Induced Ginibre Ensemble and random operations

Transcription

1 Induced Ginibre Ensemble and random operations Karol Życzkowski in collaboration with Wojciech Bruzda, Marek Smaczyński, Wojciech Roga Jonith Fischmann, Boris Khoruzhenko (London), Ion Nechita (Touluse), Benoit Collins (Ottawa), Hans-Juergen Sommers (Duisburg), Karol Penson (Paris) Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw Random Matrices, Bielefeld, December 16, 2011 KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

2 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 KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

3 How random mixed states appear in quantum physics? Reduction of random pure states 1) Consider an ensemble of random pure states ψ of a composite system distributed according to a given measure µ. 2) Perform partial trace over a chosen subsystem B to get a random mixed state ρ : = Tr B ψ ψ Depending on the structure of the composite system, the initial measure µ in the space of the pure states and the choice of the subsystem B, over which the averaging is performed one obtaines different ensembles of random mixed states. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

4 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 ) Composition of quantum operations December 16, / 34 Composed bi partite systems on H A H B Ensembles obtained by partial trace: a) induced measure 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

5 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! 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 π (1) (x) = P MP (x) = 1 2π 4 x 1 for x [0, 4] KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

6 Bures ensemble i) Consider a superposition of two pure states: a random state ψ 1 and the same state transformed by a local unitary V A, φ := (½ ½ + V A ½) ψ 1, where ψ 1 = U AB 0, 0 while V A U(N) and U AB U(N 2 ) are Haar random unitary matrices. ii) The reduced state ρ B = (½+V A )GG (½+V A ) Tr[(½+V A )GG (½+V A )] is distributed according i λ 1/2 i 1...N i<j (λ i λ j ) 2 λ i +λ j to the Bures measure, P B (λ 1,...λ N ) = CN B (Osipov, Sommers, Życzkowski, 2010) characterized by the Bures distribution, [ ( ) P B (x) = 1 4π a ( 3 x + a ) 2/3 ( ) ] 2 x 1 a ( x a ) 2/3 2 x 1 where a = 3 3. Square matrix G of size N from the Ginibre ensemble is obtained from the first column of U AB od size N 2 which acts on 0, 0. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

7 Let Generalized ensemble of random states W k,s := (U 1 + U U k ) G 1 G s where U i are independent Haar random unitary matrices, while G i are independent random Ginibre matrices. Define generalized ensemble of normalized random density matrices ρ k,s := W k,s W k,s /Tr(W k,sw k,s ) Special cases: s = 0, k = 1 s = 0, k = 2 s = 0, k = k s = 1, k = 1 s = 1, k = 2 s = s, k = 1 maximally mixed state arcsine ensemble k Kesten ensemble Hilbert-Schmidt ensemble Bures ensemble s Fuss Catalan ensemble KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

8 K.Ż., K. Penson, I. Nechita and B. Collins, J. Math. Phys. (2011) KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34 Spectral properties of ensembles analyzed Spectral density P(x) of the rescaled eigenvalue x = Nλ matrix W P(x) x 0 support mean entropy ½ π (0) {1} 0 ½ + U arcsine x 1/2 [0, 2] ln G M-P π (1) x 1/2 [0, 4] 1/2 = 0.5 (½ + U)G Bures x 2/3 [0, 3 3] ln G 1 G 2 F C π (2) x 2/3 [0, ] 5/ G 1 G s F C π (s) x s/(s+1) [0, b s ] s+1 j=2 1 j Table: Ensembles of random mixed states obtained as normalized Wishart matrices, ρ = WW /TrWW. Here b s = (s + 1) s+1 /s s and the mean entropy S = x ln xp(x)dx.

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10 B) Quantum Chaos & 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 ½ K ](σ) 0, σ M KN and K = 2, 3,... The Kraus form ρ = Φ(ρ) = i A i ρa i, where the Kraus operators satisfy i A i A i = ½, which implies that the trace is preserved allows one to represent the superoperator Φ as a matrix of size N 2 Φ = i A i Ā i. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

11 Interacting quantum dynamical systems Generalized quantum baker map with measurements a) Quantisation [ of Balazs and ] Voros applied for the asymmetric map B = F FN/K 0 N, where N/K Æ. 0 F N(K 1)/K where F N denotes the Fourier matrix of size N. Then ρ = Bρ i B b) M measurement operators projecting into orthogonal subspaces Kraus form: ρ i+1 = M i=1 P iρ P i c) vertical shift by /2 ( Loziński, Pakoński, Życzkowski 2004) Standard classical model K = 2, dynamical entropy H = ln2; Asymmetric model, K > 2, entropy decreases to zero as K. Classical limit: N with K N. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

12 Exemplary spectra of superoperator for L fold generalized baker map B L & measurement with M Kraus operators for N = 64 and M = 2: 1 a) 1 b) 1 c) K = 64, L = K = 4, L = K = L = 32 a) weak chaos (K = 64 and L = 1), b) strong chaos (K = 4 and L = 4) universal behaviour: λ 1 = 1 and uniform Girko disk of eigenvalues of radius R, (described by real Ginibre ensemble). c) weak chaos (K = 32 and L = 32). KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

13 Conjecture on spectral properties of superoperators describing non unitary dynamics In the case of strong chaos and large interaction with the environment the superoperators display universal behaviour and can be described by random matrices pertaining to real Ginibre ensemble. Real ensemble is used since the map preserves hermicity of density matrix ρ and superoperator Φ can be represented by a real matrix. For instance, in the Bloch vector τ representation any state of size N is given by ρ = ½/N + τ λ and the map ρ = Φ(ρ) reads τ = Φτ where Φ is a real matrix of size N 2 1 and τ a generalized Bloch vector of length N 2 1, while λ is a vector of generators of SU(N) with N 2 1 components. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

14 Complementary quantum maps For any quantum operation ρ = Φ(ρ) = Tr E (Uρ φ φ U ) Φ(ρ) = M i=1 K iρk i = M i=1 q iρ i M i=1 K i K i = ½ N, φ H M. one defines a complementary map Φ : H N H M Complementary channel Φ sends initial state ρ into the correlation matrix σ (state on the M-dimensional environment E) Φ(ρ) = Tr S (Uρ φ φ U ) = σ σ ij = TrK i ρk j, i, j = 1,...,M ( matrix σ ij used by G. Lindblad for his inequality in 1991) KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

15 Complementary maps and rectangular superoperators Superoperator A associated with a complementary map Φ is represented by a rectangular matrix of size M 2 N 2. Squared singular values of A (equal to eigenvalues of A A) can be described by Marchenko Pastur distribution with parameter c = M 2 /N 2. Can we diagonalize a rectangular matrix A of size M = M 2 times N = N 2?? (in the following we shall omit the primes and consider rectangular matrices of size M N.) KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

16 Quadratisation of a rectangular matrix Consider a rectangular matrix X of size M N, with M rows and N columns. Assume that M > N, so that the matrix is standing and can be splited into a square upper part Y of size M and a lower part Z or size (M N) N. We introduce a quadratisation of X defined as a square matrix G of size N such that: [ ] [ ] Y G W X = W = Z 0 Simple algebra leads to an explicit solution G = ( ½ N + 1 Y Z Z 1 ) 1/2 Y Y KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

17 Quadratisation of random rectangular matrices Let X be a complex random matrix of size M N with independent Gaussian entries of the same variance. Assume M > N and split the standing matrix X into the square upper part Y and lower rectangular Z. Then a) the matrix G = ( ½ N + 1 Z Z 1 ) 1/2 Y Y Y is distributed according to the induced Ginibre ensemble : P (2) (2) ( IndG (G) = C N,M det G G ) M N ( exp Tr G G ) b) the same distribution describes random square matrices of the form G = U X X, where U is a Haar random unitary matrix (of CUE). Fischmann, Bruzda, Khoruzhenko, Sommers, Życzkowski 2011 where explicit form for the normalisation constant C (2) N,M is derived. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

18 Complex induced Ginibre ensemble with L = M N Joint probability distribution for (complex) eigenvalues λ i : P(λ 1,...,λ N ) = 1 N!π N N j=1 1 Γ(j+L) j<k λ k λ j 2 ( N j=1 λ j 2L exp ) N j=1 λ j 2, Spectral density P(λ) [ ] P(λ) = 1 γ(l, λ 2 ) π Γ(L) γ(l+n, λ 2 ) Γ(L+N), where γ(a, z) := z 0 ta 1 e t dt is the lower incomplete gamma function. N = 64 and a) L = M N = 0, b) L = 8 and c) L = 32. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

19 Real induced Ginibre ensemble with L = M N Joint probability distribution for (complex) eigenvalues λ i : P(λ 1,...,λ N ) =... (explicit formula too long to reproduce it here...) Spectral density: a) complex density P C (x + iy) [ ] P C 2 (x + iy) = π y erfc( 2y)e 2y2 γ(l,x 2 +y 2 ) Γ(L) γ(l+n 1,x2 +y 2 ) Γ(L+N 1), b) real density P R (x) [ ] P R (x) = 1 γ(l,x 2 ) 2π Γ(L) γ(l+n 1,x2 ) Γ(N+L 1) + Jon(x) (for the last term Jon(x) consult the thesis of Jonith...) N = 64 and a) L = M N = 0, b) L = 8 and c) L = 32. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

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21 s steps propagators for perturbed baker map Φ s B Exemplary spectra of superoperator Φ s B for s steps non-unitary evolution i) spectral properties of 1 step propagator Φ coincide with these of real random Ginibre matrices (uniform disk apart of the real axis) ii) properties of s step propagators Φ s are similar as products of random matrices: KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

22 a) the radial density of complex eigenvalues r = z of Φ s B of the periodically measured baker map displays asymptotically an algebraic power law for products of s random Ginibre matrices of Burda, Jarosz, Livan, Nowak, Świȩch, 2010: P s (r) r 1+2/s with an error-function Ansatz (red line) describing the finite N effects. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

23 b) the squared singular values of Φ s B for non-unitary baker map can be described by Fuss-Catalan distribution of order t = s 1. Let x = N 2 λ, where λ is an eigenvalue of Φ s (Φ s ). Then s = 2, t = 1 (Wishart) 1 x/4 P 1 (x) = π x [0, 4], x Marchenko Pastur distrib. (with moments given by the Catalan numbers); s 3, t 2, the Fuss Catalan distrib. P t (x) for x [0, (t + 1) t+1 /t t ] (with moments given by the Fuss Catalan numbers) expicitely derived in Penson, Życzkowski, 2011 KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

24 Fuss-Catalan numbers FC s (n) Generalized Fuss-Catalan numbers ( are defined for any integer s, FC s (n) := 1 sn+n ) sn+1 n some examples of FC numbers: where n = 0,...,7. FC 1 (n) = 1, 1, 2, 5, 14, 42, 132, 427,... FC 2 (n) = 1, 1, 3, 12, 55, 273, 1428, 7752,... FC 3 (n) = 1, 1, 4, 22, 140, 969, 7084, 53820,... FC 4 (n) = 1, 1, 5, 35, 285, 2530, 23751, ,.... The family FC 1 (n) coincides with the standard Catalan numbers, and gives the moments of the Marchenko Pastur distribution. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

25 Fuss-Catalan distribution π (s) defined for an integer number s is( characterized by its moments x n π (s) (x)dx = 1 sn+n ) sn+1 n = FCs (n) equal to the generalized Fuss-Catalan numbers. The density π (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) ). The same moments decribe (asymptotically) distribution of singular values for s th power of Ginibre G s, (Alexeev, Götze, Tikhomirov 2010) KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

26 Fuss-Catalan distributions π (s) The moments of π (s) are equal to Fuss-Catalan numbers. Using inverse Mellin transform one can represent π (s) by the Meijer G function, which in this case reduces to s hypergeometric functions Exact explicit expressions for FC π (s) ( ) s = 1, π (1) (x) = 1 π x 1 F ; ; 1 4 x = s = 2, π (2) ( (x) = 3 2πx 2/3 2 F 1 1 6, 1 3 ; 2 3 ; 4x 27 = π 1 x/4 π x ) 3 2( x) x x 2 3 ( x) 1 3, Marchenko Pastur ( ) 3 6πx 1/3 2 F 1 1 6, 2 3 ; 4 3 ; 4x 27 = Fuss Catalan Arbitrary s, π (s) (x) is a superposition of s hypergeometric functions, π (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, Życzkowski, Phys. Rev. E KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

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28 Raney numbers ) Generalisation of FC( numbers: the Raney numbers R p,r (n) := r pn+r pn+r n defined for n = 0, 1,... and related in combinatorics to Raney sequences. In a particular case R s+1,1 (n) are equal to FC s (n). By definition the following property holds R p,p (n) = R p,1 (n + 1). examples of Raney numbers: R 4,2 (n) = 1, 2, 9, 52, 340, 2394, 17710, ,...(A069271) R 4,5 (n) = 1, 5, 30, 200, 1425, 10626, 81900, ,..., R 5,2 (n) = 1, 2, 11, 80, 665, 5980, 56637, ,...(A118969), R 6,3 (n) = 1, 3, 21, 190, 1950, 21576, , ,.... where n = 0,...,7. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

29 Raney distributions Result of M lotkowski (2011): If a point (r, p) satisfies inequalities p 0 and 0 < r p the Raney numbers R p,r (n) describe the moments of a measure µ p,r with a compact support. Point (1, 1) implies constant moments, R 1,1 (n) = 1, which represents a singular, Dirac delta measure, µ 1,1 = δ(x 1). If a measure µ p,r is represented by a density we will denote it by W p,r (x). ( x n W p,r (x)dx = R p,r (n) = r pn+r ) pn+r n Then W s+1,1 (x) reduces to the Fuss Catalan probability density π s (x). KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

30 Raney distributions II Set Σ of positive measures in the parameter space (p, r) M lotkowski (2010) Example: Raney distribution W 2,2 (x) gives the semicircle law centered at x = 2 with support x [0, 4] W 2,1 (x) is Marchenko-Pastur while W 3,1 (x) is not positive! KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

31 Raney distributions III exemplary Raney distributions W p,r (x) Raney distribution W 3,r (x) are positive for r = 1, 2, 3 and supported in x [0, ] Raney distribution W 4,r (x) are positive for r = 1, 2, 3, 4. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

32 Raney distributions IV Some properties: For integer values of p, q the function W p,r (x) is supported in the interval [0, p p /(p 1) p 1 ] The distribution W p,r (x) is non-negative if 1 r p For p > r the distribution W p,r (x) diplays for small x a singularity of the type x (p r)/p. For p = r the diagonal Raney distributions behave for x << 1 as W p,p (x) x 1/p. The folowing relation holds (M lotkowski 2010) W p,p (x) = x W p,1 (x) = x π p 1 (x) Exemplary formula for an intermediate Raney distribution [ 4 ( x) x 3] 2 W 3,2 (x) = π x 1 3 ( x) 2 3 KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34

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34 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. Other ensembles of random pure states + partial trace generate states with Arcsine, k Kesten, Bures, s Fuss-Catalan distrib. Real Ginibre matrices are applicable to describe superoperators associated with quantum maps under the condition of strong chaos and large interaction with the environment Complementary maps are described by rectangular matrices. Their quadratisations are described by matrices of induced real Ginibre ensemble for which densities and correlation functions are derived. s step propagators correspond to products of Ginibre matrices, the singular values of which are described by are Complementary maps are described by Fuss Catalan distributions. Explicit and exact analytical forms of the Fuss-Catalan and Raney distributions expressed as combinations of hypergeometric functions. KŻ (IF UJ/CFT PAN ) Composition of quantum operations December 16, / 34