Quantum Chaos in Composite Systems Karol Zyczkowski in - - PowerPoint PPT Presentation

Quantum Chaos in Composite Systems

Karol ˙ Zyczkowski

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

University of Hradec Kr´ alov´ e, May 10, 2017

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 1 / 28

The collaboration with Peter

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 2 / 28

The collaboration with Peter

started in 1990 in Germany...

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 2 / 28

joint papers on wave chaos

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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) ∼ sβ for s → 0, where s stands for the (normalised) level spacing, si = φi+1 − φi. see e.g. Fritz Haake, Quantum Signatures of Chaos

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 4 / 28

Classical kicked top model - Haake, Ku´ s, Sharf 1987

Discrete dynamics on a sphere: X 2 + Y 2 + Z 2 = 1 X ′ = Re(X cos p + Z sin p + iY )eikZ cos p−X sin p, Y ′ = Im(X cos p + Z sin p + iY )eikZ 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

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 5 / 28

Quantum kicked top model - Haake, Ku´ s, Sharf 1987

Discrete dynamics in Hilbert space of dimension N = 2j + 1 i) Hamiltonian H(t) = pJy + k

2j J2 z

+∞

j=−∞ δ(t − n)

ii) Unitary evolution operator U = exp[−i(k/2j)J2

z ] exp[−ipJy]

Level spacing distribution P(s) (where s = (φi+1 − φi)N/2π) a) k ∈ [0.1, 0.3] (regular dynamics) b) k ∈ [10.0, 10.5] (chaotic dynamics) N = 201

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 6 / 28

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.

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Open Systems and spectral properties of nonunitary evolution operators

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joint papers on chaotic scattering

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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) = TrE[V (ρ ⊗ ωE) V †] . where ωE is an initial state of the environment, while V is non–local unitary; VV † = ✶.

Kraus form

ρ′ = Φ(ρ) =

i AiρA† i ,

where the Kraus operators satisfy relation

• i A†

i Ai = ✶,

which implies that the trace is preserved.

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 10 / 28

Nonunitary Dynamics & mixed quantum states

Due to interaction with environment the image of a pure state becomes mixed, ρ′ = Φ(|ψψ|) = (ρ′)2

Set MN of all mixed states of size N

MN := {ρ : HN → HN; ρ = ρ†, ρ ≥ 0, Trρ = 1} example: M2 = B3 ⊂ ❘3 - Bloch ball with all pure states at the boundary

The set MN is compact and convex:

ρ =

i ai|ψiψi| where ai ≥ 0 and

i ai = 1.

*) The set of mixed quantum states has N2 − 1 real dimensions, **) For N ≥ 3 the set MN 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 MN of mixed states.

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 11 / 28

Two coupled quantum kicked tops

Miller and Sarkar (1998), Demkowicz-Dobrza´ nski and Ku´ s (2008), Lakshminarayan (2010) Two spins j1 and j2 described in space of dimension D = (2j1 + 1)(2j2 + 1) Unitary evolution operator V =U12(U1 ⊗ U2) where Ui = exp

• −i k

2j J2

zi

• exp
• −iπ

2 Jyi

• ,

i = 1, 2 U12 = exp

• −iǫ

j Jz1 ⊗ Jz2

• .

Global dynamics of two coupled spins is unitary, but the evolution of the reduced state is not ! |φ0 ∈ H1 ⊗ H2, σt = TrB

• V t|φ0φ0|(V †)t

We analyze mixed states σt obtained by partial trace over the other spin...

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 12 / 28

Coupled kicked tops - level density of reductions

Two spins j1 and j2 described each in dimension Ni = 2ji + 1. Let c = (2j1 + 1)/(2j2 + 1) denote the ratio of both dimensions

1 2 x 0.00 0.15 0.30 P(x) 0.0 1.5 3.0 x 0.00 0.25 0.50 P(x) 2 4 x 0.0 1.5 3.0 P(x) 4 8 x 0.00 0.15 0.30 P(x)

Level density P(x) of the rescaled eigenvalue x = λ1N1 for rectangularity c = 0.2, 0.5, 1.0 and 4.0 compared with Marchenko-Pastur distribution.

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K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 14 / 28

Hradec Kr´ alov´ e

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Random Matrices and Marchenko-Pastur distribution (1967)

Let G be a nonhermitian random matrix of size N1 × N2 with i.i.d. Gaussian complex variables. Then the spectrum of a positive matrix W = GG †/TrGG † has density given by Pc(x) =

1 2πx

• (x − x−) (x+ − x),

where x = N1λ, rectangularity c = N1/N2 and support x± =

• 1 ± √c

2. For square matrices c = 1 this expression reduces to the standard Marchenko – Pastur distribution P1(x) = √

1−x/4 π√x

, x ∈ [0, 4], equivalent to setting x = y2 with y distributed according to Wigner semicircle Vladimir Marchenko & Leonid Pastur (2000)

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Quantum coupled kicked top - first observations

In the case of classically chaotic motion the level density P(x) of partially reduced states, σ = Tr2

• |ψ12ψ12|
• , is described by the

Marchenko-Pastur distribution, so it conforms to predictions of random matrices. Questions concerning time evolution: – Do any initial state φ0 leads to a ’generic’ mixed state σt = TrB

• V t|φ0φ0|(V †)t

after a sufficiently long evolution time t ? – Are density matrices σt distributed uniformly (with respect to the flat, Hilbert-Schmidt measure) inside the set MN of mixed quantum states ?

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 16 / 28

Symmeterized Marchenko–Pastur distribution

Trace distance between two states DTr(ρ, σ) = ρ − σ1 = 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, fc(x) = MPc(x) ⊞ MPc(−x), where x = N1µ and ⊞ denotes free multiplicative convolution. In the case of HS measure, (rectangularity c = 1) we obtain a normalized symmetric distribution f1(x) = −1 − 3x2 +

• 1 + 3x

√ 3 + 33x2 − 3x4 + 6x 2/3 2 √ 3πx

• 1 + 3x

√ 3 + 33x2 − 3x4 + 6x 1/3 . and analogous formulae for fc(x) with an arbitrary parameter c = N1/N2 > 0.

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 17 / 28

Coupled kicked tops - level density of the differenece

Two spins j1 and j2 described each in dimension Ni = 2ji + 1 evolve by non–local unitary U. Define ρj = TrB

• U||j, j|j, j|U†

, j = 1, 2. Let µ be an eigenvalue of ρ1 − ρ2 and y = N1µ

−2 2 y 0.00 0.15 0.30 P(y) −2 2 y 0.0 0.5 1.0 P(y) −3 3 y 0.0 0.2 0.4 P(y) −6 6 y 0.00 0.07 0.14 P(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.

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co-author Zbyszek Pucha la after his research trip on tetilla law.

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Average distance between 2 random states

Take two random states σ and ρ acting on HN, generated according to the flat (HS) measure (c = 1). For large N their trace distance

Dtr(ρ, σ) → D∗ := 1

2 + 2 π ≈ 1.1366

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 → 9

16.

b) Bures distance B(ρ, σ) =

• 2(1 −
• F(ρ, σ)) →

√ 2 2 .

c) Chernoff bound Ch(ρ, σ) = Trρ

1 2 σ 1 2 →

8

3π

2 .

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 20 / 28

Coupled kicked tops - evolution in time

50 100 t 1 2 Dtr

k = 3.2 k = 3.5 k = 3.8 k = 4.0 limit value

50 100 t 1 2 Dtr

ǫ = 0.005 ǫ = 0.01 ǫ = 0.1 ǫ = 1.0 limit value

Time evolution of the trace distance Dtr(ρt, σt) between initially

• rthogonal states

a) stronger chaos (larger kicking strength k) b) stronger interaction (larger coupling strength ǫ) increases convergence to the universal asymptotic distance D∗ = 1/2 + 2/π ≈ 1.1366..

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 21 / 28

Concentration of measure in high dimensions

Consider two random states of dimension N ≫ 1 The average value of their trace distance reads Dtr(ρ, σ) = D∗ = 1/2 + 2/π , but this distribition becomes singular: for N → ∞ one has P

• Dtr(ρ, σ)
• → δ(D − D∗)

This distance converges almost surely to a single value D∗ ! How this might be possible ??? ❘ ❘

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 22 / 28

Concentration of measure in high dimensions

Consider two random states of dimension N ≫ 1 The average value of their trace distance reads Dtr(ρ, σ) = D∗ = 1/2 + 2/π , but this distribition becomes singular: for N → ∞ one has P

• Dtr(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 ❘N ? and in a unit ball in ❘3 ?

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 22 / 28

Properties of ’typical’ pure states in HN

One quantum state fixed, one random...

Fix an arbitrary state |ψ1. Generate randomly the other state |ψ2.

• What is the average angle χ between these states ?
• What is the distribution P(χ) of the angle χ := arccos |ψ1|ψ2| ?

❘

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 23 / 28

Properties of ’typical’ pure states in HN

One quantum state fixed, one random...

Fix an arbitrary state |ψ1. Generate randomly the other state |ψ2.

• What is the average angle χ between these states ?
• What is the distribution P(χ) of the angle χ := arccos |ψ1|ψ2| ?

Measure concentration phenomenon

’Fat hiper-equator’ of the sphere SN in ❘N+1... It is a consequence of the Jacobian factor for expressing the volume element of the N– sphere. Let z = cos ϑ1, so that J ∼ (sin ϑ1)N−1J2(ϑ2, . . . , ϑN) 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?

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 23 / 28

Quantitative description of Measure Concentration

Levy’s Lemma (on higher dimensional spheres)

Let h : SN → ❘ be a Lipschitz function, with the constant η and the mean value h =

• SN h(x)dµ(x).

Pick a point x ∈ SN 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
• ≤ 2 exp
• −(N + 1)α2

9π3η2

• Simple application: the distance from the ’equator’

. Take h(x1, ...xN+1) = x1. Then Levy’s Lemma says that the probability

• f finding a random point of SN outside a band along the equator of

width 2α converges exponentially to zero as 2 exp[−C(N + 1)α2]. As N >> 1 then every equator of SN is ’FAT’.

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Concentration of measure in high dimensions

What is the expected Euclidean distance between two random points in a unit ball in ❘N ? The answer is

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 25 / 28

Concentration of measure in high dimensions

What is the expected Euclidean distance between two random points in a unit ball in ❘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 D2(x, y) = √1 + 1. For two random states of large dimension N their Hilbert Schmidt (=Euclidean) distance vanishes as D2

HS(ρ, σ) = Tr(ρ − σ)2 = Trρ2 + Trσ2 − 2Trρσ → 0.

Howver, their average trace distance is larger and non-trivial,

Dtr(ρ, σ) → D∗ := 1

2 + 2 π ≈ 1.1366

Why do we care about the trace distance ?

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Distinguishing random states

Helstrom theorem (1967)

Suppose one is given a quantum state ρ ∈ {ρ1, ρ2}. Probability P of discriminating between these states is bounded by

P ≤

1 2 + 1 4Dtr(ρ1, ρ2)

For instance, for orthogonal states Dtr = 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 ≤ 1 2 + 1 4D∗ = 5 8 + 1 2π ≃ 0.784155.

• Z. Pucha

la,

• L. Pawela, K. ˙

Z., Phys. Rev. A (2016).

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 26 / 28

Concluding Remarks

Two coupled spins: offer a useful model of a composite quantum chaotic system. 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∗ = 1

2 + 2 π, valid almost surely for any states due to concentration

• f measure.

Trace distance between any two initially orthogonal states tends to D∗ with convergence time decreasing with coupling strength and degree of chaos.

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Happy Birthday

K ˙ Z (IF UJ/CFT PAN ) Quantum Chaos in Composite Systems May 10, 2017 28 / 28

Happy Birthday Dear Peter !

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