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\font\BF=cmbx10 scaled \magstep 3
\line{\hfill Preprint KUL-TF-94/2}
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\centerline{\BF Defining Quantum Dynamical Entropy}
\vskip 30pt plus30pt
\centerline{R.~Alicki$^{1,2}$ and M.~Fannes$^{1,3}$}
\vskip 12pt
\centerline{\tt fgbda26@blekul11.bitnet \qquad fgbda20@blekul11.bitnet}
\vskip 80pt plus80pt
\noindent {\bf Abstract}\hfill\break
We propose an elementary definition of the dynamical entropy for a
discrete-time quantum dynamical system. We apply our construction to
classical dynamical systems and to the shift on a quantum spin chain. In
both cases the expected results are obtained.
\noindent {\bf Mathematics Subject Classification (1991):
46L55, 28D20, 82B10}
\vskip 80pt plus 80pt
\vfootnote1
{Inst. Theor. Fysica, Universiteit Leuven, B-3001 Leuven, Belgium}
\vfootnote2
{On leave of absence from the Institute of Theoretical Physics
and Astrophysics,}
\vfootnote{}
{University of Gda\'nsk, PL-80-952 Gda\'nsk, Poland}
\vfootnote3
{Onderzoeksleider, N.F.W.O. Belgium}
\vfill\eject
\beginsection 1. Introduction
We propose a new definition of dynamical entropy for discrete-time quantum
dynamical systems. The basic ingredient is the notion of finite partition
of unity. Such partitions can be evolved with the dynamics and composed
among themselves to yield finer and finer, but still finite, partitions. A
finite partition, together with the dynamics, can be considered as a
symbolic dynamics modeling the quantum dynamical system. We can then use
the standard notion of von~Neumann entropy as a measure of the dynamical
entropy. This is very reminiscent of the classical situation where finer
and finer partitions of the ``phase space'' of the system are generated by
the dynamics. The Kolmogorov-Sinai entropy then measures the asymptotic
entropy production by the dynamics [1,2].
Several constructions of quantum dynamical entropy have been considered. It
is well-known that a truly quantum mechanical dynamics behaves badly with
respect to locality \eg a finite dimensional subalgebra together with its
single-step evolved algebra will generically generate an infinite
dimensional algebra. A first attempt to overcome this difficulty was made
in [3]. The basic object was a finite dimensional subalgebra of the
centralizer of the invariant state, which is, unfortunately, in most cases
limited to the scalars. A general theory, together with quite a number of
applications, has been developed, by mapping the system onto classical
models and computing the dynamical entropy as the supremum of the classical
entropies [4--6]. This approach is technically very involved and in order
to apply it to specific models, it is often necessary to satisfy strong
conditions.
Decompositions of unity, similar to our's, appeared in [7]. This proposal
however, assumed strong time invariance conditions on the partition,
producing hereby a severe conflict between locality and invariance.
In section~2, we introduce the notion of partition of unity and of
dynamical entropy $H$. We also describe the symbolic dynamics generated by
the partition. The aim of section~3 is to show that, for classical
dynamical systems, $H$ is the Kolmogorov-Sinai invariant. As a second
application we consider the shift on a quantum spin chain with the $d\times
d$ matrices as single site observables and show that, up to a term $\log
d$, $H$ coincides with the usual von~Neumann mean entropy of a shift
invariant state.
\beginsection 2. Defining dynamical entropy
A discrete-time dynamical system consists in a C*-algebra $\A$, an
automorphism $\Theta$ of $\A$ and a state $\om$ on $\A$ which is left
invariant by the dynamics: $\om\circ\Theta=\om$. We will also need to
specify a unital $\ast$-subalgebra $\A_0$ of $\A$, globally invariant under
$\Theta$. The elements of $\A_0$ will play the role of smooth or local
elements. In the examples of sections~3 and 4 a natural choice for $\A_0$
is obvious. We don't know however how to specify $\A_0$ for a completely
general system.
A {\it finite partition of size\/} $k$ {\it of unity\/} is a set
$\X=\{x_1,x_2,\ldots x_k\}$ of elements of $\A_0$ satisfying:
$$\sum_{i=1}^k x_i^*x_i = \idty. \eqno(2.1)$$
We can compose two partitions $\X=\{x_1,x_2,\ldots x_k\}$ and
$\Y=\{y_1,y_2,\ldots y_\ell\}$ to get a new partition $\X\circ\Y =
\{x_iy_j\mid i=1,2,\ldots k,\ \ j=1,2,\ldots\ell\}$. Also, if $\X$ is a
partition of unity $\Theta(\X)=\{\Theta(x_1),\Theta(x_2),\ldots
\Theta(x_k)\}$ is again a partition.
To any partition $\X$ of size $k$ we associate a $k\times k$ density
matrix $\rho[\X]$ with $(i,j)$ matrix elements
$$\rho[\X]_{i,j}= \om(x_j^*x_i), \quad i,j=1,2,\ldots k. \eqno(2.2)$$
We now define the dynamical entropy in terms of the von~Neumann entropy of
the density matrices generated by the partition $\X$ and the shift
$\Theta$. More precisely, let $\X$ be any finite partition.
$H_{(\om,\Theta)}(\X)$ is defined as:
$$\eqalign{
H_{(\om,\Theta)}(\X)
&= \limsup_m\ {1\over m} S(\rho[\Theta^{m-1}(\X) \circ\cdots \Theta(\X)
\circ \X]) \cr
&= \limsup_m\ {1\over m} \tr \eta\left(\rho[\Theta^{m-1}(\X) \circ\cdots
\Theta(\X) \circ \X]\right) \leq \log k. } \eqno(2.3)$$
$\eta$ is
the standard entropy function on $[0,1]$:
$$\eta(t) = \cases{-t\log t &for
$0