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.
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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)
,
Page 10
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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