] (x) dx, (1) (a = ( c 1) 2 and b = ( c + 1) 2 ).

14 Ginibre and Wishart ensembles Notation: Let γ be the full cycle γ = (1 2 k). For any permutation α, let α be its length on the Cayley graph of S k generated by transpositions. The geodesic inequality α + α 1 γ γ = k 1 holds. We define g(α), the genus of a permutation α on I, as g(α) = α + α 1 γ γ. (2) 2

15 Ginibre and Wishart ensembles Let W d be a d d Wishart matrix of parameter (d, s) and let Z d = ( Wd ds ds Id ) d be its centered and renormalized version. Theorem The moments of Z d are given by [ 1 E d Tr( Z p ) ] d = ( ) d α p/2 d 2g(α), (3) s α Sp o

16 Ginibre and Wishart ensembles The previous theorem implies that x k µ d (dx) x k π c (dx) (only genus zero terms survive):

17 Ginibre and Wishart ensembles The previous theorem implies that x k µ d (dx) x k π c (dx) (only genus zero terms survive): Theorem Assuming s/d c, with the renormalization W d / sd, µ d π c.

18 Ginibre and Wishart ensembles The previous theorem implies that x k µ d (dx) x k π c (dx) (only genus zero terms survive): Theorem Assuming s/d c, with the renormalization W d / sd, µ d π c. However the only thing we can say about supp(µ d ) at this point is lim d d d dsupp(µ d ) supp(π c ). The converse inclusion is not clear.

19 Ginibre and Wishart ensembles In general, convergence of µ d to something, is ensured by convergence of moments (under reasonable boundedness assumption).

20 Ginibre and Wishart ensembles In general, convergence of µ d to something, is ensured by convergence of moments (under reasonable boundedness assumption). In order to ensure the convergence of supp(µ d ), we need to look at moments k that grow as d grows.

21 Ginibre and Wishart ensembles In general, convergence of µ d to something, is ensured by convergence of moments (under reasonable boundedness assumption). In order to ensure the convergence of supp(µ d ), we need to look at moments k that grow as d grows. [for example, d 1 Tr diag(1 + ε, 1,..., 1) k ) behaves like d 1 Tr diag(1 + ε, 1,..., 1) k ) at k fixed, d. The situation starts to change when k >> log d this is the minimal growth of k needed to understand supp(µ d ).]

22 Ginibre and Wishart ensembles Theorem In the situation where s/d converges, and d, then the extremal eigenvalues of W d /d converge almost surely to a, b.

23 Ginibre and Wishart ensembles Theorem In the situation where s/d converges, and d, then the extremal eigenvalues of W d /d converge almost surely to a, b. And almost surely, when 1 d s, the extremal eigenvalues of ds(wd /ds 1/d) converge to ±2.

24 Application: APPT A quantum state ρ D(C d 1 C d 2 ) is absolutely PPT (or APPT) if for any unitary matrix U U(d), UρU PPT. APPT = U U(d) U(PPT )U PPT.

25 Application: APPT A quantum state ρ D(C d 1 C d 2 ) is absolutely PPT (or APPT) if for any unitary matrix U U(d), UρU PPT. APPT = U U(d) U(PPT )U PPT. APPT is a convex body, a convex compact set with non-empty interior. Known fact: ɛd + (1 ɛ) I d S PPT for some ɛ < 1 d 1.

26 Application: APPT A quantum state ρ D(C d 1 C d 2 ) is absolutely PPT (or APPT) if for any unitary matrix U U(d), UρU PPT. APPT = U U(d) U(PPT )U PPT. APPT is a convex body, a convex compact set with non-empty interior. Known fact: ɛd + (1 ɛ) I d S PPT for some ɛ < 1 d 1. There exists an explicit algebraic characterization.

27 Application: APPT Setup: d = d 1 d 2, we pick a Wishart matrix of parameter d, s and try to check the APPT property for s, d Let p = min(d 1, d 2 ).

28 Application: APPT When p = min(d 1, d 2 ), we have the following (almost sharp) threshold estimate

29 Application: APPT When p = min(d 1, d 2 ), we have the following (almost sharp) threshold estimate Theorem (C, Nechita, Ye) Let ρ be a random state according to the parameters d, s. (i) almost surely, when d and s > (4 + ε)p 2 d, the quantum state ρ is APPT; (ii) when 1 p 2 d and s < (4 ε)p 2 d, ρ is not APPT almost surely; (iii) when p 2 τd for a constant τ (0, 1], there exists a constant C τ such that whenever s < 4(C τ ε)p 2 d, ρ is not APPT almost surely.

30 Application: APPT When p = min(d 1, d 2 ) is fixed and s/d c for a constant c > 0 as d, sharp estimate on the threshold for APPT. Theorem (C, Nechita, Ye) Let ρ be a random induced state distributed according to the measure µ d,s. Almost surely, when d and s cd, one has: (i) ρ APPT, if c > (p + p 2 1) 2 ; (ii) ρ / APPT, if c < (p + p 2 1) 2.

31 Multimatrices So far we only dealt with one matrix. What if we take a bunch of i.i.d random matrices and take NC polynomials in them?

32 Multimatrices So far we only dealt with one matrix. What if we take a bunch of i.i.d random matrices and take NC polynomials in them? [E.g. understand the behaviour of W (1) d W (2) d + W (2) d W (1) d, if W (1) d, W (2) d are independent copies]

33 Multimatrices So far we only dealt with one matrix. What if we take a bunch of i.i.d random matrices and take NC polynomials in them? [E.g. understand the behaviour of W (1) d W (2) d + W (2) d W (1) d, if W (1) d, W (2) d are independent copies] In principle, the method of moments allows us to understand the limiting behaviour of µ d. But the limiting behaviour of supp(µ d ) is much more difficult to understand.

34 Multimatrices So far we only dealt with one matrix. What if we take a bunch of i.i.d random matrices and take NC polynomials in them? [E.g. understand the behaviour of W (1) d W (2) d + W (2) d W (1) d, if W (1) d, W (2) d are independent copies] In principle, the method of moments allows us to understand the limiting behaviour of µ d. But the limiting behaviour of supp(µ d ) is much more difficult to understand. C*-algebra vs von Neumann algebra.

35 Multimatrices So far we only dealt with one matrix. What if we take a bunch of i.i.d random matrices and take NC polynomials in them? [E.g. understand the behaviour of W (1) d W (2) d + W (2) d W (1) d, if W (1) d, W (2) d are independent copies] In principle, the method of moments allows us to understand the limiting behaviour of µ d. But the limiting behaviour of supp(µ d ) is much more difficult to understand. C*-algebra vs von Neumann algebra. In practice, we can t understand directly the behaviour of moments where k grows together with d for multi matrices.

36 Linearization trick

37 Linearization trick Roughly speaking:

38 Linearization trick Roughly speaking: Understanding supp(µ d ) for all non-commutative polynomials in W (1) d, W (2) d [W (1) d W (2) d + W (2) d W (1) d and all the others] is equivalent to

39 Linearization trick Roughly speaking: Understanding supp(µ d ) for all non-commutative polynomials in W (1) d, W (2) d [W (1) d W (2) d + W (2) d W (1) d and all the others] is equivalent to Understanding supp(µ d ) for all a 0 1 d + a 1 W (1) d + a 2 W (2) d for all a 0, a 1, a 2 M k (C) selfadjoint matrices, all k.

40 Linearization trick Advantage: no non-commutative multiplication ( linearization ).

41 Linearization trick Advantage: no non-commutative multiplication ( linearization ). Price to pay: (1) allow matrix coefficients (2) give up speed of convergence (global equivalence).

42 Linearization trick Advantage: no non-commutative multiplication ( linearization ). Price to pay: (1) allow matrix coefficients (2) give up speed of convergence (global equivalence). Extra bonus: we obtain for free the understanding of non-commutative polynomials with matrix coefficients [e.g. a 1 W (1) d W (2) d + a 2 W (2) d W (1) d.]

43 Linearization trick Advantage: no non-commutative multiplication ( linearization ). Price to pay: (1) allow matrix coefficients (2) give up speed of convergence (global equivalence). Extra bonus: we obtain for free the understanding of non-commutative polynomials with matrix coefficients [e.g. a 1 W (1) d W (2) d + a 2 W (2) d W (1) d.] (the bonus matters in the applications)

44 Free probability and random matrices A non-commutative probability space : unital algebra A with tracial state ϕ (Elements therein: NCRV). E.g. random matrices (M d (L (Ω, P)), E[d 1 Tr( )])

45 Free probability and random matrices A non-commutative probability space : unital algebra A with tracial state ϕ (Elements therein: NCRV). E.g. random matrices (M d (L (Ω, P)), E[d 1 Tr( )]) Let A 1,..., A k be subalgebras of A. They are free 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.

46 Free probability and random matrices A non-commutative probability space : unital algebra A with tracial state ϕ (Elements therein: NCRV). E.g. random matrices (M d (L (Ω, P)), E[d 1 Tr( )]) Let A 1,..., A k be subalgebras of A. They are free 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. The joint distribution of a k-tuple (a 1,..., a k ) of selfadjoint random variables is the collection of its NC moments.

47 Free probability and random matrices Convergence in distribution = pointwise convergence of moments.

48 Free probability and random matrices Convergence in distribution = pointwise convergence of moments. Sequences of random variables (a (d) 1 ) n,..., (a (d) k ) n are called asymptotically free as n iff the k-tuple (a (d) 1,..., a(d) k ) n converges in distribution towards a family of free random variables.

49 Free probability and random matrices Theorem (Voiculescu) Let U 1,..., U k,... be a collection of independent Haar distributed random matrices of M d (C) and (Wi d ) i I be a set of constant matrices of M d (C) admitting a joint limit distribution for large n with respect to the state d 1 Tr.

50 Free probability and random matrices Theorem (Voiculescu) Let U 1,..., U k,... be a collection of independent Haar distributed random matrices of M d (C) and (Wi d ) i I be a set of constant matrices of M d (C) admitting a joint limit distribution for large n with respect to the state d 1 Tr. Then, the family ((U 1, U1 ),..., (U k, Uk ),..., (W i)) admits a limit distribution, and is asymptotically free with respect to E(d 1 Tr).

51 Free probability and random matrices Strong convergence = in addition, convergence of the set of singular values towards the limiting set in the Hausdorff distance sense.

52 Free probability and random matrices Theorem (C, Male) Let U 1,..., U k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi d ) i I be a set of constant matrices of M d (C) admitting a STRONG joint limit distribution for large d with respect to the state d 1 Tr.

53 Free probability and random matrices Theorem (C, Male) Let U 1,..., U k,... be a collection of independent Haar distributed random matrices of M n (C) and (Wi d ) i I be a set of constant matrices of M d (C) admitting a STRONG joint limit distribution for large d with respect to the state d 1 Tr. Then the family ((U 1, U1 ),..., (U k, Uk ),..., (W i)) admits a STRONG limit distribution, and is STRONGLY asymptotically free with respect to E(d 1 Tr).

54 Application: treshold for asymetric PPT In M d M n, Banica and Nechita (arxiv ) consider the partial transpose of Wishart matrices of parameter dn, dm and prove that as d (m, n fixed), µ d converges to a free difference of Marchenko Pastur distributions of parameter m(n ± 1)/2.

55 Application: treshold for asymetric PPT In M d M n, Banica and Nechita (arxiv ) consider the partial transpose of Wishart matrices of parameter dn, dm and prove that as d (m, n fixed), µ d converges to a free difference of Marchenko Pastur distributions of parameter m(n ± 1)/2. They prove that the spectrum of this distribution is positive iff n m/4 + 1/m and m 2.

56 Application: treshold for asymetric PPT In M d M n, Banica and Nechita (arxiv ) consider the partial transpose of Wishart matrices of parameter dn, dm and prove that as d (m, n fixed), µ d converges to a free difference of Marchenko Pastur distributions of parameter m(n ± 1)/2. They prove that the spectrum of this distribution is positive iff n m/4 + 1/m and m 2. This proves that PPT does not hold with high probability if one of these conditions is violated.

57 Application: treshold for asymetric PPT In M d M n, Banica and Nechita (arxiv ) consider the partial transpose of Wishart matrices of parameter dn, dm and prove that as d (m, n fixed), µ d converges to a free difference of Marchenko Pastur distributions of parameter m(n ± 1)/2. They prove that the spectrum of this distribution is positive iff n m/4 + 1/m and m 2. This proves that PPT does not hold with high probability if one of these conditions is violated. Conversely, our result proves that PPT does hold with high probability if these conditions are satisfied.

58 Application: simultaneous behaviour of quantum channels Let π d be a (random projection) in M k M d having the following property: A M k, the pair of matrices (π d, A 1 d ) converge strongly.

59 Application: simultaneous behaviour of quantum channels Let π d be a (random projection) in M k M d having the following property: A M k, the pair of matrices (π d, A 1 d ) converge strongly. Let Φ d be the channel End(Im(π d )) M k obtained by taking the partial trace over M d. Let χ be any other quantum channel M p M q.

60 Application: simultaneous behaviour of quantum channels Let π d be a (random projection) in M k M d having the following property: A M k, the pair of matrices (π d, A 1 d ) converge strongly. Let Φ d be the channel End(Im(π d )) M k obtained by taking the partial trace over M d. Let χ be any other quantum channel M p M q. Theorem (C, Fukuda, Nechita) The collection Φ d (pure states) converges strongly to a convex body (the dual of {A, lim π d A 1 d π d 1}).

61 Application: simultaneous behaviour of quantum channels Let π d be a (random projection) in M k M d having the following property: A M k, the pair of matrices (π d, A 1 d ) converge strongly. Let Φ d be the channel End(Im(π d )) M k obtained by taking the partial trace over M d. Let χ be any other quantum channel M p M q. Theorem (C, Fukuda, Nechita) The collection Φ d (pure states) converges strongly to a convex body (the dual of {A, lim π d A 1 d π d 1}). The same holds true for for Φ d χ!

62 Application: simultaneous behaviour of quantum channels Example 1: π d is a random projection of rank tkd, t [0, 1] (allows us to obtain violation of MOE additivity as large as log 2)

63 Application: simultaneous behaviour of quantum channels Example 1: π d is a random projection of rank tkd, t [0, 1] (allows us to obtain violation of MOE additivity as large as log 2) Example 2: π d = k 1 (U (d) i U (d) j ) (complementary channel of Φ d : X k 1 U i XUi ). This allows us to prove that Φ d S1 S = 4(k 1)/k 2, k 2.

64 Application: simultaneous behaviour of quantum channels Example 1: π d is a random projection of rank tkd, t [0, 1] (allows us to obtain violation of MOE additivity as large as log 2) Example 2: π d = k 1 (U (d) i U (d) j ) (complementary channel of Φ d : X k 1 U i XUi ). This allows us to prove that Φ d S1 S = 4(k 1)/k 2, k 2. Some other applications to new example of k-positive maps...

65 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. (4) The absolute positive partial transpose property for random induced states. math/arxiv: With I. Nechita and D. Ye, to appear in RMTA (5) In preparation, with M. Fukuda and I. Nechita (6) In preparation, with P. Hayden and I. Nechita