Numerical Range of non-hermitian random Ginibre matrices and the Dvoretzky theorem

Transcription

1 Numerical Range of non-hermitian random Ginibre matrices and the Dvoretzky theorem Karol Życzkowski Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, PAS, Warsaw in collaboration with Bennoit Collins (Kyoto), Piotr Gawron (Gliwice), Sasha Litvak (Edmonton) Probability & Analysis 3, Bȩdlewo, May 18, 017 KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

2 Quantum physics & matrices Quantum states = statics To describe objects of the micro world quantum physics uses quantum states: vectors from a N-dimensional Hilbert space. In Dirac notation: a) ket: ψ = (z 1, z,..., z N ) H N, b) bra: ψ = (z 1, z,..., z N ) HN, c) bra-ket ψ φ C = (ψ, φ) is a scalar product, d) ket- bra ψ φ is an operator = a matrix of order N (often finite!). Mixed quantum states ρ = i a i ψ i ψ i where a i 0 and i a i = 1 (matrix of order N) = normalized convexed combination of projection operators P ψ = ψ ψ Discrete Dynamics: Quantum maps Quantum map Φ : ρ = Φ(ρ) a linear operation acting on matrices of order N. A map Φ can be represented by a matrix of order N. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 017 / 8

3 A) Random density matrices Mixed quantum state = density operator which is a) Hermitian, ρ = ρ, b) positive, ρ 0, c) normalized, Trρ = 1. Let M N denote the set of density operators of size N. Ensembles of random states in M N Random matrix theory point of view: Let A be matrix from an arbitrary ensemble of random matrices. Then ρ = AA TrAA forms a random quantum state fixed trace ensembles : (Hilbert Schmidt, Bures, etc.) Hans Jürgen Sommers, K.Ż, (001, 003, 004, 010) KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

4 B) Quantum Maps & Nonunitary Dynamics Quantum operation: a linear, completely positive trace preserving map Φ acting on a density matrix ρ positivity: Φ(ρ) 0, ρ M N complete positivity: [Φ 1 K ](σ) 0, σ M KN and K =, 3,... The Kraus form ρ = Φ(ρ) = i A i ρa i, where the Kraus operators satisfy i A i A i = 1, which implies that the trace is preserved allows one to represent the superoperator Φ as a (non-hermitian) matrix of size N Φ = i A i Ā i KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

5 KZ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

6 Otton Nikodym & Stefan Banach, talking at a bench in Planty Garden, Cracow, summer 1916 KZ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

7 Composed bi partite systems on H A H B Let G be a rectangular N K matrix will all independent complex Gaussian entries (Ginibre ensemble) Ensembles obtained by partial trace: a) induced measure, A = G i) natural measure on the space of pure states obtained by acting on a fixed state 0, 0 with a global random unitary U AB of size NK ψ = N K G ij i j i=1 j=1 ii) partial trace over the K dimensional subsystem B gives ρ A = Tr B ψ ψ = GG and leads to the induced measure P N,K (λ) in the space of mixed states of size N. Integrating out all eigenvalues but λ 1 one arrives (for large N) at the Marchenko Pastur distribution P c (x = Nλ 1 ) with the parameter c = K/N. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

8 Spectral properties of random matrices Non-hermitian matrix G of size N of the Ginibre ensemble Under normalization TrGG = N the spectrum of G fills uniformly (for large N!) the unit disk The so called circular law of Girko! asymptotic operator norm: G a.s. Hermitian, positive matrix ρ = GG of the Wishart ensemble Let x = Nλ i, where {λ i } denotes the spectrum of ρ. As Trρ = 1 so x = 1. Distribution of the spectrum P(x) is asymptotically given by the Marchenko Pastur law P 1 (x) = P MP (x) = 1 π 4 x 1 for x [0, 4] KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

9 product of matrices and Fuss-Catalan distribution P s ) = FCs (n) defined for an integer number s is ( characterized by its moments x n P s (x)dx = 1 sn+n sn+1 n equal to the generalized Fuss-Catalan numbers. The density P s is analitic on the support [0, (s + 1) s+1 /s s ], while for x 0 it behaves as 1/(πx s/(s+1) ). Asymptotic distribution of singular values for: a) product G 1 G G s and b) for s th power of Ginibre G s, (Alexeev, Götze, Tikhomirov 010) KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

10 Fuss-Catalan distributions P s The moments of P s are equal to Fuss-Catalan numbers. Using inverse Mellin transform one can represent P s by the Meijer G function, which in this case reduces to s hypergeometric functions Exact explicit expressions for FC P s ( ) s = 1, P 1 (x) = 1 π x 1 F 0 1 ; ; 1 4 x = ( s =, P (x) = 3 πx /3 F 1 1 6, 1 3 ; 3 ; 4x 7 = 3 3 1π 1 x/4 π x ) 3 ( x) x x 3 ( x) 1 3, Marchenko Pastur ( ) F 1 1 6, 3 ; 4 3 ; 4x 7 = 3 6πx 1/3 Fuss Catalan Arbitrary s, p s (x) is a superposition of s hypergeometric functions, P s (x) = s j=1 β ( (j) j sf s 1 a 1,..., a(j) s ; b (j) 1,..., b(j) s 1 ; α jx ). Penson, K. Ż., Phys. Rev. E 011. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

11 Wawel castle in Cracow KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

12 Ciesielski theorem KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

13 Ciesielski theorem: With probability 1 ɛ the bench Banach talked to Nikodym in 1916 was localized in η-neighbourhood of the red arrow. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

14 Plate commemorating the discussion between Stefan Banach and Otton Nikodym (Kraków, summer 1916) KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

15 Numerical Range (Field of Values) Definition For any operator A acting on H N one defines its NUMERICAL RANGE (Wertevorrat) as a subset of the complex plane defined by: Λ(A) = { ψ A ψ : ψ H N, ψ ψ = 1}. (1) In physics: Rayleigh quotient, R(A) := x A x / x x Hermitian case For any hermitian operator A = A with spectrum λ 1 λ λ N its numerical range forms an interval: the set of all possible expectation values of the observable A among arbitrary pure states, Λ(A) = [λ 1, λ N ]. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

16 Numerical range and its properties Compactness Λ(A) is a compact subset of C containing spectrum, λ i (A) Λ(A). Convexity: Hausdorff-Toeplitz theorem - Λ(A) is a convex subset of C. Example Numerical range for random matrices of order N = 6 a) normal, b) generic (non-normal) e e KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

17 Numerical range for matrices of order N =. with spectrum {λ 1, λ } a) normal matrix A Λ(A) = closed interval [λ 1, λ ] b) not normal matrix A Λ(A) = elliptical disk with λ 1, λ as focal points and minor axis, d = TrAA λ 1 λ (Murnaghan, 193; Li, 1996). [ ] 0 1 Example : Jordan matrix, J =. 0 0 Its numerical range forms a circular disk, Λ(J) = D(0, r = 1/). The set Ω = CP 1 of N = pure quantum states The set of N = pure quantum states, ψ e iα ψ H, normalized as ψ ψ = 1, forms the Bloch sphere, S = CP 1. A projection of the Bloch sphere onto a plane forms an ellipse, (which could be degenerated to an interval). KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

18 Quantum States and Numerical Range/Shadow Classical States & normal matrices Proposition 1.Let C N denote the set of classical states (probability distributions) of size N, which forms the regular simplex N 1 in R N 1. Then the set of similar images of orthogonal projections of C N on a plane is equivalent to the set of all possible numerical ranges Λ(A) of all normal matrices A of order N (such that AA = A A). Quantum States & non normal matrices Proposition.Let M N denotes the set of quantum states (hermitian positive matrices, ρ = ρ 0, of size N normalized as Tr ρ = 1), embedded in R N 1 with respect to Euclidean geometry induced by Hilbert-Schmidt distance. Then the set of similar images of orthogonal projections M N on a -plane is equivalent to the set of all possible numerical ranges Λ(A) of all matrices A of order N. Dunkl, Gawron, Holbrook, Miszczak, Pucha la, K. Ż (011) KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

19 Shadows of typical matrices of size N = 4 [Classification of numerical ranges for N = 4 is not yet complete...] KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

20 KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

21 Large N limit and random matrices I Non hermitian Ginibre matrices of size N normalized as TrGG = N I 0 1 I 0 1 I 0 1 ρ r R R R N = 10 N = 100 N = 1000 Numerical range and spectrum of random Ginibre matrices of size N. Note circular disk of eigenvalues of Girko and the non normality belt. Result I In the limit N the numerical range of a random Ginibre matrix G converges a.s. (in the Hausdorff distance) to the disk of radius. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

22 Large N limit and random matrices II Diagonalized Ginibre matrices of size N normalized as TrGG = N I 0 I 0 I 0 ρ r R R R N = 10 N = 100 N = 1000 As diagonal matrix G = ZGZ = diag ( Eig(G) ) is normal, the numerical range W (G ) forms the convex hull of its spectrum and asymptotically converges to the unit disk of Girko. Note absence of the non normality belt! KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

23 Large N limit and random matrices III Strictly tri diagonal random matrices T (with T ij = ξ for i > j) of size N normalized as TrTT = N I 0 1 I 0 1 I 0 1 r R R R N = 10 N = 100 N = 1000 The spectrum of T by definition includes the origin, {0}. Its numerical range W (T ) forms a disk with a thick non normality belt. Result II In the limit N the numerical range of a triangular matrix T converges a.s. (in the Hausdorff distance) to the disk of radius. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 017 / 8

24 Large N limit and random matrices Theorem CGLZ, J. Math. Anal. Appl. 418 (014) Let R > 0 and let {X n } n be a sequence of complex random matrices or order n such that for every θ R with probability one lim Re( e iθ ) X n = R n Then with probability one lim d ( H W (Xn ), D(0, R) ) = 0. n Examples: For random Ginibre matrix G and random triagonal matrix T one can show that the operator norm. does not depend on the phase θ and (upon the normalization used) R converges to. Hence the numerical range of G almost surely forms a disk of radius R. Relation to the Dvoretzky theorem! KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

25 Cross-sections of a n cube How a generic cross-section of a 3 cube looks like?? KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

26 Cross-sections of a n cube How a generic cross-section of a 3 cube looks like?? How a generic cross-section of a n cube looks like?? KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

27 Cross-sections of a n cube How a generic cross-section of a 3 cube looks like?? How a generic cross-section of a n cube looks like?? 3-d cross-sections of an n-cube n = 10, 500, figures by Guillaume Auburn and Jos Leys Webpage: Quand les cubes deviennent ronds KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

28 Convex sets in high dimensions Dvoretzky theorem A. Dvoretzky, Some results on convex bodies and Banach spaces (1961) (Under some technical assumptions) a generic cross-section of a compact, high dimensional, convex set almost surely forms a disk! I 0 1 I 0 1 I 0 1 ρ r R R R KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

29 Bench commemorating the discussion between Otton Nikodym and Stefan Banach (Kraków, summer 1916) Sculpture: Stefan Dousa opened in Planty Garden, Cracow, Oct. 14, 016 Fot. Andrzej Kobos KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

30 Concluding Remarks Random mixed state of size N from the induced ensemble (which leads to Marchenko-Pastur spectral density) is obtained by the partial trace of a composite system in an initially random pure state. Singular values of products of (independent) Ginibre matrices are described by s Fuss-Catalan distributions, the densities of which can be written in terms of hypergeoemtric functions. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

31 Concluding Remarks Random mixed state of size N from the induced ensemble (which leads to Marchenko-Pastur spectral density) is obtained by the partial trace of a composite system in an initially random pure state. Singular values of products of (independent) Ginibre matrices are described by s Fuss-Catalan distributions, the densities of which can be written in terms of hypergeoemtric functions. projections of the set of classical states (i.e. probability simplex) are equivalent to the set of numerical ranges of all normal matrices of a given order. The set of all possible projections of the N 1 dimensional set of quantum states of order N onto a plane is equivalent (up to shift and rescaling) to the set of numerical ranges of matrices of order N. Numerical range of a large random Ginibre matrix (independent complex Gaussian entries) forms almost surely a disk of radius R =. KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8

32 KŻ (IF UJ/CFT PAN ) Numerical range of random matrices May 18, / 8