Quantum Chaos and Nonunitary Dynamics

Transcription

1 Quantum Chaos and Nonunitary Dynamics Karol Życzkowski in collaboration with W. Bruzda, V. Cappellini, H.-J. Sommers, M. Smaczyński Phys. Lett. A 373, 320 (2009) Institute of Physics, Jagiellonian University, Cracow, Poland and Center for Theoretical Physics, Polish Academy of Sciences Krasnoyarsk, July 9, 2009 KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

2 Closed systems: Unitary dynamics Pure states in a finite dimensional Hilbert space H N N = 2; ψ = cos ϑ eiφ sin ϑ 2 0 Bloch sphere of N = 2 pure states Space of pure states for an arbitrary N: Unitary dynamics = rotation of the sphere! a complex projective space P N 1 of 2(N 1) real dimensions. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

3 Unitary evolution Fubini-Study distance in P N 1 D FS ( ψ, ϕ ) := arccos ψ ϕ Unitary evolution Let U = exp(iht). Then ψ = U ψ. Since ψ ϕ 2 = ψ U U ϕ 2 any unitary evolution is a rotation in P N 1 hence it is an isometry (with respect to any standard distance!) Classical limit: what happends for large N? How an isometry may lead to classically chaotic dynamics? The limits t and N do not commute. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

4 Closed systems, Unitary Dynamics & Quantum Chaos Quantum chaology : analogues of classically chaotic systems 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) 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 KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

5 Open Systems & Mixed Quantum States Set M N of all mixed states of size N M N := {ρ : H N H N ; ρ = ρ, ρ 0, Trρ = 1} example: M 2 = B 3 Ê 3 - Bloch ball with all pure states at the boundary The set M N is compact and convex: ρ = i a i ψ i ψ i where a i 0 and i a i = 1. It has N 2 1 real dimensions, M N Ê N2 1. How the set of all N = 3 mixed states looks like? An 8 dimensional convex set with only 4 dimensional subset of pure (extremal) states, which belong to its 7 dim boundary KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

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12 If you like to know more please have a look at I. Bengtsson and K. Życzkowski, Geometry of Quantum States (Cambridge, 2006) KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

13 Quantum maps Quantum operation: linear, completely positive trace preserving map positivity: Φ(ρ) 0, ρ M N complete positivity: [Φ ½ K ](σ) 0, 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 = ½. Kraus form ( measurement operators A i ) 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, Trρ =Trρ = 1. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

14 Quantum stochastic maps (trace preserving, CP maps) Superoperator Φ : M N M N A quantum operation can be described by a matrix Φ of size N 2, ρ = Φρ or ρ mµ = Φmµ nν ρ nν. The superoperator Φ can be expressed in terms of the Kraus operators A i, Φ = i A i Ā i. Dynamical Matrix D: Sudarshan et al. (1961) obtained by reshuffling of a 4 index matrix Φ is Hermitian, Dmn µν := Φmµ nν, so that D Φ = D Φ =: ΦR. Theorem of Choi (1975). A map Φ is completely positive (CP) if and only if the dynamical matrix D is positive, D 0. D acts on H A H B and trace preserving condition reads: Tr A D = ½. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

15 Spectral properties of a superoperator Φ Quantum analogue of the Frobenious-Perron theorem Let Φ represent a stochastic quantum map, i.e. a) Φ R = D Φ 0; (Choi theorem) b) Tr A D Φ = ½ k Φ kk = δ ij. (trace preserving condition) ij Then i) the spectrum {z i } N2 i=1 of Φ belongs to the unit disk, ii) the leading eigenvalue equals unity, z 1 = 1, iii) the corresponding eigenstate (with N 2 components) forms a matrix ω of size N, which is positive, ω 0, normalized, Trω = 1, and is invariant under the action of the map, Φ(ω) = ω. Classical case In the case of a diagonal dynamical matrix, D ij = d i δ ij reshaping its diagonal {d i } of length N 2 one obtains a matrix of size N, where S ij = D ii, of size N which is stochastic and recovers the standard F P theorem. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24 jj

16 Exemplary spectra of (typical) superoperators 1 Im(z) a) N = 2 1 Im(z) b) N = Re(z) Re(z) Spectra of several random superoperators Φ for a) N = 2 and b) N = 3 contain: i) the leading eigenvalue z 1 = 1 corresponding to the invariant state ω, ii) real eigenvalues, iii) complex eigenvalues inside the disk of radius r = z 2 1. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

17 Random quantum states How to generate a mixed quantum state at random? 1) Fix M 1 and take a N M random complex Ginibre matrix X (all elements are independent random complex Gaussian variables) 2) Write down the positive matrix Y := XX, 3) Renormalize it to get a random state ρ, ρ := Y TrY. This matrix is positive, ρ 0 and normalised, Trρ = 1, so it represents a quantum state! Special case of M = N (square Ginibre matrices) Then random states are distributed uniformly with respect to the Hilbert-Schmidt (flat) measure, e.g. for M = N = 2 random mixed states cover uniformly the interior of the Bloch ball. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

18 Random quantum operation (stochastic maps) Interaction with M dim. enviroment 1) Fix the size of ancilla M 1, 2) Chose a random unitary matrix U according to the Haar measure on U(NM), 3) Construct a random map defining ρ = Tr M [U(ρ ν ν )U ], where ν H M is an arbitrary (fixed) state of the environment. Probability distribution for random maps P(D) det(d M N2 ) δ(tr A D ½), In the special case M = N 2 the first factor is equal to unity, so there are no other constraints on the distribution of D, besides the partial trace condition, Tr A D = ½. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

19 Bloch vector representation of any state ρ ρ = N 2 1 i=0 τ i λ i where λ i are generators of SU(N) such that tr ( λ i λ j) = δ ij and λ 0 = ½/ N and τ i are expansion coefficients. Since ρ = ρ, the generalized Bloch vector τ = [τ 0,...,τ N 2 1] is real. Stochastic map Φ in the Bloch representation The action of linear the map Φ can be represented as τ = Φ(τ) = Cτ + κ, where C is a real, asymmetric contraction [ ] matrix of size N 2 1 while κ is 1 0 a translation vector. Thus Φ = and the eigenvalues of C are κ C also eigenvalues of Φ. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

20 Random maps & random matrices For large N the measure for C can be described by the real Ginibre ensemble of non-hermitian Gaussian matrices. Spectral density in the unit disk The spectrum of Φ consists of: i) the leading eigenvalue z 1 = 1, ii) the component at the real axis, the distribution of which is asymptotically given by the step function P(x) = 1 2Θ(x 1)Θ(1 x), iii) complex eigenvalues, which cover the disk of radius r = z 2 1 uniformly according to the Girko distribution. Subleading eigenvalue r = z 2 The radius r of the disk is determined by the size of the ancilla, r 1/ M, so the spectrum of the rescaled matrix Φ := MΦ covers the entire unit disk. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

21 Spectral density for random stochastic maps Numerical results for the full rank case, M = N 2 a) Distribution of complex eigenvalues of 10 4 rescaled random operators Φ = NΦ already for N = 10 can be approximated by the circle law of Girko. b) Distribution of real eigenvalues P(x) of Φ plotted for N = 2, 3, 7 and 14 tends to the step function! KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

22 Convergence to equilibrium and decoherence rate Average trace distance to invariant state ω = Φ(ω) L(t) = Tr Φ t (ρ 0 ) ω ψ, where the average is performed over an ensemble of initially pure random states, ρ 0 = ψ ψ. Numerical result confirm an exponential convergence, L(t) exp( αt). a) Average trace distance of random pure states to the invariant state of Φ as a function of time for N = 4( ), 6( ), 8( ). b) mean convergence rate α Φ scales as ln N with the system size N. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

23 Comparison with a quantum dynamical system Generalized quantum baker map with measurements [ a) Standard quantisation of Balazs and Voros B = F FN/2 0 N 0 F N/2 where F N denotes the Fourier matrix of size N. Then ρ = Bρ i B b) M measurement operators projecting into orthogonal subspaces ρ i+1 = M i=1 P iρ P i ], 1 1 Im λ 0 Im λ Re λ Re λ Numerical spectra of superoperator of baker map for N = 64, a) M = 2, r a 1/ 2, b) M = 8 r b 1/ 8, (M. Smaczyński, Master Thesis 09) KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24

24 Concluding Remarks Quantum Chaos: a) in case of closed systems one studies unitary evolution operators and characterizes their spectral properties, b) for open, interacting systems one analyzes non unitary time evolution described by quantum stochastic maps. We analyzed spectral properties of quantum stochastic maps and formulated a quantum analogue of the Frobenius-Perron theorem. A natural flat measure in the space of quantum operations (stochastic maps) is defined and an algorithm to produce them at random is given. For large N random quantum operations can be described by random matrices of the real Ginibre ensemble. Sequential action of a fixed random map brings all pure states to the invariant state exponentially fast. The convergence rate scales logarithmically with the system size N. KŻ (IF UJ/CFT PAN ) Quantum Chaos and Interacting Systems July 9, / 24