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 K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 1 / 28
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 2 , . . . , z N ) ∈ H N , b) bra : � ψ | = ( z 1 , z 2 , . . . , z N ) ∗ ∈ H ∗ N , c) bra-ket � ψ | φ � ∈ ❈ = ( 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 A map Φ can be represented by a matrix of order N 2 . order N . K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 2 / 28
A) Random density matrices Mixed quantum state = density operator which is a) Hermitian , ρ = ρ ∗ , b) positive , ρ ≥ 0, c) normalized , Tr ρ = 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 AA ∗ ρ = Tr AA ∗ forms a random quantum state fixed trace ensembles : (Hilbert–Schmidt, Bures, etc.) urgen Sommers, K. ˙ Hans–J¨ Z , (2001, 2003, 2004, 2010) K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 3 / 28
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 ⊗ ✶ K ]( σ ) ≥ 0, ∀ σ ∈ M KN and K = 2 , 3 , ... The Kraus form ρ ′ = Φ( ρ ) = � i A i ρ A ∗ i , where the Kraus operators satisfy i A ∗ � i A i = ✶ , which implies that the trace is preserved allows one to represent the superoperator Φ as a ( non-hermitian ) matrix of size N 2 A i ⊗ ¯ � Φ = A i K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices i May 18, 2017 4 / 28 .
K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 5 / 28
Otton Nikodym & Stefan Banach , talking at a bench in Planty Garden, Cracow, summer 1916 K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 5 / 28
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 N K � � | ψ � = 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 = N λ 1 ) with the parameter c = K / N . K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 6 / 28
Spectral properties of random matrices Non-hermitian matrix G of size N of the Ginibre ensemble Under normalization Tr GG ∗ = 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 = N λ i , where { λ i } denotes the spectrum of ρ . As Tr ρ = 1 so � x � = 1. Distribution of the spectrum P ( x ) is asymptotically given by the Marchenko–Pastur law � 1 4 P 1 ( x ) = P MP ( x ) = x − 1 for x ∈ [0 , 4] 2 π K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 7 / 28
product of matrices and Fuss-Catalan distribution P s defined for an integer number s is characterized by its moments 1 � sn + n � x n P s ( x ) dx = � = FC s ( 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 / ( π x s / ( s +1) ). Asymptotic distribution of singular values for: a) product G 1 G 2 · · · G s and b) for s –th power of Ginibre G s , ( Alexeev, G¨ otze, Tikhomirov 2010) K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 8 / 28
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 √ � � 1 − x / 4 1 − 1 2 ; ; 1 s = 1, P 1 ( x ) = π √ x 1 F 0 4 x = , Marchenko–Pastur π √ x √ √ � � � � 3 − 1 6 , 1 3 ; 2 3 ; 4 x 3 1 6 , 2 3 ; 4 3 ; 4 x s = 2, P 2 ( x ) = 2 π x 2 / 3 2 F 1 − 6 π x 1 / 3 2 F 1 = 27 27 √ 2 ( 27+3 √ 81 − 12 x ) 2 √ x √ √ 3 3 − 6 3 3 2 3 = Fuss–Catalan 3 ( 27+3 √ 81 − 12 x ) 1 12 π 2 3 x Arbitrary s , ⇒ p s ( x ) is a superposition of s hypergeometric functions, a ( j ) 1 , . . . , a ( j ) s ; b ( j ) 1 , . . . , b ( j ) P s ( x ) = � s � � j =1 β j s F s − 1 s − 1 ; α j x . Penson, K. ˙ Z. , Phys. Rev. E 2011. K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 9 / 28
Wawel castle in Cracow K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 10 / 28
Ciesielski theorem K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 11 / 28
Ciesielski theorem: With probability 1 − ǫ the bench Banach talked to Nikodym in 1916 was localized in η -neighbourhood of the red arrow . K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 12 / 28
Plate commemorating the discussion between Stefan Banach and Otton Nikodym ( Krak´ ow, summer 1916 ) K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 13 / 28
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 2 ≤ · · · ≤ 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 ]. K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 14 / 28
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) Im Im � 0.4 2 � � � 0.2 1 � Re � 0.5 0.5 1.0 Re � 2 � 1 1 2 � 0.2 � � � � 0.4 � � 1 � � � 0.6 � 2 � K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 15 / 28
Numerical range for matrices of order N = 2 . with spectrum { λ 1 , λ 2 } a) normal matrix A ⇒ Λ( A ) = closed interval [ λ 1 , λ 2 ] b) not normal matrix A ⇒ Λ( A ) = elliptical disk with λ 1 , λ 2 as focal Tr AA † − | λ 1 | 2 − | λ 2 | 2 � points and minor axis, d = ( Murnaghan, 1932; Li, 1996 ). � 0 1 � Example : Jordan matrix , J = . 0 0 Its numerical range forms a circular disk, Λ( J ) = D (0 , r = 1 / 2). The set Ω 2 = ❈ P 1 of N = 2 pure quantum states The set of N = 2 pure quantum states, | ψ � ∼ e i α | ψ � ∈ H 2 , normalized as � ψ | ψ � = 1, forms the Bloch sphere , S 2 = ❈ P 1 . A projection of the Bloch sphere onto a plane forms an ellipse , (which could be degenerated to an interval). K ˙ Z (IF UJ/CFT PAN ) Numerical range of random matrices May 18, 2017 16 / 28
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