Numerical Range of non-hermitian random Ginibre matrices and the - - PowerPoint PPT Presentation

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

Karol ˙ Zyczkowski

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, B¸ edlewo, May 18, 2017

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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 = (z1, z2, . . . , zN) ∈ HN, b) bra: ψ| = (z1, z2, . . . , zN)∗ ∈ H∗

N,

c) bra-ket ψ|φ ∈ ❈ = (ψ, φ) is a scalar product, d) ket- bra |ψφ| is an operator = a matrix of order N (often finite!).

Mixed quantum states

ρ =

i ai|ψiψi| where ai ≥ 0 and

i ai = 1 (matrix of order N)

= normalized convexed combination of projection operators Pψ = |ψψ|

Discrete Dynamics: Quantum maps

Quantum map Φ : ρ′ = Φ(ρ) – a linear operation acting on matrices of

• rder N.

A map Φ can be represented by a matrix of order N2.

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A) Random density matrices

Mixed quantum state = density operator which is a) Hermitian, ρ = ρ∗, b) positive, ρ ≥ 0, c) normalized, Trρ = 1.

Let MN denote the set of density operators of size N.

Ensembles of random states in MN

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–J¨ urgen Sommers, K. ˙ Z, (2001, 2003, 2004, 2010)

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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 ∈ MN complete positivity: [Φ ⊗ ✶K](σ) ≥ 0, ∀σ ∈ MKN and K = 2, 3, ...

The Kraus form

ρ′ = Φ(ρ) =

i AiρA∗ i ,

where the Kraus operators satisfy

• i A∗

i Ai = ✶, which implies that the trace is preserved

allows one to represent the superoperator Φ as a (non-hermitian) matrix of size N2 Φ =

• i

Ai ⊗ ¯ Ai .

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Otton Nikodym & Stefan Banach,

talking at a bench in Planty Garden, Cracow, summer 1916

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Composed bi–partite systems on HA ⊗ HB

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 UAB of size NK

|ψ =

N

• i=1

K

• j=1

Gij|i ⊗ |j

ii) partial trace over the K dimensional subsystem B gives ρA = TrB|ψψ| = GG ∗ and leads to the induced measure PN,K(λ) in the space of mixed states of size N. Integrating out all eigenvalues but λ1

• ne arrives (for large N) at the Marchenko–Pastur distribution

Pc(x = Nλ1) with the parameter c = K/N.

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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. 2

Hermitian, positive matrix ρ = GG ∗ of the Wishart ensemble

Let x = Nλi, where {λi} 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 P1(x) = PMP(x) =

1 2π

• 4

x − 1 for x ∈ [0, 4]

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product of matrices and Fuss-Catalan distribution Ps

defined for an integer number s is characterized by its moments

• xnPs(x)dx =

1 sn+1

sn+n

n

• = FCs(n)

equal to the generalized Fuss-Catalan numbers . The density Ps is analitic on the support [0, (s + 1)s+1/ss], while for x → 0 it behaves as 1/(πxs/(s+1)). Asymptotic distribution of singular values for: a) product G1G2 · · · Gs and b) for s–th power of Ginibre G s, (Alexeev, G¨

• tze, Tikhomirov 2010)

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Fuss-Catalan distributions Ps

The moments of Ps are equal to Fuss-Catalan numbers. Using inverse Mellin transform one can represent Ps by the Meijer G–function, which in this case reduces to s hypergeometric functions

Exact explicit expressions for FC Ps

s = 1, P1(x) =

1 π√x 1F0

• −1

2; ; 1 4x

• =

√

1−x/4 π√x

, Marchenko–Pastur s = 2, P2(x) =

√ 3 2πx2/3 2F1

• −1

6, 1 3; 2 3; 4x 27

• −

√ 3 6πx1/3 2F1

• 1

6, 2 3; 4 3; 4x 27

• =

=

3

√ 2 √ 3 12π

3

√ 2(27+3√81−12x)

2 3 −6 3

√x x

2 3 (27+3√81−12x) 1 3

Fuss–Catalan Arbitrary s, ⇒ ps(x) is a superposition of s hypergeometric functions, Ps(x) = s

j=1 βj sFs−1

• a(j)

1 , . . . , a(j) s ; b(j) 1 , . . . , b(j) s−1; αjx

• .

Penson, K. ˙ Z., Phys. Rev. E 2011.

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Wawel castle in Cracow

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Ciesielski theorem

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Ciesielski theorem: With probability 1 − ǫ the bench Banach talked to Nikodym in 1916 was localized in η-neighbourhood of the red arrow.

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Plate commemorating the discussion between Stefan Banach and Otton Nikodym (Krak´

• w, summer 1916)

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Numerical Range (Field of Values)

Definition

For any operator A acting on HN one defines its NUMERICAL RANGE (Wertevorrat) as a subset of the complex plane defined by:

Λ(A) = {ψ|A|ψ : |ψ ∈ HN, 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 λ1 ≤ λ2 ≤ · · · ≤ λN its numerical range forms an interval: the set of all possible expectation values of the observable A among arbitrary pure states, Λ(A) = [λ1, λN].

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Numerical range and its properties

Compactness

Λ(A) is a compact subset of C containing spectrum, λi(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)

• 0.5

0.5 1.0 Re 0.6 0.4 0.2 0.2 0.4 Im

• 2

1 1 2 Re 2 1 1 2 Im

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Numerical range for matrices of order N = 2.

with spectrum {λ1, λ2} a) normal matrix A ⇒ Λ(A) = closed interval [λ1, λ2] b) not normal matrix A ⇒ Λ(A) = elliptical disk with λ1, λ2 as focal points and minor axis, d =

• TrAA† − |λ1|2 − |λ2|2

(Murnaghan, 1932; Li, 1996). Example : Jordan matrix, J = 1

• .

Its numerical range forms a circular disk, Λ(J) = D(0, r = 1/2).

The set Ω2 = ❈P1 of N = 2 pure quantum states

The set of N = 2 pure quantum states, |ψ ∼ eiα|ψ ∈ H2, normalized as ψ|ψ = 1, forms the Bloch sphere, S2 = ❈P1. A projection of the Bloch sphere onto a plane forms an ellipse, (which could be degenerated to an interval).

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Quantum States and Numerical Range/Shadow

Classical States & normal matrices Proposition 1.Let CN denote the set of classical states (probability distributions) of size N, which forms the regular simplex ∆N−1 in ❘N−1. Then the set of similar images of orthogonal projections of CN on a 2–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 2.Let MN denotes the set of quantum states (hermitian positive matrices, ρ∗ = ρ ≥ 0, of size N normalized as Tr ρ = 1), embedded in ❘N2−1 with respect to Euclidean geometry induced by Hilbert-Schmidt distance. Then the set of similar images of orthogonal projections MN on a 2-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. ˙ Z (2011)

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Shadows of typical matrices of size N = 4

[Classification of numerical ranges for N = 4 is not yet complete...]

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Large N limit and random matrices I

Non–hermitian Ginibre matrices of size N normalized as TrGG ∗ = N

− √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ − √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ − √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ

r ρ

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 √ 2.

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Large N limit and random matrices II

Diagonalized Ginibre matrices of size N normalized as TrGG ∗ = N

− √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ − √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ − √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ

r ρ

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!

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Large N limit and random matrices III

Strictly tri–diagonal random matrices T (with Tij = ξ for i > j)

• f size N normalized as TrTT ∗ = N

− √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ − √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ − √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ

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 √ 2.

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Large N limit and random matrices

Theorem CGLZ, J. Math. Anal. Appl. 418 (2014) 516–533

Let R > 0 and let {Xn}n be a sequence of complex random matrices or

• rder n such that for every θ ∈ ❘ with probability one

lim

n→∞ ||Re

• eiθXn
• || = R

Then with probability one lim

n→∞ dH

• W (Xn), D(0, R)
• = 0.

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 √ 2. Hence the numerical range of G almost surely forms a disk of radius R. Relation to the Dvoretzky theorem !

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Cross-sections of a n–cube

How a generic cross-section of a 3–cube looks like??

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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??

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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, 500000 figures by Guillaume Auburn and Jos Leys Webpage: Quand les cubes deviennent ronds

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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 !

− √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ − √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ − √ 2 −1 1 √ 2 ℜ − √ 2 −1 1 √ 2 ℑ

r ρ

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Bench commemorating the discussion between Otton Nikodym and Stefan Banach (Krak´

• w, summer 1916)

Sculpture: Stefan Dousa

• Fot. Andrzej Kobos
• pened in Planty Garden, Cracow, Oct. 14, 2016

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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.

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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 N2 − 1 dimensional set of quantum states of order N onto a 2–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 = √ 2.

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