# A Survey on the Classical Limit of Quantum Dynamical Entropies

**ABSTRACT** We analyze the behavior of quantum dynamical entropies production from sequences of quantum approximants approaching their (chaotic) classical limit. The model of the quantized hyperbolic automorphisms of the 2-torus is examined in detail and a semi-classical analysis is performed on it using coherent states, fulfilling an appropriate dynamical localization property. Correspondence between quantum dynamical entropies and the Kolmogorov-Sinai invariant is found only over time scales that are logarithmic in the quantization parameter.

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**ABSTRACT:**We analyze a model quantum dynamical system subjected to periodic interaction with an environment, which can describe quantum measurements. Under the condition of strong classical chaos and strong decoherence due to large coupling with the measurement device, the spectra of the evolution operator exhibit an universal behavior. A generic spectrum consists of a single eigenvalue equal to unity, which corresponds to the invariant state of the system, while all other eigenvalues are contained in a disk in the complex plane. Its radius depends on the number of the Kraus measurement operators and determines the speed with which an arbitrary initial state converges to the unique invariant state. These spectral properties are characteristic of an ensemble of random quantum maps, which in turn can be described by an ensemble of real random Ginibre matrices. This will be proven in the limit of large dimension.Physical Review E 06/2010; 81(6 Pt 2):066209. · 2.31 Impact Factor

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arXiv:0708.2779v2 [math-ph] 13 Dec 2007

A Survey on the Classical Limit

of Quantum Dynamical Entropies∗

V. C

“Mark Kac” Complex Systems Research Centre, Uniwersytet Jagiello´ nski

ul. Reymonta 4, 30–059 Krak´ ow, Poland

and

Centrum Fizyki Teoretycznej, Polska Akademia Nauk

Al. Lotnik´ ow 32/44, 02–668 Warszawa, Poland

We analyzethe behaviorof quantumdynamicalentropiesproductionfromse-

quences of quantumapproximantsapproachingtheir (chaotic) classical limit. The

model of the quantized hyperbolic automorphisms of the 2–torus is examined in

detail and a semi–classical analysis is performed on it using coherent states, ful-

filling an appropriate dynamical localization property. Correspondence between

quantum dynamical entropies and the Kolmogorov–Sinai invariant is found only

over time scales that are logarithmic in the quantization parameter.

PACS numbers: 05.45.-a, 05.45.Mt, 05.45.Ac, 03.65.Fd

Contents

1 Introduction2

2 Dynamical systems: algebraic setting4

3 Classical limit: coherent states

3.1Anti–Wick Quantization

5

6 . . . . . . . . . . . . . . . . . . . . . .

4 Classical and quantum cat maps

4.1 Finite dimensional quantizations . . . . . . . . . . . . . . . . . .

4.2 Coherent states for cat maps . . . . . . . . . . . . . . . . . . . .

7

7

10

5 Quantum and classical time evolutions10

∗Proceedings of the 3rd Workshop on Quantum Chaos and Localisation Phenomena,

Warsaw, Poland, May 25–27, 2007

(1)

Page 2

2

V. Cappellini

6 Dynamical Entropies

6.1 Kolmogorov Metric Entropy . . . . . . . . . . . . . . . . . . . .

6.2 Quantum Dynamical Entropies . . . . . . . . . . . . . . . . . . .

6.3

ALF–entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4Comparison of dynamical entropies

12

12

14

15

17. . . . . . . . . . . . . . . .

7 Classical limit of quantum dynamical entropies18

8 Conclusions and outlook19

References19

1. Introduction

The notion of classical chaos is associated with motion on a compact phase–

space with high sensitivity to initial conditions: trajectories diverge exponentially

fast and nevertheless remain confined to bounded regions [1–7].

In an opposite way, quantization on compacts yields discrete energy spectra,

which in term entail quasi–periodic time–evolution [8].

Nevertheless, nature is fundamentally quantal and, according to the correspon-

dence principle, classical behavior must emerge in the limit ? → 0.

Also, classical and quantum mechanics are expected to overlap over times ex-

pected to scale as ?−αfor some α > 0 [7], the so–called semi–classical regime.

Actually, it turns out that this is true only for regular classical limits whereas, for

chaotic ones, classical and quantum mechanics agree over times which scale as

−log? [5–7], and footprints of the exponential separation of classical trajecto-

ries are found even on finite dimensional quantization provide that the time does

not exceed such a logarithmic upper bound [6,9]. Both time scales diverge when

? → 0, but the shortness of the latter means that classical mechanics has to be

replaced by quantum mechanics much sooner for quantum systems with chaotic

classical behavior. The logarithmic breaking time −log? has been considered by

some as a violation of the correspondence principle [10,11] and by others, see [6]

and Chirikov in [5], as the evidence that time and classical limits do not commute.

The analytic studies of logarithmic time scales have been mainly performed

by means of semi–classical tools, essentially by focusing, via coherent state tech-

niques, on the phase space localization of specific time evolving quantum observ-

ables. In the following, we shall show how they emerge in the context of quantum

dynamical entropies.

As a particular example, we shall concentrate on finite dimensional quantiza-

tions of continuous hyperbolic automorphisms of the 2–torus T2≔ R2/Z2(the

unit square with opposite sides identified), which are prototypes of chaotic behav-

ior; indeed, their trajectories separate exponentially fast with a Lyapunov exponent

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A Survey on the Classical Limit of Quantum Dynamical Entropies

3

logλ+> 0 [12,13]. If δ is an initial error along a trajectory, and δn≃ δλn

sical spreading after n steps of the (time–stroboscopic) dynamics, then boundness

of the motion imposes δn ? 1, where 1 is the diameter of the 2–torus T2. This

explain why the limit δ → 0 has necessarily to be performed before the time–limit,

and the Lyapunov exponent can be computed as

+its clas-

logλ+= lim

n→∞

1

n

lim

δ→0log

?δn

δ

?

.

(1)

Standard quantization, ` a la Berry, of hyperbolic automorphisms on T2[14, 15]

yields Hilbert spaces of a finite dimension N, this latter variable playing the role

of the semi–classical parameter and setting to 1/N the minimal size of the phase–

space grain cells. Imposing the latter bound, min{δ} ? 1/N, its evident how the

conflict between the two limits, emerging once δn≃ 1, can be transferred in the

time–step n as n ≃ logN/logλ+. In this sense, rather than a violation of the cor-

respondence principle, the logarithmic breaking–time indicates the typical scaling

for a joint time–classical limit suited to classically chaotic quantum systems.

The Kolmogorov–Sinai dynamical entropy [3] (KS–entropy, for short) is de-

finedbyassigning measures tobunches oftrajectories and computing theShannon–

entropy per time–step of the ensemble of bunches in the limit of infinitely many

time–steps: The more chaotic the time–evolution, the more the possibile bunches

and the larger their entropy. The production of different bunches of trajectories

issuing from the same bunch is typical of high sensitivity to initial conditions and

this is indeed the mechanism at the basis of the theorem of Ruelle and Pesin [16],

linking KS–entropy of a smooth, classical dynamical systems, to the sum of its

positive Lyapunov exponents.

In the quantum realm, there are different candidates for non–commutative ex-

tensions of the KS–invariant [17–21]: in this paper we shall focus on one of them,

called ALF–entropy [18], and we shall study its semi–classical limit.

The ALF–entropy is based on the algebraic properties of dynamical systems,

that is on the fact that they are describable by suitable algebras of observables, their

time evolution by linear maps on these algebras, and their states by expectations

over them.

We show that, while being bounded by log N, nevertheless over numbers of

time steps 1 ≪ n < logN, the entropy content per letter, or entropy production, is

logλ+. It thus follows that the joint limit n,N → +∞, with n ∝ logN, yields the

Kolmogorov–Sinai entropy. This confirms the numerical results in [22] and [23],

where the dynamical entropy [18] is applied to the study of the quantum kicked

top, respectively to quantum cat maps.

In this approach, the presence of logarithmic time scales indicates the typical

scaling forajoint time/classical limitsuited topreserve positive entropy production

in quantized classically chaotic quantum systems.

Page 4

4

V. Cappellini

The paper is organized as follows: Section 2 contains a brief review of the

algebraic approach to classical and dynamical systems, while Section 3 introduces

some basic semi–classical tools. Sections 4 and 5 deal with the quantization of hy-

perbolic maps on finite dimensional Hilbert spaces and the relation between clas-

sical and time limits. Section 6 gives an overview of various models of quantum

dynamical entropies present in the literature and particularly focus on the one pro-

posed by Alicki and Fannes [18,24] (ALF–entropy, where L stands for Lindblad).

Finally, in Section 7, the semi–classical behavior of quantum dynamical entropies

is studied and the emergence of a typical logarithmic time scale is showed.

2. Dynamical systems: algebraic setting

Usually, continuous classical motion is described by means of a measure space

X, the phase–space, endowed with the Borel σ–algebra and a normalized measure

µ, µ(X) = 1. The “volumes”

?

of measurable subsets E ⊆ X represent the probabilities that phase–points x ∈ X

belong to them. By specifying the statistical properties of the system, the measure

µ defines a “state” of it. In such a scheme, a reversible discrete time dynamics

amounts to an invertible measurable map T onto X such that µ ◦ T = µ, and to its

iterates {Tk| k ∈ Z}: T–invariance of the measure µ ensure that the state defined by

µ can be taken as an equilibrium state with respect to the given dynamics. Phase–

trajectories passing through x ∈ X at time 0 are then sequences {Tkx | k ∈ Z} [3].

Classical dynamical systems are thus conveniently described by triplets (X,T,µ).

In the present work we shall focus upon the following:

µ(E) =

E

µ(dx)

• X – a compact metric space:

the 2–dimensional torus T2= R2/Z2= {(x1, x2) ∈ R2(mod 1)};

• T – invertible measurable transformations from X to itself such that T−1are

also measurable;

• µ – the Lebesgue measure µ(dx) = dx1dx2on T2.

In this paper, we consider a general scheme for quantizing and dequantizing,

i.e. for taking the classical limit (see [25]). Within this framework, we focus on the

semi–classical limit of quantum dynamical entropies of finite dimensional quan-

tizations of the celebrated Arnold’s cat map and of generic maps belonging to

the so–called unimodular group on the 2–torus: in the following we simply de-

note such a family of maps cat maps family. The last denomination is perfectly

legitimate, in fact the acronym CAT stands for Continuous Automorphism of the

Torus.

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A Survey on the Classical Limit of Quantum Dynamical Entropies

5

In order to make the quantization procedure more explicit, it proves useful to

follow an algebraic approach and replace (X,T,µ) with (Mµ,Θ,ωµ) where

• Mµ– is the von Neumann algebra L∞

tially bounded µ–measurable functions on X, equipped with the so–called

essential supremum norm ? · ?∞[26];

• {Θk| k ∈ Z} – is the discrete group of automorphisms of Mµwhich im-

plements the dynamics: Θ(f) ≔ f ◦ T−1. The invariance of the reference

measure reads now ωµ◦ Θ = ωµ;

• ωµ– is the state on Mµdefined by the reference measure µ

µ(X) of (equivalence classes of) essen-

ωµ: Mµ∋ f ?−→ ωµ(f) ≔

?

X

µ(dx) f(x) ∈ R+

.

Quantum dynamical systems are described in a completely similar way by a

triple (M,Θ,ω), the critical difference being that the algebra of observables M is

no longer Abelian:

• M – is a von Neumann algebra of operators, the observables, acting on a

Hilbert space H ;

• Θ – is an automorphism of M ;

• ω – is an invariant normal state on M: ω ◦ Θ = ω .

Quantizing essentially corresponds to suitably mapping the commutative, clas-

sical triple (Mµ,Θ,ωµ) to a non–commutative, quantum triple (M,Θ,ω).

3. Classical limit: coherent states

Performing the classical limit or a semi–classical analysis consists in studying

how a family of algebraic triples (M,Θ,ω), depending on a quantization ?–like

parameter, is mapped onto (Mµ,Θ,ωµ) when the parameter goes to zero. The most

successful semi–classical tools are based on the use of coherent states (CS for

short).

For our purposes, we shall use a large integer N as a quantization parameter,

i.e. we use 1/N as the ?–like parameter. In fact, we shall consider cases where

M is the algebra MNof N–dimensional square matrices acting on CN, the quan-

tum reference state is the normalized trace1

dynamics is given in terms of a unitary operator UTon CNin the standard way:

ΘN(X) ≔ U∗

In full generality, coherent states will be identified as follows.

NTr on MN, denoted by τN, and the

TX UT.

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6

V. Cappellini

Definition 3.1 A family {|CN(x)? | x ∈ X} ∈ H of vectors, indexed by points x ∈ X,

constitutes a set of coherent states if it satisfies the following requirements

1. Measurability: x ?→ |CN(x)? is measurable on X;

2. Normalization: ?CN(x)?2= 1, x ∈ X;

3. Overcompleteness: N

?

Xµ(dx)|CN(x)??CN(x)| = 1N;

4. Localization: given ε > 0 and d0> 0, there exists N0(ǫ,d0) such that for

N ≥ N0and dX(x, y) ≥ d0one has

N|?CN(x),CN(y)?|2≤ ε.

The symbol dX(x, y) used in the localization property stands for the length of the

shorter segment connecting the two points on X. Of course the latter quantity does

depend on the topological properties of X. In particular, for the 2–torus,

dT2 (x, y) ≔ min

n∈Z2

The overcompleteness condition may be written in dual form as

??? x − y + n

???R2

.

(2)

N

?

X

µ(dx)?CN(x),XCN(x)? = TrX,

X ∈ MN.

Indeed,

N

?

X

µ(dx)?CN(x),XCN(x)? = N Tr

??

X

µ(dx)|CN(x)??CN(x)|X

?

= TrX

.

3.1. Anti–Wick Quantization

In order to study the classical limit and, more generally, the semi–classical

behavior of (MN,ΘN,τN) when N → ∞, we introduce two linear maps. The first,

γN∞, (anti–Wick quantization) associates N × N matrices of MNto functions in

Mµ= L∞

µ(X); the second one, γ∞N, maps N × N matrices to functions in L∞

Definition 3.2 Given a family { |CN(x)? | x ∈ X } of CS in CN, the anti–Wick

quantization scheme will be described by a (completely) positive unital map γN∞:

Mµ→ MN

?

The corresponding dequantizing map γ∞N : MN → Mµwill correspond to the

(completely) positive unital map

µ(X).

Mµ∋f ?→ N

X

µ(dx) f(x)|CN(x)??CN(x)| ≕ γN∞(f) ∈ MN

.

MN∋ X ?→ ?CN(x),XCN(x)? ≕ γ∞N(X)(x) ∈ Mµ

.

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A Survey on the Classical Limit of Quantum Dynamical Entropies

7

Both maps are identity preserving because of the conditions imposed on the

CS–family of and are also completely positive, since the domain of γN∞is a com-

mutative algebra as well as the range of γ∞N. The following two equivalent prop-

erties are less trivial:

Proposition 3.1 For all f ∈ Mµ

lim

N→∞γ∞N◦ γN∞(f) = f

µ–a. e.

Proposition 3.2 For all f,g ∈ Mµ

lim

N→∞τN?γN∞(f)∗γN∞(g)?= ωµ(fg) =

The previous two propositions, proved in [27,28], can be taken as requests on any

well–defined quantization/dequantization scheme for observables. In the sequel,

we shall need the notion of quantum dynamical systems (MN,ΘN,τN) tending to

the classical limit (Mµ,Θ,ωµ). We then not only need convergence of observables

but also of the dynamics. This aspect will be considered in Section 5.

?

X

µ(dx) f(x)g(x).

4. Classical and quantum cat maps

In this section, we collect the basic material needed to describe both classical

and quantum cat maps and we introduce a specific set of CS that will enable us to

perform the semi–classical analysis over such dynamical systems.

4.1. Finite dimensional quantizations

We first introduce cat maps in the spirit of the algebraic formulation introduced

in the previous sections.

Definition 4.1 Hyperbolic continuous automorphisms of the torus are generically

represented by triples (Mµ,Θ,ωµ), where

• Mµis the algebra of essentially bounded functions on the two dimensional

torus T2≔ R2/Z2=?(x1, x2) ∈ R2(mod 1)?, equipped with the Lebesgue

• {Θk| k ∈ Z} is the family of automorphisms (discrete time evolution) given

by Mµ∋ f ?→ (Θkf)(x) ≔ f(A−kx (mod 1)), where A =

entries such that ad − bc = 1, |a + d| > 2 and maps T2onto itself;

measure µ(dx) ≔ dx;

?a b

c d

?

has integer

Page 8

8

V. Cappellini

• ωµis the expectation obtained by integration with respect to the Lebesgue

measure: Mµ∋ f ?→ ωµ(f) ≔

Denoting with t ≔ Tr(A)/2 the semi–trace of A, |t| > 1, the two irrational eigen-

values of A can be written as 1 < λ+≔ t+

Distances are stretched along the direction of the eigenvector |e+?, A|e+? = λ+|e+?,

contracted along that of |e−?, A|e−? = λ−|e−? and all periodic points are hyper-

bolic [29]. Once the folding condition is added, the hyperbolic automorphisms of

the torus become prototypes of classical chaos, with positive Lyapunov exponent

logλ+.

One can quantize the associated algebraic triple (Mµ,Θ,ωµ) on either infi-

nite [30] or finite dimensional Hilbert spaces [14, 15, 31]. In the following, we

shall focus on the latter.

Given an integer N, we consider an orthonormal basis |j? of CN, where the

index jruns through theresidual class modulo N, here and inthefollowing denoted

by (Z/NZ), namely |j + N? ≡ |j?, j ∈ Z. By using this basis we define two

unitary matrices UNand VN, representing position and momentum shift operators,

as follows:

?

T2dx f(x), that is left invariant by Θ.

√t2− 1 and 1 > λ−≔ t−

√t2− 1 = λ−1

+.

UN|j? ≔ exp

?2πi

Nu

?

|j + 1?,

and

VN|j? ≔ exp

?2πi

N(v − j)

?

|j?.

(3)

In the last equation, we explicitly indicated the dependence on two arbitrary phases

(u,v) ∈ [0,1) labeling the representation and fulfilling

UN

N= e2iπu1N,

VN

N= e2iπv1N.

(4)

It turns out that

UNVN= exp

?2iπ

N

?

VNUN.

(5)

Introducing Weyl operators labeled by n = (n1,n2) ∈ Z2

WN(n) ≔ exp

?iπ

Nn1n2

?

Vn2

NUn1

N= WN(−n)∗

(6)

it follows that

WN(Nn) = eiπ(Nn1n2+2n1u+2n2v)

(7a)

WN(n)WN(m) = exp

?iπ

Nσ(n, m)

?

WN(n+ m),

(7b)

where σ(n, m) ≔ n1m2− n2m1is the so–called symplectic form.

Definition 4.2 Quantized cat maps will be identified with triples (MN,ΘN,τN)

where

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A Survey on the Classical Limit of Quantum Dynamical Entropies

9

• MNis the full N×N matrix algebra over C generated by the (discrete) group

of Weyl operators {WN(n) | n ∈ Z2}. In the following, such a group will be

denoted by Weyl group ;

• ΘN: MN?→ MNis the automorphism such that

WN(p) ?→ ΘN(WN(p)) ≔ WN(Ap)

In the definition of above, we have omitted reference to the parameters u,v in (3):

they must be chosen such that

,

p ∈ (Z/NZ)2

.

(8)

?ac

db

??u

v

?

=

?u

v

?

+N

2

?ac

bd

?

(mod 1)

.

(9)

Then, the folding condition (4) is compatible with the time evolution [15]. Therea-

son for (9) is the following: denoting with ˆ e1and ˆ e2the standard unit vectors of R2,

the representation generated by the two generators UN= WN(ˆ e1) and VN= WN(ˆ e2)

and the one generated by ΘN(UN) = WN(A ˆ e1) and ΘN(VN) = WN(A ˆ e2) must be

unitarily equivalent; in other words the two representations must be labeled by the

same u and v. According to (4), this can be expressed by

[WN(ˆ e1)]N= [WN(Aˆ e1)]N

and[WN(ˆ e2)]N= [WN(Aˆ e2)]N; (10)

the latter equation restrict the possible couples (u,v) available and leads to (9).

An important set of matrices A, originally called “set of quantizable maps”

and characterized by (u,v) = (0,0), is also important for historical reasons, indeed

it was the set used by Berry and Hannay [32] to develop the first quantization of

Cat Maps. Recent developments of Berry’s approach to quantization can be found

in [33–35].

Further, relation (7b) is also preserved since the condition det(A) = 1 guar-

antees that the symplectic form remains invariant, i.e. σ(An,Am) = σ(n, m). In-

variance of σ(·,·) , together with (7), also allows equation (8) to hold true for all

p ∈ Z2and not only for those in (Z/NZ)2.

Many other useful relations can be obtained by using the explicit expression

WN(n)|j? = exp

?iπ

N(−n1n2+ 2n1u + 2n2v)

?

exp

?

−2iπ

N

jn2

?

|j + n1?

.

(11)

In particular, from (11) one readily derives the decomposition

MN∋ X =

?

m∈(Z/NZ)2

τN

?

X WN(−m)

?

WN(m)

,

(12)

while from equation (7b) one gets

[WN(n),WN(m)] = 2isin

?π

Nσ(n, m)

?

WN(n+ m)

,

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10

V. Cappellini

which suggests that the ?–like parameter is 1/N and that the classical limit corre-

spond to N → ∞ . In the following section, we set up a CS technique suited to

study classical cat maps as limits of quantized cats.

4.2. Coherent states for cat maps

We shall construct a CS–family { |CN(x)? | x ∈ T2} on the 2–torus by means

of the discrete Weyl group. We define

|CN(x)? ≔ WN(⌊Nx⌋)|CN?

,

(13a)

where ⌊Nx⌋ = (⌊Nx1⌋,⌊Nx2⌋), 0 ≤ ⌊Nxi⌋ ≤ N − 1 is the largest integer smaller

than Nxiand the reference vector |CN? is chosen to be

|CN? =

N−1

?

j=0

CN(j)|j?

,

CN(j) ≔

1

2(N−1)/2

??N − 1

j

?

.

(13b)

Measurability and normalization are immediate, overcompleteness comes as fol-

lows. Let Y be the operator in the left hand side of Definition 3.1.3.

If τN(Y WN(n)) = τN(WN(n)) for all n = (n1,n2) with 0 ≤ ni? N − 1, then ac-

cording to (12) applied to Y it follows that Y = 1. This is indeed the case as, using

equations (7b), (13) and N–periodicity,

τN(Y WN(n)) =

?

?

T2dx?CN(x),WN(n)CN(x)?

?2πi

1

N2

p∈(Z/NZ)2

= τN(WN(n))

.

=

T2dx exp

Nσ(n,⌊Nx⌋)

?2πi

?

?CN,WN(n)CN?

?

=

?

exp

Nσ(n, p)

?CN,WN(n)CN?

(14)

In the last line we used that when x runs over T2, ⌊Nxi⌋, i = 1,2 runs over the set

of integers 0,1,...,N − 1.

The proof the localization property in Definition 3.1 is more technical and

requires several steps: the willing reader can find it in [27,28].

5. Quantum and classical time evolutions

One of the main issues in the semi–classical analysis is to compare if and how

the quantum and classical time evolutions mimic each other when a quantization

parameter goes to zero.

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A Survey on the Classical Limit of Quantum Dynamical Entropies

11

In the case of classically chaotic quantum systems, the situation is strikingly

different from the case of classically integrable quantum systems. In the former

case, classical and quantum mechanics agree on the level of coherent states only

over times which scale as −log?.

As before, let T denote the evolution on the classical phase space X and UT

the unitary single step evolution on CN, the so–called Floquet operator, which

represent its “quantization”. We formally state the semi–classical correspondence

of classical and quantum evolution using coherent states:

Condition 5.1 Dynamical localization: There exists an α > 0 such that for all

choices of ε > 0 and d0> 0 there exists an N0∈ N with the following property:

if N > N0and k ≤ αlogN, then N|?CN(x),Uk

d0.

TCN(y)?|2≤ ε whenever d(Tkx, y) ≥

Remark 5.1 The condition of dynamical localization is what is expected of a

good choice of coherent states, namely, on a time scale logarithmic in the inverse

of the semi–classical parameter, evolving CS should stay localized around the

classical trajectories. Informally, when N → ∞, the quantities

Kk(x, y) ≔ ?CN(x),Uk

should behave as if N|Kk(x, y)|2≃ δ(Tkx − y) (note that this hypothesis makes

our quantization consistent with the notion of regular quantization described in

Section V of [21]). The constraint k ≤ αlogN is typical of hyperbolic classical

behavior and comes heuristically as follows. The maximal localization of coher-

ent states cannot exceed the minimal coarse–graining dictated by 1/N; if, while

evolving, CS stayed localized forever around the classical trajectories, they would

get more and more localized along the contracting direction. Since for hyper-

bolic systems the increase of localization is exponential with Lyapunov exponent

logλ+> 0, this sets the upper bound, better known as logarithmic breaking–time,

and indicates that α ≃ 1/logλ+.

Proposition 5.1 Let (MN,ΘN,τN) be a general quantum dynamical system as

defined in Section 3 and suppose that it satisfies Condition 5.1. Let ?X?2 ≔

√τN(X∗X), X ∈ MNdenote the normalized Hilbert–Schmidt norm. In the ensuing

topology

lim

k, N→∞

k<αlog N

TCN(y)?

(15)

?Θk

N◦ γN∞(f) − γN∞◦ Θk(f)?2= 0

.

(16)

Remark 5.2 The above proposition, whose proof can be found in [27,28], can be

seen as a modification of the so–called Egorov’s property (see [36]), and gives the

Page 12

12

V. Cappellini

strength of the non–commutativity of classical and time limits when the classical

system has a positive Lyapunov exponent. The same (logarithmic) scaling for the

breaking–time has been found numerically in [37] also for discrete classical cat

maps, converging in a suitable classical limit to continuous cat maps. Analogously,

similar analysis [38] has been performed on sequences of discrete approximants of

discontinuous automorphisms on the 2–torus, known as Sawtooth maps, and the

logarithmic breaking–time has been recovered there too.

We shall not prove the dynamical localization condition 5.1 for the quantum

cat maps, but adirect derivation offormula (16), based onthe simple expression (8)

of the dynamics when acting on Weyl operators, is available in [27,28] and reads

as follows

Proposition 5.2 Let (MN,ΘN,τN) be a sequence of quantum cat maps tending

with N → ∞ to a classical cat map with Lyapunov exponent logλ+; then

lim

k, N→∞

k<log N/(2logλ+)

?Θk

N◦ γN∞(f) − γN∞◦ Θk(f)?2= 0

,

where ? · ?2is the Hilbert–Schmidt norm of Proposition 5.1.

6. Dynamical Entropies

Intuitively, one expects the instability proper to the presence of a positive Lya-

punov exponent to correspond to some degree of unpredictability of the dynamics:

classically, the metric entropy of Kolmogorov provides the link [8].

6.1. Kolmogorov Metric Entropy

For continuous classical systems (X,T,µ) such as those introduced in Sec-

tion 2, the construction of the dynamical entropy of Kolmogorov is based on

subdividing X into measurable disjoint subsets {Eℓ| ℓ = 1,2,··· ,D } such that

?

atoms T−j(Eℓ) = {x ∈ X | Tjx ∈ Eℓ}; one can then form finer partitions

ℓEℓ= X which form finite partitions (coarse graining) E.

Under the dynamical maps T : X → X , any given E evolves into Tj(E) with

E[0,n−1]≔

n−1

?

j=0

Tj(E)

= E

?

T(E)

?

···

?

Tn−1(E)

Page 13

A Survey on the Classical Limit of Quantum Dynamical Entropies

13

whose atoms

Ei0i1···in−1≔

n−1

?

j=0

T−jEij

= Ei0

?

T−1(Ei1)

?

···

?

T−n+1(Ein−1)

have volumes

µi0i1···in−1≔ µ?Ei0i1···in−1

Definition 6.1 We shall set i = {i0i1···in−1} and denote by Ωn

n tuples with ijtaking values in {1,2,··· ,D}.

The atoms of the partitions E[0,n−1]describe segments of trajectories up to time

n encoded by the atoms of E that are traversed at successive times; the volumes

µi = µ(Ei) corresponds to probabilities for the system to belong to the atoms

Ei0,Ei1,··· ,Ein−1at successive times 0 ? j ? n − 1. The n tuples i by themselves

provide a description of the system in a symbolic dynamic.

The richness in diverse trajectories, that is the degree of irregularity of the

motion (as seen with the accuracy of the given coarse-graining) correspond intu-

itively to our idea of “complexity” and can be better measured by the Shannon

entropy [39]

Sµ(E[0,n−1]) ≔ −

?·

(17)

Dthe set of Dn

?

i∈Ωn

D

µilogµi.

(18)

In the long run, E attributes to the dynamics an entropy per unit time–step

hµ(T,E) ≔ lim

n→∞

1

nSµ(E[0,n−1]) .

(19)

This limit is well defined [3] and the “average entropy production” hµ(T,E) mea-

sure how predictable the dynamics is on the coarse grained scale provided by the

finite partition E. To remove the dependence on E, the Kolmogorov–Sinai entropy

hKS

measurable partitions [3,39]:

µ(T) of (X,T,µ) (or KS–entropy) is defined as the supremum over all finite

hKS

µ(T) ≔ sup

E

hµ(T,E)

·

(20)

For the automorphisms of the 2-torus, we have the well-known result [3]:

Proposition 6.1 Let (Mµ,Θ,ωµ) be as in Definition 4.1, then hKS

µ(T) = logλ+.

Page 14

14

V. Cappellini

6.2. Quantum Dynamical Entropies

The idea behind the notion of dynamical entropy is that information can be ob-

tained by repeatedly observing a system in the course of its time evolution. Due

to the uncertainty principle, or, in other words, to non-commutativity, if observa-

tions are intended to gather information about the intrinsic dynamical properties

of quantum systems, then non-commutative extensions of the KS-entropy ought

first to decide whether quantum disturbances produced by observations have to be

taken into account or not.

Concretely, let us consider a quantum system described by a density matrix

ρ acting on a Hilbert space H. Via the wave packet reduction postulate, generic

measurement processes may be described by finite sets Y = {y1,y2,...,yD} of

bounded operators yj∈ B(H) such that?

the system caused by the corresponding measurement process:

jy∗

jyj= 1. These sets are called parti-

tions of unity (p.u., for sake of shortness) and describe the change in the state of

ρ ?→ Γ∗

Y(ρ) :=

?

j

yjρy∗

j.

(21)

It looks rather natural to rely on partitions of unity to describe the process of col-

lecting information through repeated observations of an evolving quantum sys-

tem [18]. Yet, most of these measurements interfere with the quantum evolution,

possibly acting as a source of unwanted extrinsic randomness. Nevertheless, the

effect is typically quantal and rarely avoidable. Quite interestingly, as we shall see

later, pursuing these ideas leads to quantum stochastic processes with a quantum

dynamical entropy of their own, the ALF-entropy, that is also useful in a classical

context.

An alternative approach [17] leads to the dynamical entropy of Connes, Narn-

hofer and Thirring [17](CNT–entropy). This approach lacks the operational appeal

of the ALF-construction, but is intimately connected with the intrinsic relaxation

properties of quantum systems [17, 40] and possibly useful in the rapidly grow-

ing field of quantum communication. The CNT-entropy is based on decomposing

quantum states rather than on reducing them as in (21). Explicitly, if the state ρ is

not a one dimensional projection, any partition of unity Y yields a decomposition

ρ =

?

j

Tr?ρy∗

jyj

?

√ρy∗

Tr?ρy∗

jyj√ρ

jyj?

·

(22)

When Γ∗

A different kind of wave packet reduction is the starting point for constructing

the coherent states entropy [21, 41] (in the following CS–entropy, for short), in

fact based on coherent states |CN(x)? as the ones introduced Definition 3.1.

Y(ρ) = ρ, reductions also provide decompositions, but not in general.

Page 15

A Survey on the Classical Limit of Quantum Dynamical Entropies

15

The map

I(E)(ρ) ≔ N

?

E

|CN(x)??CN(x)| ρ |CN(x)??CN(x)| µ(dx)

,

(23)

for a measurable subset E ⊂ X and an operator ρ, is called an instrument: it

describe the change in the state ρ of the system caused by an E–dependent mea-

surement process (compare with (21)), actually a double approximate measure-

ment in the phase space. Repeated measurement, taken stroboscopically during

the dynamical evolution and performed with different instrument I(Eij) labeled

by different elements Eijof a partition E, map the input state ρ into many possible

output { ρi| i ∈ Ωn

{ R+∋ ωi≔ ω(ρi) | i ∈ Ωn

correspondence between strings i ∈ Ωn

up with a probability space and a similar reasoning leading us in Section 6.1 to the

KS invariant, can now be used for constructing the CS–entropy.

D}, which in turn can be mapped into many positive numbers

D} summing up to one. Now we have once more the

Dand probability ωi, in other word we end

6.3. ALF–entropy

The idea underlying the ALF–entropy is that the evolution of a quantum dy-

namical system can be modeled by repeated measurements at successive equally

spaced times, the measurements corresponding to p.u. as in equation (21).

Such a construction associates a quantum dynamical system with a symbolic

dynamics corresponding to the right–shift along a quantum spin half–chain [42].

Generic p.u. Y = {y1,y2,...,yD} need not preserve the state, but disturbances

are kept under control by suitably selecting the subalgebra of observables M0∋ yj.

The construction of the ALF–entropy for a quantum dynamical system (M,Θ,ω)

can be resumed as follows:

• One selects a Θ–invariant subalgebra M0⊆ M and a p.u. Y = {y1,...,yD}

of finite size D with yj∈ M0. After j time steps Y will have evolved into

another p.u. from M0: Θj(Y) ≔ {Θj(y1),Θj(y2),...,Θj(yD)} ⊂ M0.

• Every p.u. Y of size D gives rise to an D–dimensional density matrix

ρ[Y]i,j≔ ω(y∗

jyi),

(24)

with von Neumann entropy Hω[Y] ≔ S(ρ[Y]) = −Tr

• Given two partitions of unit Y = {y1,y2,...,yD}, Z = {z1,z2,...,zB}, of

size D, respectively B, one gets a finer partition of unit of size BD as the set

?

ρ[Y]logρ[Y]

?

.

Y ◦ Z ≔ { y1z1,...,y1zB;y2z1,...,y2zB;...;yDz1,...,yDzB}·

(25)

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