Systems with off–diagonal disorder on a lattice Karol ˙ Zyczkowski in collaboration with Tomasz Tkocz, Marek Ku´ s (Warsaw) Marek Smaczy´ nski, Wojciech Roga (Cracow) Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw Quantum Chaos , Luchon , March 17, 2015 K ˙ Z (IF UJ/CFT PAN ) Off–diagonal disorder March 17, 2015 1 / 28
Some spectral properties of quantum systems Karol ˙ Zyczkowski in collaboration with Tomasz Tkocz, Marek Ku´ s (Warsaw) Marek Smaczy´ nski, Wojciech Roga (Cracow) Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw Quantum Chaos , Luchon , March 17, 2015 K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 2 / 28
Random matrices: applications in quantum & classical physics A) Quantum Chaos and Unitary Dynamics : ’Quantum chaology’ Quantum analogues of classically chaotic dynamical systems can be described by random matrices a) autonomous systems – Hamiltonians: Gaussian ensembles of random Hermitian matrices, (GOE, GUE, GSE) b) periodic systems – evolution operators: Dyson circular ensembles of random unitary matrices, (COE, CUE, CSE) K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 3 / 28
Random Matrices & Universality Universality classes Depending on the symmetry properties of the system one uses ensembles form orthogonal ( β = 1); unitary ( β = 2) and symplectic ( β = 4) ensembles. The exponent β determines the level repulsion, P ( s ) ∼ s β for s → 0 where s stands for the (normalised) level spacing, s i = φ i +1 − φ i . see e.g. F. Haake, Quantum Signatures of Chaos K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 4 / 28
Wigner Semicircle Law Spectral density P ( x ) for random hermitian matrices can be obtained by integrating out all eigenvalues but one from jpd. For all three Gaussian ensembles of Hermitian random matrices one obtains (asymptotically, for N → ∞ ) the Wigner Semicircle Law (1955) P ( x ) = 1 � 2 − x 2 2 π √ where x denotes a normalized eigenvalue , x i = λ i / N K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 5 / 28
Extremal eigenvalues & Tracy–Widom Law Statistics of extremal cases - the largest eigenvalue x max The normalized largest eigenvalue (” s ” of Tracy–Widom) √ N ) N − 1 / 6 s := ( x max − 2 of a GUE random matrix is (asymptotically) distributed according to the Tracy-Widom law (1994) F 2 ( s ) = det (1 − K ) , where K is the integral operator with the Airy kernel K ( x , y ) = Ai ( x ) Ai ′ ( y ) − Ai ′ ( x ) Ai ( y ) . x − y The scaling behaviour of the finite size effect (as N − 1 / 6 ) is due to Bowick & Brezin (1991) and Forrester (1991). K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 6 / 28
Tracy–Widom distributions Tracy–Widom distributions F β ( s ) Distributions F β ( s ) and the largest eigenvalue of random GUE matrices (image by A. Edelman) K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 7 / 28
Level spacing distribution P ( s ) Nearest neighbour spacing s s i = x i +1 − x i (” s ” of Wigner), where ∆ is the mean spacing ∆ a) Gaussian ensembles for N = 2 ⇒ Wigner surmise β = 1 GOE (orthogonal) P 1 ( s ) = π 2 s exp( − π 4 s 2 ) π 2 s 2 exp( − 4 β = 2 GUE (unitary) P 2 ( s ) = 32 π s 2 ) 3 6 π 3 s 4 exp( − 64 2 18 9 π s 2 ) β = 4 GSE (symplectic) P 4 ( s ) = These distributions derived for N = 2 work well also for Gaussian ensembles in the asymptotic case, N → ∞ . Random unitary matrices & Circular ensembles of Dyson Uniform density of phases along the unit circle, P ( φ ) = 1 / 2 π . Phase spacing , s i = N 2 π [ φ i +1 − φ i ] since ∆ = 2 π/ N . For large matrices the level spacing distributions for Gaussian ensembles (Hermitian matrices) and circular ensembles (unitary matrices) coincide. K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 8 / 28
K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 9 / 28
Extremal spacings for random unitary matrices . Consider a) Minimal spacing s min = min j { s j } N ( how close to degeneracy? ) j =1 b) Maximal spacing s max = max j { s j } N and j =1 Minimal spacing distribution for N = 4 random unitary matrices Two qubits & random local gates Analytical results P 2 ⊗ 2 ( t ) for CUE(2) ⊗ CUE (2) case, where t = s min P 2 ⊗ 2 ( t ) = 1 4 − cos( π t − 3 sin( π t 2 )+8 sin( π t ) − 3 sin(3 π t � � � � 2 π (1 − t ) 2 ) 2 ) 4 π CUE(4), P (2) CUE , β = 2, 4 ( t ) = ... explicit result to long to reproduce it here... P (0) 4 ( t ) = 3(1 − t ) 2 . Poisson ensemble , β = 0, CPE(4), K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 10 / 28
Minimal spacing P ( s min ) for N = 4 unitary matrices Comparison of spacing distribution P ( s min ) for a) Poisson CPE(4), b) CUE (2) ⊗ CUE (2), c) CUE(4). 3 P CPE 4 CUE 2 ⊗ 2 2 CUE 4 1 0 0 0.5 1 S min mean values: � s min � CPE 4 = 1 / 4, � s min � CUE 2 ⊗ CUE 2 ≈ 0 . 4, � s min � CUE 4 ≈ 0 . 54 s, ˙ Smaczy´ nski, Tkocz, Ku´ Zyczkowski Phys. Rev. E (2013) K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 11 / 28
Minimal spacing P ( s min ) for large unitary matrices (here N = 100) Minimal spacing distribution P ( s min ) for 0) Poisson CPE( N ), P 0 ( s min ) = A 0 Ne − Ns min 1 Ns min e − A 2 1 Ns 2 P 1 ( s min ) = 2 A 2 1) COE ( N ), min min e − A 3 2 Ns 3 P 2 ( s min ) = 3 A 3 2 Ns 2 2) CUE( N ), min . K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 12 / 28
Average minimal spacing � s min � for large unitary matrices Approximation of independent spacings Assume spacings s j described by the distribution P β ( s ) are independent. Minimal spacing Since for small spacings P β ( s ) ∼ s β so the integrated distribution � s 0 P ( s ′ ) ds ′ behaves as I β ( s ) ∼ s 1+ β I ( s ) = Matrix of order N yields N spacings s j . The minimal spacing s min occurs for such an argument that I β ( s min ) ≈ 1 / N . 1 Thus ( s min ) 1+ β ≈ 1 / N = ⇒ s min ≈ N − β +1 K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 13 / 28
Average maximal spacing � s max � Approximation of independent spacings Assume spacings s j described by the distribution P β ( s ) are independent. Mean maximal spacing for COE Since for large spacings P β ( s ) ∼ s 1 exp( − s 2 ) so the integrated distribution � s 0 P ( s ′ ) ds ′ behaves as I 1 ( s ) ∼ − exp( − s 2 ) I 1 ( s ) = Matrix of order N yields N spacings s j . The maximal spacing s max occurs for such an argument that 1 − I 1 ( s max ) ≈ 1 / N . √ Thus exp[ − ( s max ) 2 ] ≈ 1 / N = ⇒ s max ≈ ln N s, ˙ Smaczy´ nski, Tkocz, Ku´ Zyczkowski Phys. Rev. E (2013) Some of these results ( and some other ) appeared in a preprint arXiv:1010.1294 ”Extreme gaps between eigenvalues of random matrices” by Ben Arous and Bourgade . K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 14 / 28
K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 15 / 28
Classical probabilistic dynamics & Markov chains Stochastic matrices Classical states : N -point probability distribution, p = { p 1 , . . . p N } , where p i ≥ 0 and � N i =1 p i = 1 Discrete dynamics : p ′ i = S ij p j , where S is a stochastic matrix of size N and maps the simplex of classical states into itself, S : ∆ N − 1 → ∆ N − 1 . Frobenius–Perron theorem Let S be a stochastic matrix : a) S ij ≥ 0 for i , j = 1 , . . . , N , b) � N i =1 S ij = 1 for all j = 1 , . . . , N . Then i) the spectrum { z i } N i =1 of S belongs to the unit disk, ii) the leading eigenvalue equals unity, z 1 = 1, iii) the corresponding eigenstate p inv is invariant, S p inv = p inv . K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 16 / 28
B) Quantum Chaos & Nonunitary Dynamics Quantum operation: linear, completely positive trace preserving map positivity : Φ( ρ ) ≥ 0, ∀ ρ ∈ M N 1 K ]( σ ) ≥ 0, complete positivity : [Φ ⊗ ∀ σ ∈ M KN and K = 2 , 3 , ... Enviromental form (interacting quantum system !) ρ ′ = Φ( ρ ) = Tr E [ U ( ρ ⊗ ω E ) U † ] . where ω E is an initial state of the environment while UU † = 1 . Kraus form ρ ′ = Φ( ρ ) = � i A i ρ A † i , where the Kraus operators satisfy i A † � 1 , which implies that the trace is preserved. i A i = K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 17 / 28
Quantum stochastic maps (trace preserving, CP) Superoperator Φ : M N → M N A quantum operation can be described by a matrix Φ of size N 2 , ρ ′ = Φ ρ ′ or ρ m µ = Φ m µ n ν ρ n ν . The superoperator Φ can be expressed in terms of the Kraus operators A i , i A i ⊗ ¯ Φ = � A i . Dynamical Matrix D : Sudarshan et al. (1961) obtained by reshuffling of a 4–index matrix Φ is Hermitian, Φ =: Φ R . D Φ = D † D mn µν := Φ m µ n ν , so that Theorem of Choi (1975). A map Φ is completely positive (CP) if and only if the dynamical matrix D Φ is positive , D ≥ 0. K ˙ Z (IF UJ/CFT PAN ) Spectral properties March 17, 2015 18 / 28
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