Quantum chaos in composite systems

Transcription

1 Quantum chaos in composite systems Karol Życzkowski in collaboration with Lukasz Pawela and Zbigniew Pucha la (Gliwice) Smoluchowski Institute of Physics, Jagiellonian University, Cracow and Center for Theoretical Physics, Polish Academy of Sciences, Warsaw Complex Quantum Systems, Mumbai, Feb. 20, 2017 KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

2 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 (for unitary dynamics) 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. Fritz Haake, Quantum Signatures of Chaos KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

3 KZ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

4 Otton Nikodym & Stefan Banach, talking mathematics in Planty Garden, Cracow, summer 1916 KZ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

5 Classical kicked top model - Haake, Kuś, Sharf 1987 Discrete dynamics on a sphere: X 2 + Y 2 + Z 2 = 1 X = Re(X cos p + Z sin p + iy )e ikz cos p X sin p, Y = Im(X cos p + Z sin p + iy )e ikz cos p X sin p, Z = X sin p + Z cos p. linear rotation parameter: p = π/2, kicking strength k k = 2.0 k = 2.5 k = 3.0 k = 6.0 transition to chaos KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

6 Quantum kicked top model - Haake, Kuś, Sharf 1987 Discrete dynamics in Hilbert space of dimension N = 2j + 1 i) Hamiltonian H(t) = pj y + k + 2j J2 z j= δ(t n) ii) Unitary evolution operator U = exp[ i(k/2j)jz 2 ] exp[ ipj y ] Level spacing distribution P(s) (where s = (φ i+1 φ i )N/2π) a) k [0.1, 0.3] (regular dynamics) b) k [10.0, 10.5] (chaotic dynamics) N = 201 KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

7 Quantum unitary kicked top - conclusions A) In the case of classically regular motion the level spacing distribution P(s) displays level clustering, (Poisson) B) In the case of classically chaotic motion the level spacing distribution P(s) displays level repulsion, (Wigner) Unitary evolution matrices U display statistical properties of circular unitary ensemble and their eigenvectors are generic and delocalized. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

8 Open Systems and spectral properties of nonunitary evolution operators KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

9 Vistula river and Wawel castle in Cracow KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

10 Interacting Systems & Nonunitary Dynamics Due to interaction of an open system with an environment one needs to work with mixed states, which arise by averaging (partial trace) over the environment. Quantum maps: ρ = Φ(ρ) (in the space of density matrices!) Enviromental form (interacting quantum system!) ρ = Φ(ρ) = Tr E [V (ρ ω E ) V ]. where ω E is an initial state of the environment, while V is non local unitary; VV = 1. Kraus form ρ = Φ(ρ) = i A i ρa i, where the Kraus operators satisfy relation i A i A i = 1, which implies that the trace is preserved. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

11 Nonunitary Dynamics & mixed quantum states Due to interaction with environment the image of a pure state becomes mixed, ρ = Φ( ψ ψ ) (ρ ) 2 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 R 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. *) The set of mixed quantum states has N 2 1 real dimensions, **) For N 3 the set M N of mixed states is neither a polytope nor an ellipsoid with a smooth surphace. The set of pure states forms only a small (measure zero!) part of the boundary of the set M N of mixed states. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

12 Ciesielski theorem KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

13 Ciesielski theorem: With probability 1 ɛ the bench Banach talked to Nikodym in 1916 was localized in η-neighbourhood of the red arrow. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

14 Plate commemorating the discussion between Stefan Banach and Otton Nikodym (Kraków, summer 1916) KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

15 KZ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

16 Two coupled quantum kicked tops Miller and Sarkar (1998), Demkowicz-Dobrzański and Kuś (2008), Lakshminarayan (2010) Two spins j 1 and j 2 described in space of dimension D = (2j 1 + 1)(2j 2 + 1) Unitary evolution operator V =U 12 (U 1 U 2 ) where U i = exp ( i k2j ) J2zi exp ( i π ) 2 J y i, i = 1, 2 U 12 = exp ( i ɛj ) J z1 J z2. Global dynamics of two coupled spins is unitary, but the evolution of the reduced state is not! φ 0 H 1 H 2, σ t = Tr B [ V t φ 0 φ 0 (V ) t] We analyze mixed states σ t obtained by partial trace over the other spin... KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

17 Coupled kicked tops - level density of reductions Two spins j 1 and j 2 described each in dimension N i = 2j i + 1. Let c = (2j 1 + 1)/(2j 2 + 1) denote the ratio of both dimensions P (x) 0.15 P (x) x x P (x) 1.5 P (x) x x Level density P(x) of the rescaled eigenvalue x = λ 1 N 1 for rectangularity c = 0.2, 0.5, 1.0 and 4.0 compared with Marchenko-Pastur distribution. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

18 Random Matrices and Marchenko-Pastur distribution (1967) Let G be a nonhermitian random matrix of size N 1 N 2 with i.i.d. Gaussian complex variables. Then the spectrum of a positive matrix W = GG /TrGG has density given by P c (x) = 1 2πx (x x ) (x + x), where x = N 1 λ, rectangularity c = N 1 /N 2 and support x ± = ( 1 ± c ) 2. For square matrices c = 1 this expression reduces to the standard Marchenko Pastur distribution 1 x/4 P 1 (x) =, x [0, 4], π x equivalent to setting x = y 2 with y distributed according to Wigner semicircle Vladimir Marchenko & Leonid Pastur (2000) KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

19 Quantum coupled kicked top - first observations In the case of classically chaotic motion the level density P(x) of partially reduced states, σ = Tr 2 ( ψ12 ψ 12 ), is described by the Marchenko-Pastur distribution, so it conforms to predictions of random matrices. Questions concerning time evolution: Do any initial state φ 0 leads to a generic mixed state σ t = Tr B [ V t φ 0 φ 0 (V ) t] after a sufficiently long evolution time t? Are density matrices σ t distributed uniformly (with respect to the flat, Hilbert-Schmidt measure) inside the set M N of mixed quantum states? KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

20 Symmeterized Marchenko Pastur distribution Trace distance between two states D Tr (ρ, σ) = ρ σ 1 = Tr ρ σ is used to describe their distinguishability. To compute it we analyze distribution of eigenvalue µ of the difference ρ σ, where both states are random. It is given by symmetrized Marchenko Pastur distribution, f c (x) = MP c (x) MP c ( x), where x = N 1 µ and denotes free multiplicative convolution. In the case of HS measure, (rectangularity c = 1) we obtain a normalized symmetric distribution f 1 (x) = )) 2/3 ( ( 3 1 3x x + 33x 2 3x 4 + 6x 2 ( ( 3 )) 1/3. 3πx 1 + 3x + 33x 2 3x 4 + 6x and analogous formulae for f c (x) with an arbitrary parameter c = N 1 /N 2 > 0. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

21 Coupled kicked tops - level density of the differenece Two spins j 1 and j 2 described each in dimension N i = 2j i + 1 evolve by non local unitary U. Define ρ j = Tr B [ U j, j j, j U ], j = 1, 2. Let µ be an eigenvalue of ρ 1 ρ 2 and y = N 1 µ P (y) 0.15 P (y) y y P (y) 0.2 P (y) y y Level density P(y) for rectangularity c = 0.2, 0.5, 1.0 and 4.0 The case, c = 1 studied by Nica & Speicher (1998), is called tetilla law, Deya & Nourdin (2012). In limiting case c one obtaines (rescaled) semicircle. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

22 co-author Zbyszek Puchala after his research trip on tetilla law. KZ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

23 Average distance between 2 random states Take two random states σ and ρ acting on H N, generated according to the flat (HS) measure (c = 1). For large N their trace distance D tr (ρ, σ) D := π To find this we compute the integral over symmeterized MP distribution f1 (y) y dy = D which describes the spectrum of the difference ρ σ Related asymptotic results (N 1) for the average: a) root fidelity F (ρ, σ) = i λi (ρσ) 3 4 F b) Bures distance c) Chernoff bound B(ρ, σ) = 2(1 F (ρ, σ)) 2 2. Ch(ρ, σ) = Trρ 1 2 σ 1 2 ( 8 3π ) 2. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

24 Coupled kicked tops - evolution in time 2 k = 3.2 k = 3.5 k = 3.8 k = 4.0 limit value 2 ɛ = ɛ = 0.01 ɛ = 0.1 ɛ = 1.0 limit value Dtr 1 Dtr t t Time evolution of the trace distance D tr (ρ t, σ t ) between initially orthogonal states a) stronger chaos (larger kicking strength k) b) stronger interaction (larger coupling strength ɛ) increases convergence to the universal asymptotic distance D = 1/2 + 2/π KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

25 Concentration of measure in high dimensions Consider two random states of dimension N 1 The average value of their trace distance reads D tr (ρ, σ) = D = 1/2 + 2/π, but this distribition becomes singular: for N one has P ( D tr (ρ, σ) ) δ(d D ) This distance converges almost surely to a single value D! How this might be possible??? KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

26 Concentration of measure in high dimensions Consider two random states of dimension N 1 The average value of their trace distance reads D tr (ρ, σ) = D = 1/2 + 2/π, but this distribition becomes singular: for N one has P ( D tr (ρ, σ) ) δ(d D ) This distance converges almost surely to a single value D! How this might be possible??? concentration of measure! What is the expected distance between two random points in a unit ball in R N? and in a unit ball in R 3? KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

27 Properties of typical pure states in H N One quantum state fixed, one random... Fix an arbitrary state ψ 1. Generate randomly the other state ψ 2. What is the average angle χ between these states? What is the distribution P(χ) of the angle χ := arccos ψ 1 ψ 2? KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

28 Properties of typical pure states in H N One quantum state fixed, one random... Fix an arbitrary state ψ 1. Generate randomly the other state ψ 2. What is the average angle χ between these states? What is the distribution P(χ) of the angle χ := arccos ψ 1 ψ 2? Measure concentration phenomenon Fat hiper-equator of the sphere S N in R N+1... It is a consequence of the Jacobian factor for expressing the volume element of the N sphere. Let z = cos ϑ 1, so that J (sin ϑ 1 ) N 1 J 2 (ϑ 2,..., ϑ N ) Hence the typical angle χ is close to π/2 and two typical random states are orthogonal and the distribution P(χ) is close to δ(χ π/2). How close? KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

29 Quantitative description of Measure Concentration Levy s Lemma (on higher dimensional spheres) Let h : S N R be a Lipschitz function, with the constant η and the mean value h = S N h(x)dµ(x). Pick a point x S N at random from the sphere. For large N it is then unlikely to get a value of h much different then the average: ( ) P h(x) h > α (N + ) 1)α2 2 exp ( 9π 3 η 2 Simple application: the distance from the equator. Take h(x 1,...x N+1 ) = x 1. Then Levy s Lemma says that the probability of finding a random point of S N outside a band along the equator of width 2α converges exponentially to zero as 2 exp[ C(N + 1)α 2 ]. As N >> 1 then every equator of S N is FAT. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

30 Concentration of measure in high dimensions What is the expected Euclidean distance between two random points in a unit ball in R N? The answer is KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

31 Concentration of measure in high dimensions What is the expected Euclidean distance between two random points in a unit ball in R N? The answer is D(x, y) 2! as a) full measure of the ball is concentrated at the surface b) for any point at the sphere another random point will belong to the equator, so their Euclidean distance is D 2 (x, y) = For two random states of large dimension N their Hilbert Schmidt (=Euclidean) distance vanishes as D 2 HS (ρ, σ) = Tr(ρ σ)2 = Trρ 2 + Trσ 2 2Trρσ 0. Howver, their average trace distance is larger and non-trivial, D tr (ρ, σ) D := π Why do we care about the trace distance? KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

32 Distinguishing random states Helstrom theorem (1967) Suppose one is given a quantum state ρ {ρ 1, ρ 2 }. Probability P of discriminating between these states is bounded by P D tr(ρ 1, ρ 2 ) For instance, for orthogonal states D tr = 2 so that P = 1 Distinguishing two generic quantum states. Theorem. Two random states of large dimension N 1 can be distinguished in a single shot experiment with probability bounded by P D = π (1) Z. Pucha la, L. Pawela, K.Ż, Phys. Rev. A (2016). KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

33 Stefan Banach sitting at his bench close to the Wawel Castle Sculpture: Stefan Dousa Fot. Andrzej Kobos KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

34 Concluding Remarks Two coupled spins: chaotic system. offer a useful model of a composite quantum For generic parameters of the model: (strong chaos and coupling strength) reduced states conform to prediction of random matrices, and their level density is (asymptotically) described by Marchenko Pastur distribution. We derived symmetric MP distribution which describes density of the difference ρ σ and allows us to compute their average trace distance D = π, valid almost surely for any states due to concentration of measure. Trace distance between any two initially orthogonal states tends to D with convergence time decreasing with coupling strength and degree of chaos. KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31

35 Bench commemorating the discussion between Stefan Banach and Otton Nikodym (Kraków, summer 1916) Sculpture: Stefan Dousa opened in Planty Garden, Cracow, Oct. 14, 2016 Fot. Andrzej Kobos KŻ (IF UJ/CFT PAN ) Quantum chaos in composite systems Feb. 20, / 31