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\begin{document}
\title{Quantum maps}
\author{S\l awomir Klimek}
\address{Department of Mathematics\\
IUPUI\\
Indianapolis, IN\ 46202}
\email{sklimek@math.iupui.edu}
\thanks{The first author was supported in part by the National Science
Foundation
under grant DMS-9500463. The second author was supported in part by the
National Science Foundation under grant DMS-9424344}
\author{Andrzej Le\'{s}niewski}
\address{Department of Physics\\
Harvard University\\
Cambridge, MA 02138}
\email{alesniew@husc.harvard.edu}
\subjclass{Primary 47-06; Secondary 47A35, 47D45}

\begin{abstract}
We describe some results on quantization of discrete time dynamical systems
(``quantum maps''). We focus our attention on a number of examples including
the cat, Kronecker, and standard maps. Our main interest lies in studying
the ergodic properties of these quantum dynamical systems.
\end{abstract}
\maketitle

\section{Why Quantum Maps?}

In classical dynamics, systems with a discrete time variable are referred to
as \emph{maps}. In this talk, we will describe some mathematical results
concerning quantum maps (this term was coined in \cite{bbtv}), i.e. discrete
time quantum systems. Their time evolution is not governed by the
Schr\"{o}dinger equation; rather it is given by a discrete unitary group
acting on a Hilbert space.

There are several reasons for studying maps in classical and quantum
dynamics:

\begin{itemize}
\item  They arise as Poincar\'{e} section maps of flows;

\item  Often they are easier to study analytically;

\item  They serve as paradigms of various phenomena in ergodic theory;

\item  They are easier to simulate on a computer than flows;

\item  Interesting maps arise in applications, e.g. in statistical
mechanics, the theory of quantum computation and quantum information theory,
etc.
\end{itemize}

We will work within the operator algebra framework, as this is the natural
setup for addressing the structural issues of quantum dynamics. Other
approaches abound in the physics and mathematics literature, see e.g. \cite
{bbtv}, \cite{bievre}, \cite{bouz}, \cite{degli}, and references therein. We
shall focus on a somewhat restricted class of quantum dynamical systems,
namely those which arise as quantizations of classical maps.

\section{Classical dynamics}

Classical mechanics is formulated in terms of a phase space $M$ which is
usually assumed to be a symplectic manifold. Points $x=(q,p)$ on $M$
describe the state of the system. Their coordinates are canonical positions $%
q$ and canonical momenta $p$. Functions $f$ on $M$ represent classical
observables. The algebra $C^{\infty }\left( M\right) $ of smooth functions
on $M$ is equipped with a Lie structure given by the Poisson bracket $%
\left\{ \cdot ,\cdot \right\} $. There is a measure $d\mu \left( x\right) $
on $M$ which describes the distribution of states throughout $M$. It is used
to define the ensemble average of an observable $f\in C^{\infty }\left(
M\right) $:
\begin{equation}
\tau \left( f\right) =\int_{M}f\left( x\right) d\mu \left( x\right) .
\label{classensav}
\end{equation}
We will consider systems for which the total volume of the phase space is
finite, $\tau \left( M\right) =1$, so that $\mu $ has the meaning of a
probability distribution.

A map $T$ of the phase space $M$ to itself which

\begin{enumerate}
\begin{description}
\begin{itemize}
\item  is one to one,

\item  preserves the phase space volume $\mu \left( TA\right) =\mu \left(
A\right) $
\end{itemize}
\end{description}
\end{enumerate}

\noindent generates a discrete time dynamics. We think of $T$ as the
evolution of the system over one time unit. Powers of $T$, $T^{n}$ ($n$
integer), describe the evolution of the system over $n$ time units.

\noindent \textbf{Examples. }We take $M$ to be a torus. The volume element
is simply given by $d\mu (x)=dqdp$, and the observables are Fourier series
in $q$ and $p$. The ensemble average of $f$ is then equal to the term $f_{00%
\text{ }}$in the Fourier expansion of $f$.

\begin{enumerate}
\item  \emph{Baker's map:}
\begin{eqnarray*}
q &\longrightarrow &q^{^{\prime }}=\left\{
\begin{array}{lll}
2q & \text{if} & q<1/2; \\
2q-1 & \text{if} & q\geq 1/2,
\end{array}
\right.  \\
p &\longrightarrow &p^{^{\prime }}=\left\{
\begin{array}{lll}
p/2 & \text{if} & q<1/2; \\
(p+1)/2 & \text{if} & q\geq 1/2.
\end{array}
\right.
\end{eqnarray*}
Clearly, this map satisfies our requirements.

\item  \emph{Cat map:}
\begin{eqnarray*}
q &\longrightarrow &q^{^{\prime }}=aq+bp, \\
p &\longrightarrow &p^{^{\prime }}=cq+dp.
\end{eqnarray*}
Here $a,b,c,d$ are integers satisfying $ad-bc=1$. This condition guarantees
that the map preserves the phase space volume (and is one to one). We also
require that $|a+d|>2$. This means that the matrix
\begin{equation*}
\left(
\begin{array}{ll}
a & b \\
c & d
\end{array}
\right)
\end{equation*}
has two different real eigenvalues $\left| \mu _{1}\right| >1$, and $\left|
\mu _{2}\right| <1$. The cat map is expanding along the direction of the
eigenvector corresponding to $\mu _{1}$, and contracting along the direction
of the eigenvector corresponding to $\mu _{2}$.

\item  \emph{Kronecker's map: }\newline
This one is simply given by
\begin{eqnarray*}
q &\longrightarrow &q^{^{\prime }}=q+\alpha , \\
p &\longrightarrow &p^{^{\prime }}=p+\beta ,
\end{eqnarray*}
where $\alpha $ and $\beta $ are real numbers such that $1,\alpha ,\beta ,$
are linearly independent over $\mathbb{Z}$. This dynamics does not have
periodic orbits.

\item  \emph{Kicked maps:}\newline
These are maps of the following form:
\begin{eqnarray*}
p &\longrightarrow &p^{^{\prime }}=p+\gamma f\left( q\right) \qquad \left(
\text{mod }1\right) , \\
q &\longrightarrow &q^{^{\prime }}=q+p^{^{\prime }}\qquad \left( \text{mod }%
1\right) ,
\end{eqnarray*}
where $f(q)=\sum_{k\in \mathbb{Z}}f_{k}e^{2\pi ikq}$ is a continuous periodic
function satisfying the condition $\sum_{k\in \mathbb{Z}}k^{2}|f_{k}|<\infty $.
The choice $f(q)=\sin 2\pi q$ gives the \emph{standard map.}

\item  \emph{Harper's maps:}\newline
They are similar to kicked maps except that the ``kinetic part'' is
periodic:
\begin{eqnarray*}
p &\longrightarrow &p^{^{\prime }}=p+\gamma _{1}f\left( q\right) ,\qquad
\text{mod }1, \\
q &\longrightarrow &q^{^{\prime }}=q+\gamma _{2}v(p^{^{\prime }}),\qquad
\text{mod }1,
\end{eqnarray*}
where both $f,v$ are periodic and satisfy suitable regularity conditions.
\end{enumerate}

\section{Classical ergodicity}

The ergodic problem in classical mechanics consists in the following: What
can be learned about the (statistical) behavior of an ensemble of mechanical
(deterministic) systems from the long time behavior of an individual system?
Integrable systems do not exhibit any stochastic behavior as the motion
takes place along periodic trajectories. Hence, ergodicity is intimately
connected to classical non-integrability. We list below some fundamental
concepts and results of classical ergodic theory.

\begin{enumerate}
\item  \emph{Recurrence theorem }(Poincar\'{e}) If $U\subset M$ has positive
measure, then there is a subset $U_{0}$ of measure zero such that for each $%
x\in U\setminus U_{0}$ there is $k$ with the property that $T^{k}x\in U$

\item  \emph{Ergodic theorem} (Boltzmann, Birkhoff, von Neumann) states that
for each observable $f$%
\begin{equation*}
\text{time average of }f=\text{ ensemble average of }f\text{,}
\end{equation*}
or, for almost all initial conditions,
\begin{equation*}
\lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=0}^{n-1}f\left(
T^{k}x_{0}\right) =\int_{M}f\left( x\right) d\mu \left( x\right)
\end{equation*}
($f$ is an observable). This theorem holds for systems which have the
following property: No non-trivial subset of the phase space is invariant
under the dynamics. Baker's maps, Kronecker's map, and the cat map are all
ergodic. The ergodicity of the kicked and Harper's maps is a more
complicated issue. For small values of the parameter $\gamma $, they are not
ergodic (a consequence of the KAM theorem). For large values of $\gamma $,
the ``islands of ergodicity'' are getting smaller and smaller. No theorems
are known, and the numerical evidence is inconclusive.

\item  Stronger than ergodicity is the \emph{mixing property}: A system is
mixing if
\begin{equation*}
\lim_{n\rightarrow \infty }\mu \left( A\cap T^{n}B\right) =\mu \left(
A\right) \mu \left( B\right) .
\end{equation*}
This property means that the dynamics spreads the set of initial states
uniformly throughout the phase space. Not all ergodic systems are mixing.
For instance, Kronec$\ker $'s map is ergodic but not mixing. The mixing
property is equivalent to the following fact about the long time behavior of
the ensemble average of a product of observables:
\begin{equation*}
\lim_{n\rightarrow \infty }\int f\left( x\right) g\left( T^{n}x\right) d\mu
\left( x\right) =\int f\left( x\right) d\mu \left( x\right) \,\int g\left(
x\right) d\mu \left( x\right) .
\end{equation*}

\item  Kolmogorov-Sinai (KS) entropy measures how strongly mixing is the
system. It is constructed as follows. Cover the phase space $M$ with a
measurable covering $\mathcal{A}=\left\{ A_{1},\ldots ,A_{k}\right\} $. With
this partition we associate its Shannon entropy:
\begin{equation*}
S\left( \mathcal{A}\right) =-\sum_{j=1}^{k}\mu \left( A_{j}\right) \log \mu
\left( A_{j}\right) .
\end{equation*}
Let us now see what happens to this covering if we wait one time unit. A new
partition of $M$ arises, this time given by $T\mathcal{A}=\left\{
TA_{1},\ldots ,TA_{k}\right\} $. To measure the resulted mixing of the phase
space we compute the Shannon entropy of the partition $\mathcal{A}\vee T%
\mathcal{A}$ obtained by intersecting the elements of the original partition
with the elements of the new partition. Keep on doing it. The limit
\begin{equation*}
S\left( T,\mathcal{A}\right) =\lim_{n\rightarrow \infty }\frac{1}{n}S\left(
\mathcal{A}\vee T\mathcal{A}\vee \ldots \vee T^{n-1}\mathcal{A}\right)
\end{equation*}
exists, and its supremum over all choices of the initial partition
\begin{equation*}
S\left( T\right) =\sup_{\mathcal{A}}S\left( T,\mathcal{A}\right)
\end{equation*}
is called the KS entropy. The KS entropy is a measure of chaos in a system as
\end{enumerate}

\begin{itemize}
\item  it is zero for periodic systems;

\item  it is zero for ergodic but not mixing systems (e.g. Kronecker's
dynamics);

\item  it is related to the Lyapunov exponents (Pesin's theorem).
\end{itemize}

For Kronecker's map the KS entropy is zero, for bakers map,
\begin{equation*}
S\left( T_{bak}\right) =\log 2,
\end{equation*}
while for the cat map,
\begin{equation*}
S\left( T_{cat}\right) =\log \left| \mu _{1}\right| .
\end{equation*}
For kicked and Harper's maps, the KS entropy is unknown.

\section{Quantum mechanics}

\subsection{Quantization}

In quantum mechanics, the commutative world of classical mechanics is
replaced by the non-commutative world of operators on Hilbert spaces
(Heisenberg, Born, Jordan, Schr\"{o}dinger, Dirac, von Neumann,...). The
quantum phase space is no longer a set of points. Rather, it is a
non-commutative space defined in terms of a non-commutative algebra of
observables. In the simplest case of the quantized flat space, this algebra
is generated by the canonical position and momentum operators.

Quantization of a dynamical system has two components: kinematic and
dynamic. The kinematic component involves the construction of a suitable
quantized phase space of the system. This quantized phase space is given in
terms of a non-commutative algebra $\mathfrak{A}_{\hslash }$ of observables. In
the language of non-commutative geometry, $\mathfrak{A}_{\hslash }$ is an
algebra of functions on the quantized phase space. Specific choices of the
structure of $\mathfrak{A}_{\hslash }$ can be made: a
$\mathbb{C}^{*}$-algebra, a
von Neumann algebra, or some suitably defined locally convex algebra.
Throughout this talk, we will assume that $\mathfrak{A}_{\hslash }$ is a von
Neumann algebra with a countable predual. In other words,
$\mathfrak{A}_{\hslash
}$ acts on a separable Hilbert space, an assumption usual made in physics. A
classical observable $f$ is mapped onto a quantum observable $Q_{\hbar
}(f)\in \mathfrak{A}_{\hslash }$.

The ``suitability'' of the choices made, namely that of the algebra
$\mathfrak{A}%
_{\hslash }$ and of the time evolution, is settled by the correspondence
principle. This amounts to showing that limits of the quantized objects, as $%
\hslash \rightarrow 0$, yield the corresponding classical objects.
Quantization is a highly non-unique procedure, and the correspondence
principle is the only physical principle allowing one to decide whether a
particular procedure is correct. A natural mathematical framework for
quantization is ``strict deformation quantization'' proposed by Rieffel \cite
{rieffel1}. The key requirement is that
\begin{equation*}
\lim_{\hbar \rightarrow 0}\left\| \frac{1}{i\hbar }\left[ Q_{\hbar
}(f),Q_{\hbar }(g)\right] -Q_{\hbar }(\{f,g\})\right\| _{\hbar }=0,
\end{equation*}
where $f,g\in C^{\infty }\left( M\right) $.

The dynamic component of quantization consists in defining a time evolution
on the quantized phase space. A natural way of doing this is to find a
suitable one parameter group of automorphisms $\alpha _{t}$ of $\mathfrak{A}%
_{\hslash }$, where the parameter (discrete or continuous) has the meaning
of time. For maps, $\alpha _{n}=\alpha ^{n}$. In examples, $\alpha $ is
often implemented by a unitary operator $F$, $\alpha \left( O\right)
=F^{-1}OF$. The correspondence principle takes the form of the requirement
\begin{equation*}
\lim_{\hbar \rightarrow 0}\left\| F^{-n}Q_{\hbar }(f)F^{n}-Q_{\hbar }(f\circ
T^{n})\right\| _{\hbar }=0.
\end{equation*}
Statements of this kind are similar to Egoroff's theorem in the theory of
pseudodifferential operators.

The ensemble average of a quantum system is given by a state $\tau _{\hbar }$
over the algebra $\mathfrak{A}_{\hslash }.$ For technical reasons, we will
assume that this state is faithful and normal. Physically, this means that
an ensemble average is given by a density matrix whose pure components form
a separating set for $\mathfrak{A}_{\hslash }$.

\begin{definition}
A quantum map is a triple $\left( \mathfrak{A}_{\hslash },\alpha ,\tau _{\hbar
}\right) $ arising as a quantization of a discrete time dynamical system in
the sense described above.
\end{definition}

This definition is somewhat tentative, and we make it here merely for the
sake of convenience. We leave out, for example, the issue of whether each
meaningful quantum system arises as a quantization of a classical system.

\subsection{Flat space}

In the case of a flat space, we choose to work with the Bargmann
representation of the Hilbert space of states, i.e. the space of analytic
functions $\varphi \left( z\right) ,\psi \left( z\right) $, with an inner
product
\begin{equation*}
\left\langle \varphi ,\psi \right\rangle =\int_{\mathbb{C}}\overline{\varphi
\left( z\right) }\psi \left( z\right) \,d\mu _{\hbar }\left( z\right) ,
\end{equation*}
where $d\mu _{\hbar }\left( z\right) =\frac{\hbar }{\pi }\,e^{-\left|
z\right| ^{2}/\hbar }d^{2}z$. We denote this Hilbert space by $\mathcal{H}%
^{2}\left( \mathbb{C}\right) .$ It carries a projective unitary representation
of the group of translations $z\rightarrow U\left( z\right) $:
\begin{equation*}
U\left( z\right) \varphi \left( w\right) =e^{-\left| z\right| ^{2}/2\hbar +%
\overline{z}w/\hbar }\varphi \left( w-z\right) .
\end{equation*}
For concreteness, the quantization map $Q_{\hbar }$ is taken to be the
Toeplitz quantization. As a suitable class of symbols for the Toeplitz
operators one may take almost periodic functions on the plane \cite{bc1},
\cite{bc2}.

\subsection{Quantum torus}

We study quantized discrete time systems whose classical phase space is a
torus. A toroidal phase space can be quantized by replacing classical
functions by unitary operators $U=e^{2\pi iq}=U\left( -i\sqrt{2}\pi \hbar
\right) $ and $V=e^{2\pi ip}=U\left( \sqrt{2}\pi \hbar \right) $ acting on
the Hilbert space $\mathcal{H}^{2}\left( \mathbb{C}\right) $ defined above.
They satisfy the following commutation relation
\begin{equation*}
UV=e^{4\pi ^{2}\hbar i}VU.
\end{equation*}
Since the torus is compact, Planck's constant must obey an integrality
condition $\hbar =1/2\pi N$, $N$ positive integer. The algebra of
observables $\mathfrak{A}_{\hslash }$ is defined as the von Neumann algebra
generated by $U,V$. For a summary of results concerning this algebra (with
and without the integrality condition), see \cite{rieffel2}. The ensemble
average of an observable $O$ is given by the following state on $\mathfrak{A}%
_{\hslash }$:
\begin{equation*}
\tau _{\hbar }\left( O\right) =\int_{\mathbb{T}^{2}}\left\langle \varphi
,U\left( z\right) ^{\dagger }OU\left( z\right) \varphi \right\rangle d^{2}z,
\end{equation*}
where $\varphi $ is an arbitrary normalized element of $\mathcal{H}%
^{2}\left( \mathbb{C}\right) $. It is well known that $\tau _{\hbar }$ is, in
fact, a faithful normal trace on $\mathfrak{A}_{\hslash }$. As $\hbar
\rightarrow 0$, this trace reproduces the classical ensemble average given
by (\ref{classensav}). It is characterized by the property that
\begin{equation*}
\tau _{\hbar }\left( U^{m}V^{n}\right) =\delta _{m0}\delta _{n0}.
\end{equation*}

\begin{theorem}
\cite{rieffel1}, \cite{rieffel2} This algebra satisfies the conditions of
strict deformation quantization.
\end{theorem}

We introduce the following notation:
\begin{equation*}
X=U\left( -i/\sqrt{2}\right) ,\quad Y=U\left( 1/\sqrt{2}\right) ,
\end{equation*}
and observe that
\begin{equation*}
\left[ X,\ Y\right] =0.
\end{equation*}
The operators $X$ and $Y$ generate an action of the group $\mathbb{Z}^{2}$
on $%
\mathcal{H}^{2}\left( \mathbb{C},d\mu _{\hslash }\right) $. We also verify
easily that,
\begin{eqnarray*}
\left[ X,\ Y\right] &=&0,\quad \left[ X,\ V\right] =0, \\
\left[ Y,\ U\right] &=&0,\quad \left[ Y,\ V\right] =0,
\end{eqnarray*}
and so $X$ and $Y$ are in the commutant of $\mathfrak{A}_{\hslash }$. Also,
\begin{equation*}
X=U^{N},\quad Y=V^{N}
\end{equation*}
We shall call a holomorphic function $\phi $ on $\mathbb{C}$ a
$\mathbb{Z}^{2}$-%
\textit{automorphic form} if
\begin{eqnarray*}
X\phi \left( z\right) &=&e^{2\pi i\theta _{1}}\phi \left( z\right) , \\
Y\phi \left( z\right) &=&e^{2\pi i\theta _{2}}\phi \left( z\right) ,
\end{eqnarray*}
where $\theta =(\theta _{1},\ \theta _{2})\in \mathbb{T}^{2}$. In other
words, $%
\mathbb{Z}^{2}$-automorphic forms are simultaneous generalized eigenvectors
of $%
X$ and $Y$. Let $\mathcal{H}_{\hslash }\left( \theta \right) $ denote the
space of all $\mathbb{Z}^{2}$-automorphic forms with fixed $\theta $.
Clearly, $%
\phi \in \mathcal{H}_{\hslash }\left( \theta \right) $ is uniquely
determined once defined on the fundamental domain $D=\left[ 0,\ 1\right]
\times \left[ 0,\ 1\right] \subset \mathbb{R}$. The space $\mathcal{H}_{\hslash
}\left( \theta \right) $ has a natural inner product defined as an integral
over this domain:
\begin{equation*}
\left\langle \phi _{1},\ \phi _{2}\right\rangle =\int_{D}\overline{\phi
_{1}(z)}\phi _{2}\left( z\right) d\mu _{\hslash }\left( z\right) .
\end{equation*}
(Note a similar integral over the entire complex plane \textit{does not
converge}, hence the $\mathbb{Z}^{2}$-automorphic forms are not in
$\mathcal{H}%
^{2}\left( \mathbb{C}\right) $.) This inner product is a $\mathbb{Z}^{2}$
version
of the Petersson inner product. In the following theorem we construct a
natural orthonormal basis for the space $\mathcal{H}_{\hslash }\left( \theta
\right) $.

\begin{theorem}
$\left( 1\right) $ The following functions are elements of $\mathcal{H}%
_{\hslash }\left( \theta \right) $:
\begin{eqnarray*}
\phi _{m}^{\left( \theta \right) }\left( z\right) =C_{m}\left( \theta
\right) e^{-N\pi z^{2}+2\sqrt{2}\pi \left( \theta _{1}+m\right) z} \\
\sum_{k\in \mathbb{Z}}e^{-N\pi k^{2}-2\pi \left( \theta _{1}+i\theta
_{2}+m\right) k+2\sqrt{2}N\pi kz},
\end{eqnarray*}
where
\begin{equation*}
C_{m}(\theta )=(2/N)^{1/4}e^{-\pi (\theta _{1}+m)^{2}/N-2\pi i\theta
_{2}m/N}.
\end{equation*}
They are periodic in $m$,
\begin{equation*}
\phi _{m+N}^{\theta }=\phi _{m}^{\theta },
\end{equation*}
and furthermore,
\begin{equation*}
\phi _{0}^{\theta },\ldots ,\phi _{N-1}^{\theta }
\end{equation*}
are orthonormal vectors in $\mathcal{H}_{\hslash }\left( \theta \right) $.%
\newline
$\left( 2\right) $ The space $\mathcal{H}_{\hslash }\left( \theta \right) $
has dimension $N$. Consequently, the functions $\phi _{n}^{\theta
},\,n=0,\ldots ,N-1,$ form an orthonormal basis for $\mathcal{H}_{\hslash
}\left( \theta \right) $.\newline
$\left( 3\right) $ There is an isomorphism
\begin{equation*}
\kappa :\,\mathcal{H}\left( \mathbb{C}\right) \longrightarrow \int_{\mathbb{T}%
^{2}}^{\oplus }\mathcal{H}_{\hslash }\left( \theta \right) \,d\theta ,
\end{equation*}
such that
\begin{eqnarray*}
\kappa U\kappa ^{-1}\phi _{m}\left( \theta ,z\right) =e^{2\pi i\left( \theta
_{1}+m\right) /N}\phi _{m}\left( \theta ,z\right) , \\
\kappa V\kappa ^{-1}\phi _{m}\left( \theta ,z\right) =e^{2\pi i\theta
_{2}/N}\phi _{m}\left( \theta ,z\right) .
\end{eqnarray*}
\end{theorem}

This is the kinematic part of quantization of the torus.

\subsection{More complicated geometries}

Phase spaces of more complicated geometry can be quantized in an analogous
way. Various techniques have been developed (geometric quantization,
deformation quantization, Toeplitz quantization, ...). Explicit
constructions are known, for example, for compact and non-compact Hermitian
symmetric spaces, Riemann surfaces, compact Kahler manifolds, a large class
of Hermitian symmetric supermanifolds, etc. An approach based on a
holomorphic representation was initiated in \cite{berezin} and has been
developed by a number of authors.

\subsection{Quantization of maps}

Now, we quantize some of the the toroidal maps introduced before.

$\left( 1\right) $ The \emph{cat map} is quantized by means of a single time
unit evolution operator $F$ on Bargmann space such that
\begin{eqnarray*}
U &\longrightarrow &U^{^{\prime }}=F^{-1}UF=e^{2\pi ^{2}iab\hbar }U^{a}V^{b},
\\
V &\longrightarrow &V^{^{\prime }}=F^{-1}VF=e^{2\pi ^{2}icd\hbar }U^{c}V^{d}.
\end{eqnarray*}
This defines an automorphism of $\mathfrak{A}_{\hslash }$. The evolution
operator $F$ can be written down explicitly in terms of Gauss sums.

\begin{theorem}
\cite{klmr}The matrix elements of the operator $F$,
\begin{equation*}
\left\langle \phi _{m}^{\left( \theta ^{*}\right) },F\phi _{n}^{\left(
\theta \right) }\right\rangle =\int_{D}\overline{\phi _{m}^{\left( \theta
^{*}\right) }\left( z\right) }\,F\phi _{n}^{\left( \theta \right) }\left(
z\right) \,d\mu _{\hbar }\left( z\right) ,
\end{equation*}
where $\theta ^{*}=T^{-1}\theta -\left( Nbd/2,Nac/2\right) $, is given by
\begin{eqnarray*}
\left\langle \phi _{m}^{\left( \theta ^{*}\right) },F\phi _{n}^{\left(
\theta \right) }\right\rangle =\left( Nb\right) ^{-1/2}e^{i\nu /2}e^{2\pi
i\left( m\widetilde{\theta }_{2}-n\theta _{2}\right) /N} \\
\sum_{r=0}^{|b|-1}e^{-2\pi ir\theta _{2}}e^{i\pi \Phi \left( m+\widetilde{%
\theta }_{1},n+Nr+\theta _{1}\right) },
\end{eqnarray*}
where $e^{i\nu }=-i\alpha /|\alpha |$, and where
\begin{equation*}
\Phi \left( x,y\right) =ax^{2}-2xy+dy^{2}.
\end{equation*}
\end{theorem}

$\left( 2\right) $ \emph{Kronecker's map} is easy to quantize \cite{klmr}:
\begin{eqnarray*}
U &\longrightarrow &U^{^{\prime }}=F^{-1}UF=e^{2\pi i\alpha }U \\
V &\longrightarrow &V^{^{\prime }}=F^{-1}VF=e^{2\pi i\beta }V
\end{eqnarray*}
One can write down explicit expressions for $F$ in the representation given
by
\begin{equation*}
\left\langle \phi _{m}^{\left( \widetilde{\theta }\right) },F\phi
_{n}^{\left( \theta \right) }\right\rangle =e^{2\pi i\beta \left( \theta
_{1}-N\alpha /2\right) }\delta _{mn}.
\end{equation*}

$\left( 3\right) $ Quantum \emph{kicked maps }are given by the following
automorphism of $\mathfrak{A}_{\hbar }$ \cite{lrs}:
\begin{eqnarray*}
U &\longrightarrow &U^{^{\prime }}=F^{-1}UF=e^{-2\pi ^{2}iab\hbar
}V^{^{\prime }}U, \\
V &\longrightarrow &V^{^{\prime }}=F^{-1}VF=Ve^{2\pi i\gamma \widetilde{f}%
\left( U\right) },
\end{eqnarray*}
where
\begin{equation*}
\widetilde{f}\left( U\right) =\sum \frac{1-e^{-4\pi ^{2}\hbar ik}}{4\pi
^{2}\hbar ki}f_{k}U^{k}.
\end{equation*}

$\left( 4\right) $ Quantization of Harper's maps is similar, and I will skip
the details.

The quantum dynamics defined above obey the correspondence principle.

\begin{theorem}
For the quantum cat, Kronecker, kicked, and Harper's maps,
\begin{equation*}
\lim_{\hbar \rightarrow 0}\left\| F^{-n}Q_{\hbar }(f)F^{n}-Q_{\hbar }(f\circ
T^{n})\right\| _{\hbar }=0.
\end{equation*}
\end{theorem}

\section{Quantum ergodicity}

\subsection{Quantum recurrence}

Much of classical ergodic theory can be extended to the quantum mechanical
context. The first fundamental result is the Poincar\'{e} recurrence
theorem. The classical recurrence theorem states that the state of a system
returns arbitrarily close to the initial point if one waits sufficiently
long. There is a simple quantum analog of this theorem. Let $\psi _{1},\psi
_{2}\ldots ,$ be a sequence of normalized vectors in the Hilbert space of
states, and let
\begin{equation*}
\rho =\sum_{n}p_{n}\rho _{n}
\end{equation*}
be the corresponding density matrix. Here $\rho _{n}=P_{\psi _{n}}$ is the
projection operator onto the vector $\psi _{n}$. It is easiest to state the
quantum recurrence theorem for the case of flows rather than maps.

\begin{theorem}
Let $\rho $ be a density matrix, and let $F$ have a purely discrete
spectrum. Then for any $\epsilon >0$ there is $T=T\left( \epsilon \right) >0$
such that every interval of length $T$ contains at least one $\tau $ with
the property that $\left\| \rho \left( t\right) -\rho \right\| _{HS}\leq
\epsilon .$
\end{theorem}

This theorem merely states that the function $t\longrightarrow \mathrm{tr}%
\left( \rho \left( t\right) \rho \right) $ is almost periodic. It is a
slight extension of a theorem proved in the fifties by Bocchieri and Loigner
\cite{bocchieri} for the case of pure states.

\subsection{Ergodicity and mixing}

We define the Hilbert space $\mathcal{K=}L^{2}\left( \mathfrak{A},\tau
\right) $
associated with the algebra $\mathfrak{A}$ as the completion of
$\mathfrak{A}$ in
the norm given by the inner product $\left( A,B\right) =\tau \left(
A^{\dagger }B\right) $. It is natural to regard this space as the quantum
version of the Koopman Hilbert space, as it reduces to the latter in the
classical case. We would like to emphasize that the analogy often drawn in
the literature between the classical Koopman space and the quantum
mechanical Hilbert space of states $\mathcal{H}$ is misleading: it is the
space $\mathcal{K}$ that is a natural scene for quantum ergodic theory. As a
consequence of the time invariance of $\tau $, $\alpha $ defines a unitary
operator on $\mathcal{K}$ which we will continue to denote by the same
symbol. This operator is the quantum version of the classical Koopman
operator.

A quantum map $\left( \mathfrak{A},\alpha ,\tau \right) $ is called:\newline
\emph{mixing}, if for all $A,B\in \mathfrak{A}$,
\begin{equation*}
\lim_{N\rightarrow \infty }\tau \left( \alpha ^{n}\left( A\right) B\right)
=\tau \left( A\right) \tau \left( B\right) ;
\end{equation*}
\emph{weak mixing}, if for all $A,B\in \mathfrak{A}$,
\begin{equation*}
\lim_{N\rightarrow \infty }\frac{1}{N}\sum_{0\leq n\leq N-1}\left| \tau
\left( \alpha ^{n}\left( A\right) B\right) -\tau \left( A\right) \tau \left(
B\right) \right| ^{2}=0;
\end{equation*}
\emph{ergodic}, if for all $A\in \mathfrak{A},$%
\begin{equation*}
\lim_{N\rightarrow \infty }\frac{1}{N}\sum_{0\leq n\leq N-1}\alpha
^{n}\left( A\right) =\tau \left( A\right) I,
\end{equation*}
strongly on $\mathcal{K}$.

For quantum maps, we have the usual hierarchy: mixing $\Rightarrow $ weak
mixing $\Rightarrow $ ergodicity.

Ergodic, weakly mixing and mixing systems can be characterized in terms of
the properties of the spectrum of the automorphism $\alpha $. We will say
that $\alpha $ has continuous spectrum if $1$ is its only eigenvalue and the
corresponding eigenvectors are the multiples of the identity operator.

\begin{theorem}
\begin{itemize}
\item[(i)]  A quantum map is ergodic if and only if $1$ is an eigenvalue of $%
\alpha $ and the corresponding eigenvectors are multiples of $I$;

\item[(ii)]  A quantum map is weakly mixing if and only if the spectrum of $%
\alpha $ is continuous;

\item[(iii)]  A weakly mixing quantum map is mixing if the spectrum of $%
\alpha $ is absolutely continuous.
\end{itemize}
\end{theorem}

Hence, quantum maps for which $\alpha $ has pure point spectrum cannot be
mixing.

\begin{theorem}
\begin{itemize}
\item[(i)]  Quantum Kronecker's dynamic is ergodic but not mixing;

\item[(ii)]  Quantum cat dynamics is mixing.
\end{itemize}
\end{theorem}

\noindent In fact, quantum Kronecker's maps are uniformly ergodic \cite{klmr}%
.

\subsection{Connes-Stormer entropy (quantum KS entropy)}

Given an algebra of observables $\mathfrak{A}$, a faithful normal trace
$\tau $,
and an automorphism $\alpha $, there is a construction of an entropy
associated with mixing of the quantum phase space resulting from the time
evolution. This entropy, denoted here by $H(\alpha )$, is called the
Connes-Stormer (CS) entropy. The construction of the CS entropy is, roughly,
parallel to the construction of the KS entropy. The CS entropy is a measure
of chaos in a quantum dynamical system, very much like the KS entropy is a
measure of chaos in a classical system.

For simple dynamics, like the cat, Kronecker, and baker's dynamics, the CS
entropy can be calculated explicitly. The result is that the quantum entropy
equals the classical entropy \cite{kl1}.

\begin{theorem}
The CS entropies of the quantized cat, Kronecker's and baker's maps are
equal to the KS entropies of the corresponding classical dynamics.
\end{theorem}

This means that, in these systems, chaotic behavior persists quantization.

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\end{document}