\overfullrule=0pt
\magnification 1200
\baselineskip=12pt
\hsize=15.3truecm \vsize=22 truecm \hoffset=.1truecm
\parskip=14 pt

% BLACKBOARD BOLD
\def\idty{{\leavevmode{\rm 1\ifmmode\mkern -5.4mu\else
                                            \kern -.3em\fi I}}}
\def\Ibb #1{ {\rm I\ifmmode\mkern -3.6mu\else\kern -.2em\fi#1}}
\def\Ird{{\hbox{\kern2pt\vbox{\hrule height0pt depth.4pt width5.7pt
    \hbox{\kern-1pt\sevensy\char"36\kern2pt\char"36} \vskip-.2pt
    \hrule height.4pt depth0pt width6pt}}}}
\def\Irs{{\hbox{\kern2pt\vbox{\hrule height0pt depth.34pt width5pt
       \hbox{\kern-1pt\fivesy\char"36\kern1.6pt\char"36} \vskip -.1pt
       \hrule height .34 pt depth 0pt width 5.1 pt}}}}
\def\Ir{{\mathchoice{\Ird}{\Ird}{\Irs}{\Irs} }}
\def\ibbt #1{\leavevmode\hbox{\kern.3em\vrule
     height 1.5ex depth -.1ex width .2pt\kern-.3em\rm#1}}
\def\ibbs#1{\hbox{\kern.25em\vrule
     height 1ex depth -.1ex width .2pt
                   \kern-.25em$\scriptstyle\rm#1$}}
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     height .7ex depth -.1ex width .2pt
                   \kern-.22em$\scriptscriptstyle\rm#1$}}
\def\ibb#1{{\mathchoice{\ibbt #1}{\ibbt #1}{\ibbs #1}{\ibbss #1}}}
\def\Nl{{\Ibb N}} \def\Cx {{\ibb C}} \def\Rl {{\Ibb R}}

% THEOREMS  : allow items in proclaim
\def\lessblank{\parskip=5pt \abovedisplayskip=2pt
          \belowdisplayskip=2pt }
\outer\def\iproclaim #1. {\vskip0pt plus20pt \par\noindent
     {\bf #1.\ }\begingroup \interlinepenalty=250\lessblank\sl}
\def\eproclaim{\par\endgroup\vskip0pt plus10pt\noindent}
\def\proof#1{\par\noindent {\bf Proof #1}\          % Use as "\proof:"
         \begingroup\lessblank\parindent=0pt}
\def\QED {\hfill\endgroup\break
     \line{\hfill{\vrule height 1.8ex width 1.8ex }\quad}
      \vskip 0pt plus20pt}

% OPERATORS
\def\cfc{C*-finitely correlated}
\def\cp{completely positive}
\def\eg{e.g.\ }
\def\gx{{\Gamma_\X}}
\def\gtx{\Gamma_{(\X,\Theta)}}
\def\id{\mathop{\rm id}\nolimits}
\def\ie{i.e.\ }
\def\ot{{\otimes}}
\def\tr{\mathop{\rm Tr}\nolimits}

% LETTERS
\def\A{{\cal A}}
\def\C{{\bf C}}
\def\cha{\raise.5ex\hbox{$\chi$}}
\def\D{{\bf D}}
\def\F{{\cal F}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\om{\omega}
\def\X{{\bf X}}
\def\Y{{\bf Y}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\font\BF=cmbx10 scaled \magstep 3
\line{\hfill Preprint KUL-TF-94/2}
\voffset=2\baselineskip
\hrule height 0pt
\vskip 80pt plus80pt

\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 $[0,1]$:
$$\eta(t) = \cases{-t\log t &for
$0<t\leq1$;  \cr 0      &for  $t=0$.\cr} \eqno(2.4)$$
The {\it dynamical entropy} $H_{(\om,\Theta)}$ is obtained by taking the
supremum over all partitions of unity:
$$H_{(\om,\Theta)} = \sup_\X\ H_{(\om,\Theta)}(\X). \eqno(2.5)$$

We will now give an interpretation of $\rho[\Theta^{m-1}(\X) \circ\cdots
\Theta(\X) \circ \X]$ in terms of either a quantum stochastic process on
$\M_k$ or of a spin chain in quantum statistical mechanics. Therefore the
construction of above can be considered to be a symbolic dynamics for
$(\A,\Theta,\om)$ with the shift as single step time evolution. So, choose
a fixed partition $\X$ of size $k$ of the identity and let $\M_k$ denote
the algebra of complex $k\times k$ matrices. An arbitrary element
$A\in\M_k\ot\A$ is in fact a $k\times k$ matrix $[a_{ij}]$ with entries in
$\A$. We can now consider the map
$$\gx: \M_k\ot\A\to\A: A\mapsto \sum_{i,j=1}^k x_i^*a_{ij} x_j.
\eqno(2.6)$$

\iproclaim Lemma 2.1.
$\gx$ extends to a unity preserving \cp\ map from $\M_k\ot\A$ to $\A$.
\eproclaim

\proof:
In order to prove that $\gx$ is \cp\ , we have to show that for each choice
of $p=1,2,\ldots$ and $A^\ell\in\M_k\ot\A$, $\ell=1,2,\ldots p$ the
$p\times p$ matrix $[\gx((A^\ell)^*A^m)]$ is positive. Now,
$$\eqalign{
[\gx((A^\ell)^*A^m)]
&= \left[\sum_{i,j=1}^k x_i^* \left(\sum_{r=1}^k (a^\ell_{r,i})^* a^m_{r,j}
    \right) x_j \right]\cr
&= \sum_{r=1}^k \left[ \left(\sum_{i=1}^k a^{\ell}_{r,i} x_i\right)^*
\left(\sum_{j=1}^k a^m_{r,j} x_j\right)\right]\cr
&\geq 0.
}$$
Clearly $\gx$ is unity preserving due to the fact that $\X$ is a partition
of unity.
\QED

As $\Theta$ is an automorphism of $\A$, $\gtx = \Theta\circ\gx$ is also a
unity preserving \cp\ map. The map $\gtx$, tensored with suitable identity
maps on $\M_k$, can be iterated to yield maps
$$\gtx^{(m)}: (\M_k)^{\ot m}\ot\A\to\A : \gtx^{(m)} = \gtx \circ (\id\ot\gtx)
\circ \cdots (\id\ot\id\ot\cdots\gtx). \eqno(2.7)$$
The maps $\gtx^{(m)}$ can now be used to construct a state $\om_\X$ on the
half-chain algebra $(\M_k)^\Nl = \ot_{i=0}^\infty \M_k$. On local elements
$M\in\M_k^{[0,m-1]}$, $\om_\X$ is given by:
$$\om_\X(M) = \om(\gtx^{(m)}(M\ot\idty_\A)). \eqno(2.8)$$
The necessary positivity, normalization and compatibility conditions for
$\om_\X$ are satisfied because it is a composition of unity preserving \cp\
maps. The construction (2.8) of a state on a spin chain is in fact the
starting point for studying the class of \cfc\ states on quantum spin
chains [8]. Alternatively (2.8) could be considered to specify the
multitime correlation functions of a stochastic process on $\M_d$. The
reduced density matrices $\rho^{(m)}_\X$, corresponding to the restriction
of $\om_\X$ to the first $m$ tensor factors of $(\M_k)^\Nl$ can easily be
computed. Let $M\in\M_k^{\ot [0,m-1]}$, and let ${\bf i}$ denote the
multiindex $(i_0,i_1,\ldots i_{m-1})$, then:
$$\eqalign{
&\tr_{(\Cx^k)^{\ot m}} \rho^{(m)}_\X M =
 \om_\X(M) =
 \om(\gtx^{(m)}(M\ot\idty_\A)) = \cr
& \sum_{\bf i,j}
   M_{\bf i,j}\,
   \om\left(\Theta(x_{i_0}^*)\Theta^2(x_{i_1}^*) \cdots
   \Theta^m(x_{i_{m-1}}^*x_{j_{m-1}}) \cdots \Theta^2(x_{j_1})
   \Theta(x_{j_0})\right) = \cr
& \sum_{\bf i,j}
   M_{\bf i,j}\,
   \om\left(x_{i_0}^*\Theta(x_{i_1}^*) \cdots
   \Theta^{m-1}(x_{i_{m-1}}^*x_{j_{m-1}}) \cdots \Theta(x_{j_1})
   x_{j_0}\right) .
}$$
Therefore:
$$(\rho^{(m)}_\X)_{(i_0,\ldots i_{m-1}),(j_0,\ldots j_{m-1})} =
\om\left(x_{j_0}^*\Theta(x_{j_1}^*) \cdots
\Theta^{m-1}(x_{j_{m-1}}^*x_{i_{m-1}}) \cdots \Theta(x_{i_1})
x_{i_0}\right),$$
or, using the notation of (2.2):
$$\rho^{(m)}_\X = \rho[\Theta^{m-1}(\X) \circ\cdots \Theta(\X) \circ
\X]. \eqno(2.9)$$

The state $\om_\X$, defined in (2.8) is generally not shift invariant as
$$\om(y)\neq\sum_{i=1}^k \om(x_i^*yx_i), \quad y\in\A. \eqno(2.10)$$
We can therefore, in general, not ascertain the existence of $\lim_m
{1\over m} S(\rho^{(m)}_\X)$ in (2.3). In specific examples however, using
an appropriate choice of $\A_0$, $\om_\X$ approaches sufficiently fast
a shift invariant state and this guarantees the existence of the limit.
In [7] attention was restricted to those partitions of unity for which
equality in (2.10) holds. This is an extremely strong condition in a truly
quantum mechanical situation where the reduced density matrices
$\rho_\om^\Lambda$ don't commute amongst themselves.

\beginsection 3. Kolmogorov-Sinai entropy

Consider a probability space $(\hbox{M},\F,\mu)$, $\hbox{M}$ is a
measurable space, $\F$ the $\sigma$-algebra of measurable subsets of
$\hbox{M}$ and $\mu$ a probability measure on $\hbox{M}$. We are also given
an automorphism $T$, which is a one to one transformation of M, such that
both $T$ and $T^{-1}$ map $\F$ into itself and
$$\mu(A) = \mu(T(A)) = \mu(T^{-1}(A)), \quad A\in\F.$$
If $\C = \{C_1,C_2,\ldots C_k\}$ is a partition of M into disjoint,
measurable subsets, the entropy $h(\C)$ of $\C$ is computed as
$$h(\C) = \sum_{i=1}^k \eta(\mu(C_i)), \eqno(3.1)$$
$\eta$ defined as in (2.4). If $\C$ and $\D$ are two partitions of M,
$\C\vee\D$ is the partition of M generated by $\C$ and $\D$. Choosing a
fixed partition $\C$, it can be shown that the following limit exists:
$$h_{(\mu,T)}(\C) = \lim_{m\to\infty} {1\over m}\ h\left(T^{m-1}(\C)\vee\cdots
T(\C)\vee\C\right). \eqno(3.2)$$
The Kolmogorov-Sinai entropy $h_{(\mu,T)}$ [1] is defined to be:
$$h_{(\mu,T)} = \sup_\C\ h_{(\mu,T)}(\C). \eqno(3.3)$$

We now apply the scheme of section~2 to this particular case. Let
$\A=\L^{\infty}(\hbox{M},\F,\mu)$ and let $\A_0$ be the subalgebra of $\A$
consisting of all step functions \ie all finite linear combinations of
characteristic functions of measurable subsets of M. The map $T$ on M
defines an automorphism $\Theta$ of $\A$ through the relation
$$\Theta(f) = f \circ T^{-1}, \quad f\in\L^{\infty}(\hbox{M},\F,\mu).
\eqno(3.4)$$
Clearly $\A_0$ is globally invariant under $\Theta$. Furthermore,
the state $\om(f) = \int d\mu\, f$ is invariant under  $\Theta$.

\iproclaim Theorem 3.1.
With the notation of above,
$$H_{(\om,\Theta)}=h_{(\mu,T)}.$$
\eproclaim

\proof:
We first show that $H_{(\om,\Theta)} \leq h_{(\mu,T)}$. Let $\X =
\{x_1,x_2,\ldots x_k\}$ be a partition of unity in the sense of (2.1). As
each $x_i$ is a step function we can find a finite partition $\C =
\{C_1,C_2,\ldots C_\ell\}$ of M into disjoint sets and complex numbers
$\psi_i^\alpha$, such that:
$$x_i = \sum_{\alpha=1}^\ell \psi_i^\alpha\, \cha_{C_\alpha}, \quad
i=1,2,\ldots k,$$
$\cha_C$ denoting the characteristic function of a measurable set $C$.
$\X$ is a partition of unity iff for all $\alpha$
$$\sum_{i=1}^k \overline{\psi_i^\alpha}\,\psi_i^\alpha = 1. \eqno(3.5)$$
We then compute the reduced density matrices $\rho^{(m)}_\X$ of (2.7):
$$\eqalign{
&(\rho^{(m)}_\X)_{(i_0,\ldots i_{m-1}),(j_0,\ldots j_{m-1})} = \cr
&(\rho[\Theta^{m-1}(\X) \circ\cdots
\Theta(\X)\circ\X])_{(i_0,\ldots i_{m-1}),(j_0,\ldots j_{m-1})} = \cr
& \sum_{\alpha_0,\ldots\alpha_{m-1}}
\overline{\psi_{j_{0}}^{\alpha_{0}}}
\,\overline{\psi_{j_1}^{\alpha_1}}
\cdots \overline{\psi_{j_{m-1}}^{\alpha_{m-1}}}\,
\psi_{i_0}^{\alpha_0}\, \psi_{i_1}^{\alpha_1}
\cdots \psi_{i_{m-1}}^{\alpha_{m-1}}\,
\om\left(\Theta^{m-1}(\cha_{C_{\alpha_{m-1}}}) \cdots
\Theta(\cha_{C_{\alpha_1}}) \cha_{C_{\alpha_0}}\right) = \cr
& \sum_{\alpha_0,\ldots\alpha_{m-1}} \overline{\psi_{j_0}^{\alpha_0}}
\,\overline{\psi_{j_1}^{\alpha_1}}
\cdots \overline{\psi_{j_{m-1}}^{\alpha_{m-1}}}\,
\psi_{i_0}^{\alpha_0}\, \psi_{i_1}^{\alpha_1}
\cdots \psi_{i_{m-1}}^{\alpha_{m-1}}\,
\mu\left(T^{m-1}(C_{\alpha_{m-1}}) \cap \cdots
T(C_{\alpha_1}) \cap C_{\alpha_0}\right).
}$$
Due to condition (3.5), the vector $\Psi^\alpha\in\Cx^k$, with
components $(\psi^\alpha_1,\psi^\alpha_2,\dots \psi^\alpha_k)$
is normalized. Hence $(\rho^{(m)})_\X$ now be written as:
$$\rho^{(m)}_\X = \sum_{\alpha_0,\ldots\alpha_{m-1}}
\mu\left(T^{m-1}(C_{\alpha_{m-1}}) \cap \cdots  C_{\alpha_0}\right) \,
\mid\Psi^{\alpha_0}\ot\cdots \Psi^{\alpha_{m-1}}\rangle\,
\langle\Psi^{\alpha_0}\ot\cdots \Psi^{\alpha_{m-1}}\mid.$$
By [9], formula (2.4), we have:
$$S(\rho^{(m)}_\X) \leq \sum_{\alpha_0,\ldots\alpha_{m-1}}
\eta\left(\mu\left(T^{m-1}(C_{\alpha_{m-1}}) \cap \cdots
C_{\alpha_0}\right)\right) = h\left(T^{m-1}(\C)\vee\cdots
T(\C)\vee\C\right).$$
Dividing both sides by $m$, and taking limes suprema and suprema on both
sides we obtain
$$H_{(\om,\Theta)} \leq h_{(\mu,T)}.$$

\noindent
The Kolmogorov-Sinai entropy is reached by taking the supremum of the
$H_{(\om,\Theta)}(\X)$ over a subclass of the partitions of the unity,
namely those consisting in characteristic functions only.
\QED

\beginsection 4. Dynamical entropy for the shift on a quantum spin chain

As the second application we consider a quantum dynamical system
$(\A,\Theta,\om)$ where $\A=(\M_d)^\Ir$ is the spin chain with single site
algebra $\M_d$. The dynamics $\Theta$ is given by the right shift and $\om$
is an arbitrary translation invariant state on $\A$. For $\A_0$ we choose
the local observables \ie
$$\A_0 = \bigcup_{\Lambda\subset\Ir \atop \#(\Lambda)<\infty} (\M_d)^{\ot
\Lambda}.$$

For any finite $\Lambda\subset\Ir$ the restriction of $\om$ to $(\M_d)^{\ot
\Lambda}$ is given by a reduced density matrix $\rho_\om^\Lambda$
$$\om(A) = \tr_{\Cx^{\ot \Lambda}} \rho_\om^\Lambda\, A, \quad
A\in(\M_d)^{\ot \Lambda}.$$
The entropy $S^\Lambda(\om)$ of $\om$ restricted to $\Lambda$ is defined to
be
$$S^\Lambda(\om) = \tr \eta(\rho_\om^\Lambda),\eqno(4.1)$$
and the entropy density $\sigma(\om)$ of $\om$ is
$$\sigma(\om) = \lim_{\Lambda\to\Ir}\ {1\over\#(\Lambda)}
S^\Lambda(\om),\eqno(4.2)$$
where the limit can be taken in the sense of van~Hove.

\iproclaim Theorem 4.1.
With the notation of above,
$$H_{(\om,\Theta)}=\sigma(\om) + \log d.$$
\eproclaim

\proof:
Choose a size $k$ partition $\X=\{x_1,x_2,\ldots x_k\}\subset\A_0$ of
unity. By locality the $x_i$ belong to some finite volume algebra. As $\om$
is translation invariant we can always assume that the $x_i$ belong to
$(\M_d)^{\ot [0,\ell-1]}$ for some $\ell=1,2,\ldots$. Let $\{e_1,e_2,\ldots
e_k\}$ be an orthonormal basis for $\Cx^k$ and denote by $w$ the mapping:
$$w: (\Cx^d)^{\ot \ell}\to\Cx^k\ot(\Cx^d)^{\ot \ell}:
\phi\mapsto\sum_{i=1}^k e_k\ot x_k\phi.$$
It is straightforward to check that $w^*$ is given by
$$w^*: \Cx^k\ot(\Cx^d)^{\ot \ell}\to(\Cx^d)^{\ot \ell}:
\sum_{i=1}^k e_i\ot\phi_i\mapsto \sum_{i=1}^k x_i\phi_i.$$
Condition (2.1), which expresses that $\X$ is a partition of unity is
equivalent to $w$ being an isometry, $w^*w=\idty$. The reduced $m$-site
density matrices $\rho_\X^{(m)}$ of the state $\om_\X$ on $(\M_k)^\Nl$ can
be expressed in terms of the reduced $(m+\ell-1)$-point density matrices
$\rho_\om^{(m+\ell-1)}$ of the state $\om$ on the chain $(\M_d)^\Ir$. If
$X\in(\M_k)^{\ot m}$, $\gtx^{(m)}(X\ot\idty_\A)$ will actually belong to
$(\M_d)^{\ot (m+\ell-1)}$ and:
$$\eqalign{
&\om_\X(X) = \tr_{(\Cx^k)^{\ot m}} \rho_\X^{(m)}\,X = \cr
&\om(\gtx^{(m)}(X\ot\idty_\A)) = \tr_{(\Cx^d)^{\ot (m+\ell-1)}}
\rho_\om^{(m+\ell-1)}\, \gtx^{(m)}(X\ot\idty) = \cr
&\tr_{(\Cx^d)^{\ot (m+\ell-1)}} \rho_\om^{(m+\ell-1)}\,
(w^{(m)})^*\, X\ot(\idty_d)^{\ot (m+\ell-1)}\, w^{(m)}.
}$$
In this formula, $w^{(m)}$ maps
$$(\Cx^d)^{\ot (m+\ell-1)}\to(\Cx^k)^{\ot m}\ot(\Cx^d)^{\ot
(m+\ell-1)}.$$
Up to a reshuffling of tensor factors $\Cx^d$ and $\Cx^k$,
$$w^{(m)} = (\idty_k^{\ot (m-1)} \ot \idty_d^{\ot (m-1)} \ot w)
\cdots (\idty_k\ot\idty_d\ot
w\ot\idty_d^{\ot (m-2)})(w\ot\idty_d^{\ot (m-1)}).$$
Clearly $w^{(m)}$ is also isometric and
$$\eqalign{
\rho_\X^{(m)}
&= \tr_{(\Cx^d)^{\ot (m+\ell-1)}} w^{(m)}\,
\rho_\om^{(m+\ell-1)}\, (w^{(m)})^* \cr
&=\tr_{(\Cx^d)^{\ot (m+\ell-1)}} R^{(m)} .
}$$
The density matrix $R^{(m)}$ on $(\Cx^k)^{\ot m}\ot(\Cx^d)^{\ot
(m+\ell-1)}$ has, up to multiple eigenvalues 0, the same spectrum as
$\rho_\om^{(m+\ell-1)}$. From [10], Theorem~2~(c), we then get
$$S(\rho_\X^{(m)}) \leq S^{(m+\ell-1)}(\om) + (m+\ell-1)\log d.$$
Dividing both sides by $m$, taking limsup for $m\to\infty$ and the
supremum over all finite partitions of unity, we
obtain
$$H_{(\om,\Theta)} \leq \sigma(\om) + \log d.$$

\noindent
To get the converse inequality, consider the partition $\X =
\{{1\over\sqrt d}e_{ij}\mid i,j=1,2,\ldots d\}$ of unity, where the
$e_{ij}$ are matrix units in $\M_d$ living say at the site 0 of the chain
$(\M_d)^{\Ir}$. One can easily compute the reduced $m$-site density matrix
$\rho_\X^{(m)}$ on the half chain $(\M_{d^2})^{\Nl}$, writing
$\M_{d^2}=\M_d \ot \M_d$:
$$ \rho_\X^{(m)} = \rho_\om^{(m)}\ot ({1\over d}\idty_d)^{\ot m},$$
and therefore $S(\rho_\X^{(m)}) = S^{(m)}(\om) + m\log d$.
\QED

\vskip 20pt
\noindent
{\bf Acknowledgements}

Part of this work was done while R.A. was visiting the Department of
Mathematics of the University of Nottingham and completed while he was
Visiting Professor at the KU~Leuven. He also acknowledges financial
support from S.E.R.C. (project GR/H72960), K.B.N. (project PB
1436/2/91) and KUL project O.T. 92/9.
\vfill\eject

\noindent
{\bf References}

\item{[1]}
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\item{[2]}
  M.~Ohya and D.~Petz:
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  York (1993)
\item{[3]}
  G.G.~Emch:
  ``Positivity of the K-entropy on non-abelian K-flows'',
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  241--252 (1974)
\item{[4]}
  A.~Connes and E.~St\o rmer:
  ``Entropy for automorphisms of II$_1$ von~Neumann algebras'',
  {\it Acta Math.\/} {\bf 134}, 289--306 (1975)
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