A New Proof of the Monotonicity of Quantum Relative Entropy for Finite Dimensional Systems
Igor Bjelakovi\'c
Technische Universit\"at Berlin
Germany
We give a self-contained, new proof of the monotonicity of the quantum relative entropy which seems to be natural from the point of view of quantum information theory. It is based on the quantum version of Stein's lemma which provides an operational interpretation of the quantum relative entropy.
quant-ph0307170.tex
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\author{Igor Bjelakovi\'c, Rainer Siegmund-Schultze\footnote{e-mail:
    \{igor, siegmund\}@math.tu-berlin.de}\\
Technische Universit\"at Berlin\\ Germany}

\title{A New Proof of the Monotonicity of Quantum Relative
  Entropy for Finite Dimensional Systems}

\newcommand{\hr}{{\cal H}}
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\newcommand{\cc}{{\mathbb C}}
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\newcommand{\nn}{{\mathbb N}}
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\newcommand{\zz}{{\mathbb Z}}
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\newcommand{\eps}{{\varepsilon}}        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\vphi}{{\varphi}}           % Frequently used abbreviations   %
\newcommand{\om}{\omega}
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\newtheorem{theorem}{Theorem}[section]         %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
\maketitle
\begin{abstract}We give a self-contained, new proof of the
  monotonicity of the quantum relative entropy which seems to be
  natural from the point of view of quantum information theory. It
  is based on the quantum version of Stein's lemma which provides an
  operational interpretation of the quantum relative entropy.
\end{abstract}
%---------------------------INTRODUCTION--------------------------
\section{Introduction and Definitions}
The monotonicity of the quantum relative entropy is one of its deeper properties. 
It has been proved by Lindblad in \cite{lindblad} for finite quantum
systems and by Uhlmann in \cite{uhlmann} for the most general situation.
Lindblad based his proof on the strong subadditivity of the von
Neumann entropy which was shown by Lieb and Ruskai in \cite{lieb-ruskai}. At the present time there are several proofs 
of these entropic properties. The most prominent proof is perhaps 
 Simon's elegant
 derivation of the famous Lieb's concavity (cf. \cite{simon}), which implies the joint 
convexity of the quantum relative entropy (cf. \cite{niel}). This in turn suffices to
show the monotonicity of the quantum relative entropy and the strong
subadditivity of the von Neumann entropy
(cf. \cite{niel}). All these proofs are analytical in
spirit. It is the intention of this article to provide another proof
of the monotonicity in the finite dimensional situation which could be more appealing from the point of view
of quantum information theory. The guiding idea in our approach is
that (in complete analogy to classical information theory) the notion of
relative entropy obtains its full information theoretical justification
through an appropriate law of large numbers or ergodic theorem. Such
ergodic theorem (cf. Theorem \ref{iid-lim-beta=s} in the next section)
 has been formulated and partially proved for the mean relative
 entropy of a
completely ergodic state with respect to a stationary product state by Hiai and Petz in \cite{petz}. Later
on Ogawa and Nagaoka (cf. \cite{ogawa}) proved that result in the case of stationary
product states. However, they used deep properties of the so called
quasi-entropies, which are analogous to the monotonicity of the
relative entropy. In the next
section we will present an elementary proof of this ergodic theorem in
the case of \emph{stationary product} states. An ergodic version of that
result (which was conjectured by Hiai and Petz) may be found in
\cite{wir2}. It was not possible to avoid the usage of monotonicity and joint convexity in the proof of that ergodic theorem in \cite{wir2}, but these properties were only needed to estimate entropies of $l$-ergodic components on local algebras, which is unnecessary for the stationary product case. We present a complete proof here in the product
situation which does not make use of these properties. \newline
Once we have this theorem in the
stationary product situation it is almost trivial to obtain the
monotonicity. In fact, according to this result the relative entropy equals a thermodynamic parameter which trivially decreases if the underlying algebra of observables is replaced by a subalgebra. In view of a well-known representation theorem for
completely positive unital maps discovered by Kraus
(cf. \cite{kraus}) this implies monotonicity.
The derivation of the monotonicity is the content of the third section of this article.\newline
In the appendix we explain the very nice approximation result of Hiai
and Petz upon which our whole result rests.

We close this section with the review of some basic definitions and
notions. Let ${\mathcal A}:={\mathcal B}(\hr)$ be the algebra of operators
acting on a finite dimensional Hilbert space $\hr$ and let $d=\dim
\hr$. States on ${\mathcal A}$ are in one-to-one correspondence with density operators acting on the Hilbert space $\hr$.
We consider the quasilocal algebra ${\mathcal A}^{\infty}$ which
is defined in the following way: To each site $x\in \zz$ we assign a
copy ${\mathcal A}_{x}$ of the algebra ${\mathcal A}$. Then for each
finite set $\Lambda \subset \zz$ we define ${\mathcal
  A}_{\Lambda}:=\bigotimes_{x\in\Lambda}{\mathcal A}_{x}$, i.e. the
algebra associated with $\Lambda$. For finite subsets
$\Lambda,\tilde\Lambda$ of $\zz$ fulfilling
$\Lambda\subset\tilde\Lambda$
there is a natural inclusion of ${\mathcal A}_{\Lambda}$ into
${\mathcal A}_{\tilde\Lambda}$ given by $a\mapsto
a\otimes\idn_{\tilde\Lambda\setminus\Lambda}$. The quasilocal algebra
${\mathcal A}^{\infty} $ is given by the norm closure of the
algebra $\bigcup_{\Lambda\subset\zz:\#\Lambda <\infty }{\mathcal A}_{\Lambda}$. A state $\tau$ on ${\mathcal A}^{\infty}$ is a
linear, positive and unital map of ${\mathcal A}^{\infty}$ into $\cc$.
Each state $\tau$ on $ {\mathcal A}^{\infty}$ defines a family of
states $\{ \tau^{(\Lambda)}\}_{\Lambda \subset \zz,\#\Lambda<\infty}$
on the local algebras ${\mathcal A}_{\Lambda}$ which fulfill
$\tau^{(\tilde\Lambda)}\!\upharpoonright\!{\mathcal
  A}_{\Lambda}=\tau^{(\Lambda)}$ for $\Lambda\subset\tilde\Lambda$. Since we will deal
only with stationary states we confine ourselves to the sets of the
form $\Lambda (n):=\{0,\ldots ,n-1\}$, $n\in\nn$. In what follows we
will always abbreviate $\tau^{(\Lambda(n))}$ by $\tau^{(n)}$ and
${\mathcal A}_{\Lambda(n)}$ by ${\mathcal A}^{(n)}$. A
detailed discussion of the quasilocal algebras may be found in the
monograph \cite{bratteli}.

For the convenience of the reader we recall the definition of the relative entropy of a state $\sigma$ on a finite dimensional $ C^{\ast}$-algebra ${\mathcal C}$ with respect to a state $\tau$:
\[ S(\sigma,\tau):=\left\{ \begin{array}{ll}  \textrm{tr}(D_{\sigma}(\log D_{\sigma}-\log D_{\tau})) & \textrm{if } \textrm{supp}(\sigma) \le \textrm{supp}(\tau)\\
\infty & \textrm{otherwise}
\end{array} \right.  \]
where $D_{\sigma}$ respectively $D_{\tau}$ denote the density
operators of $\sigma$ respectively $\tau$ and $\textrm{supp}(\sigma)$ is the
support projection of the state $\sigma$. The mean relative entropy of
the stationary state $\om_{1}$ on $ {\mathcal A}^{\infty}$ with respect to the
stationary \emph{product} state $\om_{2}$ is given by
\[
s(\om_{1},\om_{2}):=\lim_{n\to\infty}\frac{1}{n}S(\om_{1}^{(n)},\om_{2}^{(n)})=\sup_{n}\frac{1}{n}S(\om_{1}^{(n)},\om_{2}^{(n)}),\]
where the last equation holds by the superadditivity of the relative entropy.
%------------------------Law of Large Numbers------------------------------
\section{A Law of Large Numbers for the Quantum Relative Entropy} 
Our proof of the monotonicity of the relative entropy will be based on
a law of large numbers for the quantum relative entropy.
Let $\psi_{\infty},\varphi_{\infty}$ be stationary product
states and let $\eps\in (0,1)$. We define
\begin{eqnarray}\label{eq:beta}
\beta_{\eps,n}(\psi_{\infty},\varphi_{\infty}):=\min\{\log
\varphi^{(n)}(q):q\in {\mathcal A}^{(n)}\textrm{ projection,
}\psi^{(n)}(q)\ge 1-\eps \}.
  \end{eqnarray}
The projection $q_{0}^{(n)}$ at which the minimum in (\ref{eq:beta}) is
attained represents an ideal measurement that distinguishes the
states $\psi^{(n)}$ and $\varphi^{(n)}$ in an optimal way. The
relationship of the quantity defined in (\ref{eq:beta}) and the
quantum relative entropy is the content of the following theorem.
\begin{theorem}\label{iid-lim-beta=s}
Let $\psi_{\infty},\varphi_{\infty}$ be stationary product
states and let $\eps\in (0,1)$. Suppose that $S(\psi,\varphi)<\infty $
holds. Then we have
\begin{eqnarray}\label{eq:the-limit}
\lim_{n\to\infty}\frac{1}{n}\beta_{\eps,n}(\psi_{\infty},\varphi_{\infty})=-S(\psi,\varphi),
  \end{eqnarray}
for all $\eps \in (0,1)$.
\end{theorem}
\textbf{Remark} Recall that the mean relative entropy of the
stationary product state $\psi_{\infty}$ with respect to the
stationary product state $\varphi_{\infty}$ fulfills
\[s(\psi_{\infty}, \varphi_{\infty})=S(\psi,\varphi).\]
\textbf{Proof of Theorem \ref{iid-lim-beta=s}:} We choose an $\eta
>0$. Then according to Theorem \ref{hiai-petz-lemma} in the
appendix there exists
an integer $l\ge 1$ such that for the abelian algebra ${\mathcal
  B}_{l}$ considered in that theorem 
\begin{eqnarray}\label{eq:firstapprox}
 \frac{1}{l}S(\psi^{(l)}\!\upharpoonright\!{\mathcal
  B}_{l},\varphi^{(l)}\!\upharpoonright\!{\mathcal B}_{l})\ge S(\psi,\varphi)-\eta,
\end{eqnarray}
holds. We consider the quasilocal algebra ${\mathcal B}_{l}^{\infty}$ and
define 
\[ P:=\psi_{\infty}\!\upharpoonright\!{\mathcal
  B}_{l}^{\infty}\textrm{ and }Q:=\varphi_{\infty}\!\upharpoonright\!{\mathcal
  B}_{l}^{\infty}.\]
Note that these states are i.i.d. states on ${\mathcal
  B}_{l}^{\infty} $. We will think of this abelian algebra as
continuous functions on the maximal ideal space $B_{l}^{\infty}$ which
in turn may be represented as the infinite cartesian product of a
finite set $B_{l}$. Let $D_{M}$ denote the classical relative mean entropy
 of the probability measure $P$ with respect to the probability
measure $Q$. Then we have by (\ref{eq:relentropyreduction}) 
\begin{eqnarray}\label{eq:class-rel-finite}
D_{M}=S(\psi^{(l)}\!\upharpoonright\!{\mathcal
  B}_{l},\varphi^{(l)}\!\upharpoonright\!{\mathcal B}_{l})\le
S(\psi^{(l)},\varphi^{(l)})= lS(\psi,\varphi)<\infty,
  \end{eqnarray}
since the projective measurements increase the von Neumann entropy
(cf. \cite{niel}). We choose a $\delta>0$ and define
\begin{eqnarray}\label{eq:reltypicalset}
C^{(n)}:=\{\om_{n}\in B_{l}^{n}:e^{n(D_{M}-\delta)}< \frac{P^{(n)}(\om_n)}{Q^{(n)}(\om_n)}<e^{n(D_{M}+\delta)}\}.  
\end{eqnarray}
By the law of large numbers we have
$\lim_{n\to\infty}P^{(n)}(C^{(n)})=1$. Consider the projection
$r_{ln}\in {\mathcal A}^{(ln)}$ corresponding to the characteristic
function  $1_{C^{(n)}}$. Then
we have
\begin{eqnarray}\label{eq:phiestimate}
 \varphi^{(ln)}(r_{ln})=Q^{(n)}(C^{(n)})&<&
 P^{(n)}(C^{(n)})e^{-n(D_{M}-\delta)}\nonumber \\
&\le & e^{-n(D_{M}-\delta)}\nonumber \\
& \le & e^{-n(l(S(\psi,\varphi)-\eta)-\delta)}\textrm{ by (\ref{eq:firstapprox})}
\end{eqnarray}
For each integer $k\ge 1$ we set $k=nl+j$, $0\le j<l$, and define
\[ q_{ln}:=r_{ln}\textrm{ and } q_{k}:=q_{ln}\otimes
\idn_{[ln,k-1]}.\]
If $n$ is large enough we have by (\ref{eq:reltypicalset}) and the law
of large numbers
\[\psi^{(k)}(q_{k})=\psi^{(ln)}(q_{ln})=\psi^{(ln)}(r_{ln})=P^{(n)}(C^{(n)})\ge
1-\eps.\]
Hence the condition in the definition (\ref{eq:beta}) is fulfilled. On
the other hand we obtain
\begin{eqnarray}
  \beta_{\eps,k}(\psi_{\infty},\varphi_{\infty})&\le&\log\varphi^{(k)}(q_{k})\nonumber \\
&\le& \log e^{-n(l(S(\psi,\varphi)-\eta)-\delta)}\quad\textrm{ (by
  (\ref{eq:phiestimate}))}\nonumber \\
& = & -nl (S(\psi,\varphi)-\eta-\frac{\delta}{l}).\nonumber
\end{eqnarray}
This shows that
\begin{eqnarray}\label{eq:iid-lim-beta<s}
\limsup_{n\to\infty}\frac{1}{k}\beta_{\eps,k}(\psi_{\infty},\varphi_{\infty})\le
-S(\psi,\varphi),
\end{eqnarray}
since $k\to \infty$ iff $n\to\infty$,
$\lim_{n\to\infty}\frac{nl}{nl+j}=1$ and $ \eta,\delta >0$ were chosen
arbitrarily. 

In the next step we show the existence of the limit in
(\ref{eq:iid-lim-beta<s}) and that this limit equals $-S(\psi,\varphi)$.
Let $(t_{y})$ be a sequence of projections in ${\mathcal
  A}^{(y)}$ such that $y\to \infty$ and
\[ \liminf_{y\to\infty}\frac{1}{y}\log\varphi^{(y)}(t_{y})<-S(\psi,\varphi).\]
Then there exists an $a>0$ and a subsequence, which we denote by
$(t_{y})$ again for notational simplicity, with the property 
\begin{eqnarray}\label{eq:toosmall}
 \varphi^{(y)}(t_{y})<e^{-y(S(\psi,\varphi)+a)}. 
\end{eqnarray}
Let $h=H(P^{(1)})$ denote the mean entropy of the i.i.d. measure
$P$ and consider the set
\begin{eqnarray}\label{eq:entropytypical}
 F^{(n)}:=\{\om_{n}\in B_{l}^{n}:e^{-n(h+\delta)}<P^{(n)}(\om_{n})<e^{-n(h-\delta)}\}, 
\end{eqnarray}
then we have 
\begin{eqnarray}\label{eq:law-of-large-numbers}
\lim_{n\to\infty}P^{(n)}( F^{(n)})=1\qquad \textrm{ by the law of
  large numbers.} 
\end{eqnarray}
Let $q_{n}$ be the projection in ${\mathcal A}^{(ln)}$ corresponding
to some $\om_{n}\in C^{(n)}\cap F^{(n)}$ (cf. (\ref{eq:reltypicalset}))
then we have
\begin{eqnarray}\label{eq:onedim-estimate}
  \varphi^{(ln)}(q_{n})&=&Q^{(n)}(\om_{n})>e^{-n(D_{M}+h+2\delta)}\textrm{ (by definition of $ C^{(n)}\textrm{ and } F^{(n)}$)}\nonumber \\ 
&>& e^{-n(lS(\psi,\varphi)+l(S(\psi)+\eta)-E_{l}+2\delta)},
\end{eqnarray}
where
$E_{l}:=\sum_{i=1}^{a_{l}}\textrm{tr}(D_{\psi^{(l)}}q_{i,l})\log\textrm{tr}(q_{i,l})$
and $a_{l}$ and $q_{i,l}$ were defined in (\ref{eq:Bn}).
The validity of the last inequality follows on the one hand from
(\ref{eq:class-rel-finite}). On the other hand we have
\begin{eqnarray}
  h&= &S(\psi^{(l)}\!\upharpoonright\!{\mathcal
    B}_{l})=-S(\psi^{(l)}\!\upharpoonright\!{\mathcal
    B}_{l},\varphi^{(l)}\!\upharpoonright\!{\mathcal
    B}_{l})-\textrm{tr}_{{\mathcal B}_{l}}(D_{\psi^{(l)}\!\upharpoonright {\mathcal
    B}_{l}}\log D_{\varphi^{(l)}\!\upharpoonright {\mathcal
    B}_{l} })   \nonumber \\
&\le & -l(S(\psi,\varphi)-\eta)-\textrm{tr}(D_{\psi^{(l)}}\log
D_{\varphi^{(l)}})-E_{l}\quad\textrm{ (by
  (\ref{eq:firstapprox}) and
  (\ref{eq:appendix2}))}\nonumber \\ 
&=& l(S(\psi)+\eta)-E_{l},\nonumber
\end{eqnarray}
which shows (\ref{eq:onedim-estimate}). Note that each minimal
projection $s_{n}\in {\mathcal B}_{l}^{(n)}$ can be written as
$\otimes_{j=1}^{n}q_{i_{j},l}$. We define the following set:
\begin{eqnarray}\label{eq:dim-typical}
L^{(n)}:=\{s_{n}\in {\mathcal B}_{l}^{(n)}: s_{n}\textrm{ minimal
  projection},
\frac{1}{n}\log\textrm{tr}(s_{n})< E_{l}+\delta \}.
  \end{eqnarray}
Then the law of large numbers implies that
\begin{eqnarray}\label{eq:L-limit}
  \lim_{n\to\infty}P^{(n)}(L^{(n)})=1,
\end{eqnarray}
after obvious identification of the minimal projections in ${\mathcal
  B}_{l}^{(n)}$ with elementary outcomes of $B_{l}^{n}$. (Observe that by definition $E_{l}$ is the expectation of $\log\textrm{tr}(q_{(.),l})$ with respect to $P$.)
Let $p_{ln}\in {\mathcal A}^{(ln)}$ denote the projection
corresponding to the characteristic function of the set $C^{(n)}\cap F^{(n)}\cap L^{(n)}$. Then for all
sufficiently large $n$ we have $\psi (p_{ln}) \ge 1-\eps$ by
(\ref{eq:law-of-large-numbers}), (\ref{eq:L-limit}) and (\ref{eq:reltypicalset}). Consider the embedding $ e_{ln}$ of the projections $t_{y}$
fulfilling (\ref{eq:toosmall}) into an appropriately chosen ${\mathcal
  A}^{(ln)}$. For example we choose the unique $n\in\nn$ with
$(n-1)l\le y<nl$. Then we have 
\begin{eqnarray}\label{eq:eln-ty}
\psi^{(ln)}(e_{ln})=\psi^{(y)}(t_{y})
\end{eqnarray}
and
\begin{eqnarray}\label{eq:eln-estimate}
 \psi^{(ln)}(e_{ln})&=&\psi^{(ln)}(e_{ln}p_{ln})+\psi^{(ln)}(e_{ln}(\idn-p_{ln}))\nonumber \\
&\le & \psi^{(ln)}(e_{ln}p_{ln})+\eps \quad (\textrm{by }\psi
(p_{ln}) \ge 1-\eps) \nonumber \\
&=& \psi^{(ln)}(p_{ln}e_{ln}p_{ln})+
\psi^{(ln)}((\idn-p_{ln})e_{ln}p_{ln}) +\eps\nonumber \\
&\le& \psi^{(ln)}(p_{ln}e_{ln}p_{ln})+2\eps\nonumber\\
&=& \textrm{tr}(p_{ln}D_{\psi^{(ln)}}p_{ln}e_{ln})+2\eps.
\end{eqnarray}
In the next step we will estimate the first term in the last line
above. Consider the spectral decomposition of
$p_{ln}D_{\psi^{(ln)}}p_{ln}$ into one-dimensional projections
\[p_{ln}D_{\psi^{(ln)}}p_{ln}=\sum_{i=1}^{b_{ln}}\lambda_{i,ln}r_{i,ln},\]
where $b_{ln}=\textrm{tr}(p_{ln})$.
For $\delta >0$ we define the set
\[T_{\delta,ln}:=\{i\in\{1,\ldots ,b_{ln}\}: \lambda_{i,ln}\le
e^{-ln(S(\psi)-\delta)}\},\]
and
\[r_{\delta,ln}:=\sum_{i\in T_{\delta,ln}}r_{i,ln}.\]
Then we have
\begin{eqnarray}\label{eq:pDp-entropy}
\lim_{n\to\infty}\psi^{(ln)}(p_{ln}-r_{\delta,ln})=0\qquad \textrm{for
  all }\delta>0.
  \end{eqnarray}
In fact, observe that
\[1\ge\sum_{i\in T_{\delta,ln}^{c}}\lambda_{i,ln}\ge
e^{-ln(S(\psi)-\delta)}\#T_{\delta,ln}^{c}=
e^{-ln(S(\psi)-\delta)}\textrm{tr}(p_{ln}-r_{\delta,ln}). \]
Hence we have
\[\textrm{tr}(p_{ln}-r_{\delta,ln})\le e^{ln(S(\psi)-\delta)}.\]
Thus, if there would be a number $c>0$ and a $\delta_{0}>0$ such that
$\limsup_{n\to\infty}\psi^{(ln)}(\idn-r_{\delta_{0},ln})=c>0$ there
would be a contradiction to the Proposition 2.1 in \cite{wir}, which
implies that there is no sequence $(q_{n})$ of projections in
${\mathcal A}^{(n)}$ with $\psi^{(n)}(q_{n})\ge f>0$ and
$\limsup_{n\to\infty}\frac{1}{n}\log\textrm{tr}(q_{n})<S(\psi)$. This
shows our claim (\ref{eq:pDp-entropy}).

The limit assertion (\ref{eq:pDp-entropy}) implies the following
estimate for sufficiently large $n\in \nn$:
\begin{eqnarray}\label{eq:pDp-estimate}
\textrm{tr}(p_{ln}D_{\psi^{(ln)}}p_{ln}e_{ln})&\le &  e^{-ln(S(\psi)-\delta)}\sum_{i\in T_{\delta,ln}}\textrm{tr}(r_{i,ln}e_{ln})+\eps\nonumber \\
&\le &  e^{-ln(S(\psi)-\delta)}\sum_{i=1}^{b_{ln}}\textrm{tr}(r_{i,ln}e_{ln})+\eps \nonumber \\
&=& e^{-ln(S(\psi)-\delta)}\textrm{tr}(p_{l\n}e_{l\n})+\eps.
\end{eqnarray}
Note that each projection $p_{ln}$ can be represented as a sum of
minimal projections $s_{i,n}\in {\mathcal B}_{l}^{(n)}$:
\[ p_{ln}=\sum_{i\in I(p_{ln})}s_{i,n},\]
for some index set $I(p_{ln})$. Recall that by construction the
operators $s_{i,n}$ are projections onto eigensubspaces of
$D_{\varphi^{(ln)}}$. Let $\{\mu_{i,n}\}_{i=1}^{a_{l}^{n}}$ be the
eigen-values corresponding to the set of minimal projections $\{s_{i,n}\}_{i=1}^{a_{l}^{n}}$. Then by (\ref{eq:onedim-estimate}) we have
\begin{eqnarray}\label{eq:pe-estimate}
  \textrm{tr}(p_{ln}e_{ln}) &\le&
  e^{n(S(\psi,\varphi)l+(S(\psi)+\eta)l-E_{l}+2\delta)}\sum_{i\in I(p_{ln})
}\varphi^{(ln)}(s_{i,n})\textrm{tr}(s_{i,n}e_{ln})\nonumber \\
&=&e^{n(S(\psi,\varphi)l+(S(\psi)+\eta)l-E_{l}+2\delta)}\sum_{i\in I(p_{ln})
}e^{n(\frac{1}{n}\log\textrm{\footnotesize tr}(s_{i,n}))}\mu_{i,n}\textrm{tr}(s_{i,n}e_{ln})\nonumber \\
&\le&
e^{n(S(\psi,\varphi)l+(S(\psi)+\eta)l+3\delta)}\varphi^{(ln)}(p_{ln}e_{ln})\quad
\textrm{ (by (\ref{eq:dim-typical}))}\nonumber \\
&\le& e^{n(S(\psi,\varphi)l+(S(\psi)+\eta)l+3\delta)}\varphi^{(ln)}(e_{ln})\nonumber \\
&=&
e^{n(S(\psi,\varphi)l+(S(\psi)+\eta)l+3\delta)}\varphi^{(y)}(t_{y})\quad\quad\quad
\textrm{ (by (\ref{eq:eln-ty}))}\nonumber \\
&\le & e^{n(S(\psi,\varphi)l+(S(\psi)+\eta)l+3\delta)}e^{-y(S(\psi,\varphi)+a)}\quad\textrm{ (by (\ref{eq:toosmall}))},
\end{eqnarray}
Taking into account (\ref{eq:pDp-estimate}) and (\ref{eq:pe-estimate})
we obtain
\[\textrm{tr}(p_{ln}D_{\psi^{(ln)}}p_{ln}e_{ln})\le
e^{-ln((\frac{y}{ln}-1)S(\psi,\varphi) +\frac{y}{ln}a-\delta-\eta-\frac{3\delta}{l})}+\eps.\]
Observe that $\lim_{y\to\infty}\frac{y}{ln}=1$ and $a>0$ hold. Hence
if $\delta,\eta>0$ are chosen appropriately small we can achieve that
the exponent in the last inequality becomes negative for large
$n$. This fact combined with (\ref{eq:eln-estimate}) and
(\ref{eq:eln-ty}) shows that
\[\lim_{y\to\infty}\psi(t_{y})=0\]
holds. This limit assertion toghether with (\ref{eq:iid-lim-beta<s})
shows (\ref{eq:the-limit}).$\qquad\Box$
%------------------------------MONOTONICITY OF THE RELATIVE ENTROPY-----------
\section{Monotonicity of the Quantum Relative Entropy}
Let $T:{\mathcal A}\to {\mathcal A}$ be a unital, completely positive
linear map of a finite dimensional $C^{\ast}$-algebra ${\mathcal A} $
into itself. There is a dual map $T_{\#}$ with respect to the inner product defined by the
unique trace $\textrm{tr}$ (which extends canonically to the state space ${\mathcal
  S}({\mathcal A})$) given by
\[ \textrm{tr}(D_{\psi}T(a))=\textrm{tr}(T_{\#}(D_{\psi})a)\quad
\textrm{for all } a\in{\mathcal A}\textrm{ and all } \psi\in {\mathcal
  S}({\mathcal A}).\] 
In \cite{kraus} K. Kraus showed that for each unital, completely
positive map $T$ there exist a finite dimensional Hilbert space
$\hr_{a}$, a state $\sigma$ on ${\mathcal B}(\hr_{a})$ and a unitary
operator $U:\hr\otimes\hr_{a}\to\hr\otimes\hr_{a}$ such that
\begin{eqnarray}\label{eq:second-kraus}
T_{\#}(D_{\psi})=\textrm{tr}_{\hr_{a}}(U(D_{\psi}\otimes D_{\sigma})U^{\ast}),
  \end{eqnarray}
where $\textrm{tr}_{\hr_{a}}$ denotes the partial trace with respect
to the Hilbert space $\hr_{a}$.
\begin{theorem}[Monotonicity of the relative entropy]\label{mono} 
Let $\psi,\varphi$ be states on ${\mathcal A}$ and let $T:{\mathcal A}\to {\mathcal
    A}$ be a unital, completely positive map of ${\mathcal A}$ into
  itself. Then it holds
  \begin{eqnarray}\label{eq:monotonicity1}
S(\psi,\varphi)\ge S(\psi\circ T,\varphi\circ T),
    \end{eqnarray}
or equivalently
\begin{eqnarray}\label{monotonicity2}
 S(D_{\psi},D_{\varphi})\ge S(T_{\#}(D_{\psi}),T_{\#}(D_{\varphi})).
\end{eqnarray}
\end{theorem}
\textbf{Proof of Theorem \ref{mono}:} The monotonicity holds
trivially in the case $ S(\psi,\varphi)=\infty$. We consider the case
$S(\psi,\varphi)<\infty$ and define two states
$\tilde\psi,\tilde\varphi$ on ${\mathcal A}\otimes{\mathcal
  B}(\hr_{a})$ by their density operators: 
\[D_{\tilde\psi}:=U(D_{\psi}\otimes D_{\sigma})U^{\ast}\textrm{ and } D_{\tilde\varphi}:=U(D_{\varphi}\otimes D_{\sigma})U^{\ast},\]
where the unitary operator $U:\hr\otimes\hr_{a}\to\hr\otimes\hr_{a}$
and the density operator $D_{\sigma}\in{\mathcal B}(\hr_{a})$ are
chosen according to the representation (\ref{eq:second-kraus}) of the
completely positive, unital map $T$.
Observe that according to the representation (\ref{eq:second-kraus})
we have
\begin{eqnarray}\label{eq:psitilde-a}
D_{\tilde\psi\!\upharpoonright\!{\mathcal A}}=\textrm{tr}_{\hr_{a}}(U(D_{\psi}\otimes D_{\sigma})U^{\ast})=T_{\#}(D_{\psi}).
\end{eqnarray}
and
\begin{eqnarray}\label{eq:varphitilde-a}
D_{\tilde\varphi\!\upharpoonright\!{\mathcal A}}=T_{\#}(D_{\varphi}).  
\end{eqnarray}
Consider the quasilocal algebra $({\mathcal A}\otimes{\mathcal
  B}(\hr_{a}))^{\infty}$ and the stationary product states
$\tilde{\psi}_{\infty}$ with $\tilde{\psi}^{(1)}=\tilde{\psi}$,
$\tilde{\varphi}_{\infty}$ with $\tilde{\varphi}^{(1)}=\tilde{\varphi}$ on $({\mathcal A}\otimes{\mathcal
  B}(\hr_{a}))^{\infty}$. Note that the algebra ${\mathcal
  A}^{\infty}$ is a subalgebra of $({\mathcal A}\otimes{\mathcal
  B}(\hr_{a}))^{\infty}$. The definition (\ref{eq:beta}) implies that
\[ \beta_{\eps,n}(\tilde\psi_{\infty}\!\upharpoonright\!{\mathcal
  A}^{\infty},\tilde\varphi_{\infty}\!\upharpoonright\!{\mathcal
  A}^{\infty})\ge
\beta_{\eps,n}(\tilde\psi_{\infty},\tilde\varphi_{\infty}),\]
and hence by Theorem \ref{iid-lim-beta=s}
\begin{eqnarray}
  -S(\tilde\psi\!\upharpoonright\!{\mathcal A},\tilde\varphi\!\upharpoonright\!{\mathcal A})&=&\lim_{n\to\infty}\frac{1}{n}\beta_{\eps,n}(\tilde\psi_{\infty}\!\upharpoonright\!{\mathcal
  A}^{\infty},\tilde\varphi_{\infty}\!\upharpoonright\!{\mathcal
  A}^{\infty})\nonumber\\
&\ge&\lim_{n\to\infty}\frac{1}{n}\beta_{\eps,n}(\tilde\psi_{\infty},\tilde\varphi_{\infty})\nonumber\\
&=& -S(\tilde\psi,\tilde\varphi).\nonumber
\end{eqnarray}
Using (\ref{eq:psitilde-a}) and (\ref{eq:varphitilde-a}) this last
inequality chain shows that
\begin{eqnarray}
  S(T_{\#}(D_{\psi}),T_{\#}(D_{\varphi}))&\le&
  S(\tilde\psi,\tilde\varphi)\nonumber \\
&=& S(U(D_{\psi}\otimes D_{\sigma})U^{\ast},U(D_{\varphi}\otimes
D_{\sigma})U^{\ast})\nonumber \\
&=&S(D_{\psi},D_{\varphi}),\nonumber
\end{eqnarray}
since the relative entropy is invariant under the action of the
unitaries and simultaneous tensor multiplication with
$D_{\sigma}$. $\qquad \Box$


%----------------------APPENDIX-----------------------------------------
\section{Appendix: The Hiai-Petz-Approximation}
In this section we give a proof of the result established by Hiai and
Petz in \cite{petz}. At the same time we simplify slightly their
arguments. Let $\varphi$ be a stationary product state on the
quasilocal algebra ${\mathcal A}^{\infty}$.
Consider the spectral resolution of $D_{\varphi^{(1)}}$
\[ 
D_{\varphi^{(1)}}=\sum_{i=1}^{d}\lambda_{i}e_{i},\]
where $\lambda_{i}$ are the eigenvalues and $e_{i}$ are projectors
onto a complete set of eigenvectors of $D_{\varphi^{(1)}} $. The
density operator $D_{\varphi^{(n)}}$ can be written as
\[
D_{\varphi^{(n)}}=\sum_{i_{1},\ldots,i_{n}=1}^{d}\lambda_{i_{1}}\cdot\ldots\cdot \lambda_{i_{n}} e_{i_{1}}\otimes\ldots\otimes e_{i_{n}},    
\] 
or after collecting all one dimensional projectors
$e_{i_{1}}\otimes\ldots\otimes e_{i_{n}}$ which correspond to the same
eigenvalue of $D_{\varphi^{(n)}}$ as 
\begin{eqnarray}\label{eq:domultipl}
D_{\varphi^{(n)}}=\sum_{ n_{1},\ldots n_{d}: n_{1}+\ldots +n_{d}=n}\left(\prod_{k=1}^{d}\lambda_{k}^{n_{k}}\right)p_{n_{1},\ldots n_{d}},  
\end{eqnarray}
with 
\[ p_{n_{1},\ldots n_{d}}:=\sum_{(i_{1},\ldots, i_{n})\in I_{n_{1},\ldots n_{d}}}e_{i_{1}}\otimes\ldots\otimes e_{i_{n}},\]
 where
\[ I_{n_{1},\ldots n_{d}}:=\{(i_{1},\ldots, i_{n}):\#\{j:i_{j}=k\}=n_{k}\textrm{ for } 1\le k\le d\}.\]
We define the map
\[ E_{n}:{\mathcal A}^{(n)}\to \bigoplus_{n_{1},\ldots ,n_{d}:\ n_{1}+\ldots +n_{d}=n} p_{n_{1},\ldots ,n_{d}}{\mathcal A}^{(n)}p_{n_{1},\ldots ,n_{d}},\]
by 
\begin{eqnarray}\label{eq:tracecondexp}
 E_{n}(a):=\sum_{n_{1},\ldots ,n_{d}:\ n_{1}+\ldots +n_{d}=n}p_{n_{1},\ldots ,n_{d}} a \ p_{n_{1},\ldots ,n_{d}}.
\end{eqnarray}
The next theorem is due to Hiai and Petz, Lemma 3.1 and Lemma 3.2 in \cite{petz}.
\begin{theorem}[Hiai and Petz]\label{hiai-petz-lemma}
If $\psi$ is a stationary state on ${\mathcal A}^{\infty}$ and ${\mathcal B}_{n}$ is the abelian subalgebra of ${\mathcal A}^{(n)}$ generated by $\{p_{n_{1},\ldots n_{d}}D_{\psi^{(n)}}p_{n_{1},\ldots n_{d}}\}_{n_{1},\ldots ,n_{d}}\cup \{p_{n_{1},\ldots n_{d}}\}_{n_{1},\ldots ,n_{d}}$ then
\begin{eqnarray}\label{eq:relentropyreduction}
  S(\psi^{(n)},\varphi^{(n)})= S(\psi^{(n)}\upharpoonright {\mathcal B}_{n},\varphi^{(n)}\upharpoonright {\mathcal B}_{n}) +S(\psi^{(n)}\circ E_{n} )-S(\psi^{(n)}),
\end{eqnarray}
and
\begin{eqnarray}\label{eq:relentropyapprox}
  S(\psi^{(n)}\circ E_{n} )-S(\psi^{(n)})\le d\log(n+1),
\end{eqnarray}
and consequently we have
\begin{eqnarray}\label{eq:abel-approx} 
\lim_{n\to\infty}\frac{1}{n}S(\psi^{(n)}\upharpoonright {\mathcal B}_{n},\varphi^{(n)}\upharpoonright {\mathcal B}_{n})=s(\psi,\varphi).  
\end{eqnarray}
\end{theorem}
\textbf{Proof of Theorem \ref{hiai-petz-lemma}:} We will prove the
theorem only in the case where the density operator $D_{\varphi^{(1)}}$
 is invertible, this extends trivially to the general situation. Since the algebra ${\mathcal B}_{n}$ is abelian it is
 representable as:
 \begin{eqnarray}\label{eq:Bn}
{\mathcal B}_{n}=\bigoplus_{i=1}^{a_{n}}\cc\cdot q_{i,n}.
 \end{eqnarray}
Using the definition of the algebra $ {\mathcal
  B}_{n}$ and the representation (\ref{eq:Bn}) it can be
easily seen that
\begin{eqnarray}\label{eq:appendix1}
 D_{\psi^{(n)}\!\upharpoonright {\mathcal B}_{n}}=\sum_{i=1}^{a_{n}}\textrm{tr}(D_{\psi^{(n)}}q_{i,n})q_{i,n},
\end{eqnarray}
i.e. the density operator $D_{\psi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}}\in {\mathcal B}_{n} $ corresponding to $\psi^{(n)}\!\upharpoonright
{\mathcal B}_{n}$ with respect to the trace $\textrm{tr}_{{\mathcal
    B}_{n}}$ of ${\mathcal B}_{n}$ is
given by (\ref{eq:appendix1}). Similarily we have also
\begin{eqnarray}\label{eq:phirestriction}
D_{\varphi^{(n)}\!\upharpoonright {\mathcal B}_{n}}=\sum_{i=1}^{a_{n}}\textrm{tr}(D_{\varphi^{(n)}}q_{i,n})q_{i,n}.
\end{eqnarray}
We have the following equation for the difference of the relative entropies: 
\begin{eqnarray}\label{relentropiedifference}
S(  D_{\psi^{(n)}},D_{\varphi^{(n)}})&-&S(D_{\psi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}},D_{\varphi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}})=\nonumber\\
&-& 
S(D_{\psi^{(n)}})-\textrm{tr}(D_{\psi^{(n)}}\log D_{\varphi^{(n)}})+S(D_{\psi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}}) \nonumber\\
&+& \textrm{tr}_{{\mathcal B}_{n}}(D_{\psi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}}\log D_{\varphi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}}).\nonumber
\end{eqnarray}
Since $D_{\varphi^{(n)}}$ is invertible $\log D_{\varphi^{(n)}\!\upharpoonright {\mathcal
    B}_{n} }$ is
well defined and contained in $ {\mathcal
    B}_{n}$. All projections $q_{i,n}$
commute with  $\log D_{\varphi^{(n)}}$. Thus we obtain
\begin{eqnarray}\label{eq:appendix2}
 \textrm{tr}_{{\mathcal B}_{n}}(D_{\psi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}}\log D_{\varphi^{(n)}\!\upharpoonright {\mathcal
    B}_{n} })&=& \textrm{tr}(D_{\psi^{(n)}}\log
D_{\varphi^{(n)}})\nonumber\\
&+&\sum_{i=1}^{a_{n}}\textrm{tr}(D_{\psi^{(n)}}q_{i,n})\log
\textrm{tr}(q_{i,n}),
\end{eqnarray}
where we have used (\ref{eq:appendix1}) and
(\ref{eq:phirestriction}). And hence
\begin{eqnarray}\label{eq:difference2}
S(D_{\psi^{(n)}},D_{\varphi^{(n)}})-S(D_{\psi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}},D_{\varphi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}})&=&S(D_{\psi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}})-S(D_{\psi^{(n)}})\nonumber\\
&+&\sum_{i=1}^{a_{n}}\textrm{tr}(D_{\psi^{(n)}}q_{i,n})\log
\textrm{tr}(q_{i,n}).\nonumber \\
\end{eqnarray}
On the other hand we can easily see that
\[E_{n}(D_{\psi^{(n)}})=\sum_{i=1}^{a_{n}}\frac{\textrm{tr}(D_{\psi^{(n)}}q_{i,n})}{\textrm{tr}(q_{i,n})}q_{i,n},\]
 and therefore
\[S(E_{n}(D_{\psi^{(n)}}))=S(D_{\psi^{(n)}\!\upharpoonright {\mathcal
    B}_{n}})+\sum_{i=1}^{a_{n}}\textrm{tr}(D_{\psi^{(n)}}q_{i,n})\log
\textrm{tr}(q_{i,n}).\]
Hence by (\ref{eq:difference2}) we have established the first equality of the theorem:
\[S(\psi^{(n)},\varphi^{(n)})= S(\psi^{(n)}\upharpoonright {\mathcal B}_{n},\varphi^{(n)}\upharpoonright {\mathcal B}_{n}) +S(\psi^{(n)}\circ E_{n} )-S(\psi^{(n)}).
\]
In order to establish (\ref{eq:relentropyapprox}) we need the following
entropy inequality:
\[ S(\sum_{i=1}^{k}a_{i}D_{i})\le\sum_{i=1}^{k}a_{i}
S(D_{i})+H(\{a_{i}\}_{i=1}^{k}),\]
where $D_{i}$ are density operators, $a_{i}$ are non negative real
numbers summing up to $1$ and $H(\{a_{i}\}_{i=1}^{k})$ denotes the
Shannon entropy of the probability distribution $ \{a_{i}\}_{i=1}^{k} $. An elementary proof of this inequality can
be found in \cite{niel}. Consider a spectral resolution of the
density operator $D_{\psi^{(n)}}=\sum_{i=1}^{d^{n}}\mu_{i,n}p_{i,n}$
into one-dimensional projections. Then we have
\begin{eqnarray}
S(E_{n}(D_{\psi^{(n)}}))-S(D_{\psi^{(n)}})&\le&
\sum_{i=1}^{d^{n}}\mu_{i,n}S(E_{n}(p_{i,n}))+H(\{\mu_{i,n}\}_{i=1}^{d^{n}})\nonumber
\\
&-&S(D_{\psi^{(n)}})\nonumber\\
&=& \sum_{i=1}^{d^{n}}\mu_{i,n}S(E_{n}(p_{i,n}))\nonumber \\
&\le& d\log (n+1),\nonumber
\end{eqnarray}
since for each projector $p_{i,n}$ the operator $p_{n_{1},\ldots
  ,n_{d}}(p_{i,n})p_{n_{1},\ldots
  ,n_{d}} $ has
 at most one dimensional range and hence the density operator
 $E_{n}(p_{i,n})$ can have at most $(n+1)^{d}$ dimensional range. Here we used an obvious estimate for the cardinality of partitions of the number $n$ into $d$ numbers. $\qquad \Box$

\textbf{Ackowledgement:} We are grateful to our colleagues Tyll
Kr\"uger, Ruedi Seiler and Arleta Szko\l a for their constant
interest and helpful comments. We are also thankful to Andreas Winter
for an interesting discussion stimulating the current work.\newline
This research was supported by DFG via SFB 288 ``Differentialgeometrie
und Quantenphysik'' at the TU Berlin.

%--------------------LITERATUR----------------------------------------------------
\begin{thebibliography}{99}
\bibitem{wir}
I. Bjelakovi\'c, T. Kr\"uger, Ra. Siegmund-Schultze, A. Szko\l a, The Shannon-McMillan Theorem
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\bibitem{bratteli}
O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical
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F. Hiai, D. Petz, The Proper Formula for Relative
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\bibitem{kraus}
K. Kraus, States, Effects and Operations. Fundamental Notions
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E. H. Lieb, M. B. Ruskai, Proof of the Strong Subadditivity of Quantum-
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\bibitem{lindblad}
G. Lindblad, Completely Positive Maps and Entropy Inequalities,
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\bibitem{niel} 
M. A. Nielsen, I. L. Chuang, Quantum Computation and
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\bibitem{simon}
B. Simon, Trace Ideals and Their Application, Cambridge University
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\end{thebibliography}

\end{document}

quant-ph0307170.extracted_bib
\begin{thebibliography}{99}
\bibitem{wir}
I. Bjelakovi\'c, T. Kr\"uger, Ra. Siegmund-Schultze, A. Szko\l a, The Shannon-McMillan Theorem
for Ergodic Quantum Lattice Systems, arXiv.org, math.DS/0207121, to
appear in \emph{Invent. Math.} (2003)

\bibitem{wir2}
I. Bjelakovi\'c, Ra. Siegmund-Schultze, An Ergodic Theorem for the
Quantum Relative Entropy, arXiv.org, quant-ph/0306094, submitted to
\emph{ Commun. Math. Phys.}

\bibitem{bratteli}
O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical
Mechanics I, Springer, New York 1979


\bibitem{petz} 
F. Hiai, D. Petz, The Proper Formula for Relative
  Entropy and its Asymptotics in Quantum Probability,
  \emph{Commun. Math. Phys.} 143, 99-114 (1991)

\bibitem{kraus}
K. Kraus, States, Effects and Operations. Fundamental Notions
of Quantum Theory, Lecture Notes in Physics 190, Springer-Verlag,
Berlin 1983 

\bibitem{lieb-ruskai}
E. H. Lieb, M. B. Ruskai, Proof of the Strong Subadditivity of Quantum-
Mechanical Entropy, \emph{J. Math. Phys.} 14, 1938-1941 (1973)

\bibitem{lindblad}
G. Lindblad, Completely Positive Maps and Entropy Inequalities,
\emph{Commun. Math. Phys.} 40, 147-151 (1975)


\bibitem{niel} 
M. A. Nielsen, I. L. Chuang, Quantum Computation and
  Quantum Information, Cambridge University Press, Cambridge 2000

\bibitem{ogawa}
T. Ogawa, H. Nagaoka, Strong Converse and Stein's Lemma in Quantum Hypothesis Testing, \emph{IEEE Trans. Inf. Th.}, vol. 46, No. 7, 2428-2433 (2000)

\bibitem{simon}
B. Simon, Trace Ideals and Their Application, Cambridge University
Press, Cambridge 1979

\bibitem{uhlmann}
A. Uhlmann, Relative Entropy and the Wigner-Yanase-Dyson-Lieb
Concavity in an Interpolation Theory, \emph{Commun. Math. Phys.} 54, 21-32 (1977)

\end{thebibliography}