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\centerline{\fontA Computation of Mutual Entropy}
\smallskip
\centerline{\fontA in Quantum Markov Chains}
\Bigskip
\centerline{\it$\hbox{ LUIGI ACCARDI }^\dag, \hbox{ MASANORI OHYA }^{\dag\dag}
\hbox{ and HIROKI SUYARI }^{\dag\dag}$}
\bigskip
$$\vbox{\it\settabs 2\columns
  \+ \dag Dipartimento di Matematica     & \dag\dag  Department of         
            \cr
  \+ \,\,Centro Matematico V.Volterra   & \quad Information Sciences       
       \cr
  \+ \,\,Universit\'a di Roma II, Italy & \quad Science University of
Tokyo, Japan \cr}
$$
\Bigskip
%\centerline{\bf Abstract}
%\bigskip
\item{}\noindent
There are several damping phenomena in quantum optics.
Such phenomena have been usually explained by open systems.
In statistical physics, open system dynamics have been used
to study the irreversibility and the approach to equilibrium.
In this paper the dynamical change of the mutual entropy
for an open system, frequently studied in the quantum optics literature,
is rigorously computed through a model of
quantum Markov chain. Especially, the concrete formula of Stinespring
expression for such a model is obtained and applied to the derivation
of the mutual entropy, and some computational results aer presented.
 
\Bigskip
\leftline{\fontB 1 Introduction}
\bigskip
The purpose of this paper is to show the use of the mutual entropy
and of quantum Markov chains (QMC for short)
in the study of an irreversible process arising from 
a model, well known in quantum optics (cf. section 2)
and used to describe \lq\lq dissipation" or \lq\lq damping".
The irreversibility of the evolution arises from taking partial trace
on a unitary evolution.
In section 3, we review some properties of the mutual entropy
and of the generalized QMC, introduced in [14] and [1], respectively.
As shown in [3], notions can be treated
in the common framework of \lq\lq lifting".
In section 4, we construct a QMC associated to the Hamiltonian model
of section 2.
The existence of Stinespring-Kraus expression of a 
transition expectation can be deduced from abstract theorems, 
but its concrete expression is missing.
In section 5, we rigorously derive this concrete expression for our model.
In section 6, we apply this expression to the computation of the mutual entropy
 in QMC.
In section 7, some computational results are presented.
In section 8, some further extensions of the present study are outlined :
their full discussion will be given in a forthcoming paper.
\Bigskip
\leftline{\fontB 2 Model for Open Systems with Harmonic Oscillators}
\bigskip
In the statistical mechanics of an open systems, the state of a system changes
under the influence of the outside environment (reservoir)
This means that one considers an interaction between the system
and the reservoir and , after having considered the associated unitary
evolution, one traces away the reservoir degrees of freedom, thus obtaining
the reduced state of the system.
A mathematical expression of this procedure is given as follows.
 
Let a system $\Sigma_0$ be described by a Hilbert space ${\Cal H}_0$,
which interacts with an external system $\Sigma_1$ described by another
Hilbert space ${\Cal H}_1$, and let the initial states of $\Sigma_0,\;\Sigma_1$
be $\rho \in {\frak S}_0,\;\omega \in {\frak S}_1$
(${\frak S}_i$ is the set of all states on ${\Cal H}_i$ ($i=0, 1$)),
respectively.
Then the combined state $\tilde \rho \in {\frak S}_0\otimes {\frak S}_1$
after the interaction between the two systems is given by 
$$\tilde \rho =U_t\left( {\rho \otimes \omega } \right)U_t^*\eqno(2.1)$$
where $U_t=\exp \left( {-itH} \right)$ with a total Hamiltonian $H$ on 
${\Cal H}_0\otimes{\Cal H}_1$. Here we took $\hbar = 1$.
 
In quantum optics, a total Hamiltonian $H$ frequently used to describe
two weakly coupled oscillators in quantum optics is given by [12]:
$$H=H_0+H_1+V\eqno(2.2)$$
$$H_0=a^*a,\quad H_1=\sum\limits_j {b_j^*b_j},\quad
V=\sum\limits_j {\left( {\varepsilon _jb_ja^*+\varepsilon _j^*b_j^*a}
\right)}\eqno(2.3)$$
where $H_0,\;H_1,\;V$ are the Hamiltonians for the observed system $\Sigma_0$,
the external system (reservoir) $\Sigma_1$ and the interaction between
$\Sigma_0$ and
$\Sigma_1$, respectively. $a,a^*$ on ${\Cal H}_0$ and $b,b^*$ on ${\Cal
H}_1$ are
pairs of anihilation and creation operators, respectively, and 
$\varepsilon _j\left( {j\in N} \right)$ are the coupling constants.
 
In this paper we assume, for simplicity, that the reservoir has a single
mode, that is,
the total hamiltonian $H$ is given by 
$$H=H_0+H_1+V=a^*a+b^*b+\varepsilon
 \left( {a^*b+ab^*} \right)\eqno(2.4)$$
 
In the model, the evolution of the initial state $\rho$ after the interaction
is mathematically expressed as
$$\rho \mapsto \bar \rho =tr_{{\Cal H}_1}U_t^*\left( {\rho \otimes \omega }
\right)U_t.\eqno(2.5)$$
 
\Bigskip
\leftline{\fontB 3 Mutual Entropy in Quantum Markov Chain}
\bigskip
In communication theory, it is very important to measure the efficiency 
in the transmission of the information from an input system to an output one.
There are some numerical functions giving an estimate of this eficiencies,
such as entropy, mutual entropy,  capacity, error probability, signal-noise
ratio.
On the other hand, the entropies have played an important role to explain
the irreversibility in physics. In particular, the mutual entropy has the above
significant meaning in both communication and physics, that is, the decrease of
the mutual entropy w.r.t. time represents the loss of the initial information
in communication processes
and also explains the irreversibile behavior of the observed system [17].
In this section we associate a quantum Markov chain (QMC) to the
Hamiltonian model,
described by (2.5) and we study its mutual entropy.
 
It is convenient to review some definitions and fundamental results :
\bigskip
{\bf Definition 3.1} [14] :
A completely positive map
$\Lambda^*:{\frak S}\left( {\Cal H} \right)\to {\frak S}\left( {\Cal K} \right)$
is called a quantum mechanical channel.
Here ${\frak S}\left( {\Cal H} \right)$ is the set of states on a Hilbert
space ${\Cal H}$.
\medskip
The quantum mechanical channel $\Lambda ^*:{\frak S}_0\to {\frak S}_0$ in
our model
 can be written as :
$$\Lambda^*\rho =tr_{{\Cal H}_1}U_t^*\left( {\rho \otimes \omega }
\right)U_t\eqno(3.1)$$
The concept of channel is related to the concept of \lq\lq lifting"
introduced in [3].
\bigskip
{\bf Definition 3.2} [3] :
Let ${\frak S}_0,{\frak S}_1$ be the set of all states on 
${\Cal H}_0,{\Cal H}_1$  respectively. A lifting from ${\frak S}_0$ to
${\frak S}_0\otimes{\frak S}_1$ is a continuous map
$${\Cal E}^*:{\frak S}_0\to {\frak S}_0\otimes{\frak S}_1\eqno(3.2)$$
\medskip
For example the correspondence  $\rho \mapsto \tilde \rho $
in (2.1) is obviously a lifting,
that is, we can take this lifting ${\Cal E}^*:{\frak S}_0\to {\frak
S}_0\otimes{\frak S}_1$
by
$${\Cal E}^*\rho =U_t\left( {\rho \otimes \omega } \right)U_t^*\eqno(3.3)$$
The correspondence between a quantum mechanical channel $\Lambda ^*$
and a lifting ${\Cal E}^*$  is explained in [3].
 
Let the Schatten decomposition of $\rho \in {\frak S}\left( {\Cal H} \right)$ be
$$\rho =\sum\limits_k {\lambda _kE_k}\eqno(3.4)$$
$$\lambda _1\ge \lambda _2\ge \cdots \ge \lambda _n\ge \cdots $$
$$E_i\bot E_j\quad \left( {i\ne j} \right)$$
where $\lambda _k$ is an eigenvalue of $\rho$ and $E_k$ is the associated
onedimensional
projection.
 
Using a quantum mechanical channeles in Definition 3.1,
the following two compound states [14] are introduced :
$${\Cal E}^*_0\rho\equiv\sigma _0=\rho \otimes \Lambda ^*\rho ,
\quad {\Cal E}^*_E\rho\equiv\sigma _E=\sum\limits_k
 {\lambda _kE_k\otimes \Lambda ^*E_k}\eqno(3.5)$$
\noindent
Clearly both ${\Cal E}^*_0$ and ${\Cal E}^*_E$ are (nonlinear) liftings.
The defnition of mutual entropy $I\left( {\rho \,;\,\Lambda ^*} \right)$ 
is given as follows :
\bigskip
{\bf Definition 3.3} [14] :
Using the above notations (3.4) and (3.5), the mutual entropy
$I\left( {\rho \,;\,\Lambda ^*} \right)$ is defined by
$$I\left( {\rho \,;\,\Lambda ^*} \right)=
\mathop {\sup p}\limits_E\left\{ {S\left( {\sigma _E,\;\sigma _0} \right)\;
;\;E=\left\{ {E_k} \right\}} \right\}\eqno(3.6)$$
where ${S\left( {\sigma _E,\;\sigma _0} \right)}$
is the relative entropy :
$$S\left( {\sigma _E,\;\sigma _0} \right)=
tr\sigma _E\left( {\log \sigma _E-\log \sigma _0} \right)\eqno(3.7)$$
By applying (3.4), (3.5) and (3.7), we have :
\bigskip
{\bf Proposition 3.4} [14] :
$$S\left( {\sigma _E,\;\sigma _0} \right)=
\sum\limits_n {\lambda _nS\left( {\Lambda ^*E_n,\;\Lambda ^*\rho } \right)}
\eqno(3.8)$$
The following fundamental theorem is well known in communication theory.
\bigskip
{\bf Theorem 3.5} [14] :
For an input state $\rho$ and a channel $\Lambda^*$,
the following inequalities hold.
$$0\le I\left( {\rho \,;\,\Lambda ^*} \right)\le \min \left\{ {S\left( \rho
 \right),
\;S\left( {\Lambda ^*\rho } \right)} \right\}\eqno(3.9)$$
\medskip
This theorem means that the information correctly transmitted through a
channel $\Lambda^*$
is always less than the initial information.
In other words the mutual entropy is useful as an efficiency of communication
processes [7, 23, 24]. Other properties of the mutual entropy are discussed
in [17, 19].
 
Then, before reviewing of QMC, we need the concept of a transition expectation :
\bigskip
{\bf Definition 3.6} [4] :
Let ${\Cal B}_0,{\Cal B}_1$ be the algebras of all bounded operators
on Hilbert spaces ${\Cal H}_0,{\Cal H}_1$ respectively, and
let ${\Cal B}_0\otimes{\Cal B}_1$ be a fixed tensor product of ${\Cal B}_0$
and ${\Cal B}_1$.
A transition expectation from ${\Cal B}_0\otimes{\Cal B}_1$ to ${\Cal B}_0$
is a completely
positive map ${\Cal E}^*:{\Cal B}_0\otimes{\Cal B}_1\to {\Cal B}_0$ satisfying
$${\Cal E}\left( {1\otimes 1} \right)=1\eqno(3.10)$$
When ${\Cal B}={\Cal B}_0={\Cal B}_1$ we say that ${\Cal E}$ is a transition
expectation on ${\Cal B}$.
 
By using the transition expectations, a generalized QMC is defined as follows : 
\bigskip
{\bf Definition 3.7} [4] : Let $\left\{ {{\Cal E}_n} \right\}_{n\ge 0}$ be
a sequence of transition
 expectations on $\Cal B$. Then there exists a unique completely positive
identity
preserving map $E_{0]}:\otimes^n{\Cal B}\to {\Cal B}$ such that for each
integer $n$ and for each
$A_0,A_1,\cdots ,A_n\in {\Cal B}$, one has
$$E_{0]}\left( {A_0\otimes A_1\otimes \cdots \otimes A_n\otimes 1\otimes
\cdots } \right)
={\Cal E}_0\left( {A_0\otimes {\Cal E}_1\left( {A_1\otimes \cdots \otimes
{\Cal E}_n
\left( {A_n\otimes 1} \right)\cdots } \right)} \right)\eqno(3.11)$$
Let $\varphi_0$ be a state on ${\Cal B}$ and define a state $\varphi$ on
$\otimes _N{\Cal B}$ by 
$$\varphi =\varphi _0\circ E_{0]}\eqno(3.12)$$
Then $\varphi$ satisfies
$$\varphi \left( {A_0\otimes A_1\otimes \cdots \otimes A_n\otimes 1\otimes
\cdots } \right)
=\varphi _0\left( {{\Cal E}_0\left( {A_0\otimes {\Cal E}_1\left(
{A_1\otimes \cdots
 \otimes {\Cal E}_n\left( {A_n\otimes 1} \right)\cdots } \right)} \right)}
\right)\eqno(3.13)$$
The state $\varphi$, characterized by (3.12) is called the generalized
Markov chain
associated to the pair $\left( {\varphi _0,\;\left\{ {{\Cal E}_n} \right\}}
\right)$.
If for each $n$
$${\Cal E}\equiv {\Cal E}_n={\Cal E}_0,\eqno(3.14)$$
then the state $\varphi$ is called homogeneous.
 
Clearly the dual of a \lq\lq linear" lifting is a transition expectation,
therefore to any linear lifting one can associate a QMC in the standard way
as explained in [3].
If the lifting is of convex product type, then we can take an advantage of
the special structure to extend the construction of QMC to the case of a
not necessarily
linear lifting. Then, according to the procedure shown in [3], at first we can construct
a lifting of convex product type, associated to the channel (3.1) in our
model as follows :
$${\Cal E}^*\rho =\sum\limits_k {\lambda _k\rho _k\otimes }\Lambda^*\rho
_k\eqno(3.15)$$
for a properly defined decomposition : $\rho =\sum\limits_k {\lambda_k\rho _k}$
(e.q., spectral decompositon, Schatten decomposition).
 
Then if the initial state $\rho\in{\frak S}$ has an above decomposition
$$\rho =\sum\limits_{k_1} {\lambda _{k_1}^{(1)}\rho _{k_1}}\eqno(3.16)$$
then by (3.15) ${\Cal E}^*\rho$ has the form :
$${\Cal E}^*\rho =\sum\limits_{k_1} {\lambda _{k_1}^{(1)}\rho _{k_1}\otimes }
\Lambda ^*\rho _{k_1}\eqno(3.17)$$
If we apply ${\Cal E}^*$ to $\Lambda ^*\rho _{k_1}$ in (3.17),
then we obtain the following state on $\left( {\otimes {\frak S}} \right)^3$:
$$\eqalignno{&\quad \quad \sum\limits_{k_1} {\lambda _{k_1}^{(1)}\rho
_{k_1}\otimes }
{\Cal E}^*\left( {\Lambda ^*\rho _{k_1}} \right)\cr
  &=\sum\limits_{k_1} {\lambda _{k_1}^{(1)}\rho _{k_1}\otimes }\sum\limits_{k_2}
 {\lambda _{k_2}^{(1,2)}\rho _{k_2}\otimes \Lambda ^*\rho _{k_2}}\cr
  &=\sum\limits_{k_1,k_2} {\lambda _{k_1}^{(1)}\lambda _{k_2}^{(1,2)}
\rho _{k_1}\otimes \rho _{k_2}\otimes \Lambda ^*\rho _{k_2}}&(3.18)\cr}$$
where $\Lambda ^*\rho _{k_1}$ is the same type decomposition :
$\Lambda^*\rho _{k_1}=\sum\limits_{k_2} {\lambda _{k_2}^{(1,2)}\rho _{k_2}}$.
At the n-th iteration we can obtain the state ${\Cal E}_{n]}^*\rho \in
\left( {\otimes {\frak S}}
 \right)^{n+1}$ defined by
$${\Cal E}_{n]}^*\rho =\sum\limits_{k_1,\cdots ,k_n} {\lambda _{k_1}^{(1)}
\lambda_{k_2}^{(1,2)}\cdots \lambda _{k_n}^{(n-1,n)}\rho _{k_1}\otimes \rho
_{k_2}
\otimes \cdots \otimes \rho_{k_{n-1}}\otimes \Lambda^*\rho_{k_n}}\eqno(3.19)$$
where 
$$\Lambda ^*\rho _{k_i}=\sum\limits_{k_{i+1}} {\lambda _{k_{i+1}}^{(i,i+1)}
\rho _{k_{i+1}}}\quad \left( {i=1,\;\cdots \;,n-1} \right)\eqno(3.20)$$
The form such as  (3.5) and (3.15) is a natural generalization of a joint
probability
in classical systems. It follows that   is a kind of a compound state introduced
in [14]. However it is not a joint probability in the sense of classical system,
because in general a joint probability in the sense of classical case does
not exist
in quantum systems. The difference between classical probability and quantum
probability is discussed in [7].
 
By (3.19) the state ${\Lambda^*}^n\rho \in {\frak S}$
after n-th iteration of the interaction between the system $\Sigma_0$ and
the reservoir
$\Sigma_1$ can be written by  
$${\Lambda^*}^n\rho =tr_{[1,n]}{\Cal E}_{n]}^*\rho \eqno(3.21)$$
where $tr_{[1,n]}$ is the partial trace w.r.t.$\otimes^n{\Cal H}_0$.
 
Using the above definitions (3.19), (3.20) and (3.21) we will investigate
the dynamical change of mutual entropy 
$\mathop {\lim}\limits_{n\to \infty }I\left( {\rho \,;\,{\Lambda^*}^n} \right)$
in the following paper [6].
 
These formulations are applicable to other types of quantum Markov chains.
 
\Bigskip
\leftline{\fontB 4 A model of Quantum Markov Chain for Quantum Optics}
\bigskip
In order to construct our model of QMC, it is useful to describe a
particular class
of Markov chains [1] and its physical meaning.
We start to review some technical terms to define the QMC :
\bigskip
{\bf Definition 4.1} [22,21] :
Let ${\Cal A}$ be a $^*$--algebra and ${\Cal A}_0\subseteq {\Cal A}$
a $^*$--subalgebra. A conditional expectation from ${\Cal A}$
to ${\Cal A}_0$ is a linear map $E:{\Cal A}\rightarrow {\Cal A}_0$ such that
$$\eqalignno{
&a\geq 0\Rightarrow E(a)\geq 0\; ;\; a\in {\Cal A}&(4.1)\cr
&E(a_0 a)=a_0 E(a)\; ;\;a_0\in {\Cal A}_0,\ a\in {\Cal A}&(4.2)\cr
&E(1)=1&(4.3)\cr
&E(a)^*E(a)\leq E(a^*a)\; ;\;a\in{\Cal A}&(4.4)\cr}$$
\bigskip
{\bf Definition 4.2} [4] :
Let ${\Cal B} \subseteq {\Cal A}$ be 
$C^*$-algebras and let $E:{\Cal A} \rightarrow {\Cal B}$ be a 
conditional expectation. An $E$--conditional density amplitude
is an operator $K\in {\Cal A}$ such that $$E(K^*K)=1 \eqno (4.5)$$
\bigskip
{\bf Definition 4.3} [4] :
A conditional density amplitude from ${\Cal B}_0\otimes{\Cal B}_1$ to
${\Cal B}_0$
is an operator $K\in {\Cal B}_0 \otimes {\Cal B}_1$ such that, for every 
density operator $\rho\in {\Cal B}_0$  
$$ K (\rho\otimes 1)  K^* \eqno(4.6)$$
is a density operator in $ {\Cal B}_0 \otimes {\Cal B}_1$.
If ${\Cal B}={\Cal B}_0 = {\Cal B}_1$ , then $K$ is called a
conditional density amplitude on ${\Cal B}$.
The condition for $K$ to be a conditional density amplitude
$$
tr_{{\Cal H}_1}K^*K=1\eqno(4.7)
$$
\medskip
 
Let $K\in{\Cal B}_0 \otimes {\Cal B}_1$ be a conditional amplitude.
Then $\Cal E$ defines a transition expectation from ${\Cal B}_0\otimes{\Cal
B}_1$
to ${\Cal B}_0$ by
$$
{\Cal E}(x)=tr_{{\Cal H}_1}K^*xK\quad x\in{\Cal B}_0 \otimes {\Cal B}_1
\eqno(4.8)
$$
\bigskip
A two parameter family 
$$K = (K(m,n))\eqno(4.9)$$
$m,n \in {\bold N} \ ,\ (m<n)$  of operators in ${\Cal A}\equiv\otimes_{\bold N}
{\Cal B}$ is called a right multiplicative functional (with respect
to the localization ${\Cal A}_{[m,n]}\equiv\otimes^n_{i=m}{\Cal B}_i$ )  if :
$$K(m,n) K(n,p) = K(m,p) \qquad ; \ m<n<p\eqno(4.10)$$
$$K(m,n) \in {\Cal A}_{[m,n]}\eqno(4.11)$$
If each $(K(m,n))$ is a conditional amplitude on
${\Cal A}_{[m,n]} \cong  \otimes ^{n-m+1}{\Cal B}$, then
the multiplicative functional $K = (K(m,n)) $ is said to be normalized.
\medskip
It is clear that a normalized multiplicative functional $K = (K(m,n)) $
has the form
$$K(0,n) = K( 0, 1) \cdot K( 1, 2)\cdot \ldots \cdot K( n-1, n)\eqno(4.12)$$
There is a conditional density amplitude $K_n$
on  ${\Cal B} $ for each $n$ such that
$$K( n-1, n) = (j_{ n-1} \otimes j_n) (K_{n-1} ) \eqno(4.13)$$
where $j_n:{\Cal B}\rightarrow \otimes_{{\bold N}}{\Cal B} $ is 
the natural embedding onto the $n$-th factor:
$$j_n:b\in{\Cal B}\rightarrow\underbrace{1\otimes 1\otimes \dots\otimes 1}_{n-1}
\otimes \underbrace{b}_{ n }\otimes 1\otimes\dots \ \in
\otimes_{{\bold N}}{\Cal B}$$
\medskip
{\bf Remark 4.4} : Let $\psi_0$ be a state on ${\Cal B}$
and let $\phi$ be the product state on ${\Cal A}=\otimes_{\bold N} {\Cal B}$.
Let $E_0:{\Cal B}\otimes {\Cal B} \rightarrow {\Cal B}$ denote the 
conditional expectation characterized by
$$E_0(a\otimes b) = \psi_0(b) a \otimes 1 \ \ \ \ ; \ \ \ \ a,b \in {\Cal B}
\eqno(4.14)$$
and let $K\in {\Cal B}\otimes {\Cal B}$ be an $E_0$-conditional
density amplitude and define $K_{n-1,n}$ as
in (4.13). Then there exists a unique state $\phi_K$ on ${\Cal A}=\otimes_
{\bold N}{\Cal B}$ such that, for each $k\in {\bold N} $ and
$a\in {\Cal A}_{[0,k]}$, one has
$$\phi_K=\lim_{n \to \infty} \phi(K^*_{n-1,n}\cdots K^*_{0,1}\cdot a\cdot
K_{0,1}\cdots K_{n-1,n}) \eqno(4.15)$$
\bigskip
Then according to the above remark, the model of QMC [10]
can be constructed.
 
As in section 2, take ${\frak S}={\frak S}_0\otimes{\frak S}_1$
and ${\Cal B}={\Cal B}_0\otimes{\Cal B}_1$.
Let $\varphi_0$ and $\varphi_1$ be states on ${\Cal B}_0$ and ${\Cal B}_1$
respectively and let $u_0$ be the partial shift on
$\otimes_{\bold N}\left( {{\Cal B}_0\otimes {\Cal B}_1} \right)
\cong \otimes_{\bold N}{\Cal B}\cong
 \left( {\otimes_{\bold N}{\Cal B}_0} \right)\otimes
 \left( {\otimes_{\bold N}{\Cal B}_1} \right)\equiv {\Cal A}$,
that is, $u_0$ acts trivially on $\otimes_{\bold N}{\Cal B}_0$
and as the usual shift on $\otimes_{\bold N}{\Cal B}$.
Let $U\in{\Cal B}_0\otimes {\Cal B}_1$ be a unitary operator, and define
$$U\left( {n-1,n} \right)=u_0^{n-1}\left( U \right)\,;\;n\ge 1\eqno(4.16)$$
If $\varphi =\otimes_{\bold N}\left( {\varphi _0\otimes \varphi _1} \right)$
is a product state on $\Cal A$, and if $\varphi_U$ is the quantum Markov chain
constructed as in the above Remark 4.4,  then the restriction of
$\varphi_U$ to the algebra ${\Cal B}_0\otimes
 \left( {\otimes_{\bold N}{\Cal B}_1} \right)$
(${\Cal B}_0$ is the "time zero algebra") gives a quantum Markov chain.
This class of Markov chains was studied by K\"ummerer [10].
 
The algebra ${\Cal B}_0$ is interpreted as the algebra of a given system,
and the algebra ${\otimes_{\bold N}{\Cal B}_1}$ is,
as the algebra of an i.i.d. quantum process representing either
a sequence of independent disturbances or a sequence of independent reservoirs.
The $U(n-1, n)$ represents the coupling of the system at time zero with
the disturbance at time $n$ (resp. with the $n$-th reservoir).
This model can be considered as a discrete approximation of the coupling
of an initial space to quantum Brownian motion.
 
\Bigskip
\leftline{\fontB 5 A Concrete Example of Stinespring Expression in Our Model}
\bigskip
In this section the Stinespring expression is concretely obtained in our model.
The technique obtained in this section can be applied to several
other models.
 
The Stinespring theorem is well known in the following form :
 
\bigskip
{\bf Theorem 5.1} [20] :
Let $\Cal A$ be a C*-algebra with a unit and ${\Cal B}(\Cal H)$ be a set
of the bounded operators on a Hilbert space $\Cal H$.
For any completely positive map $\Lambda : {\Cal A}\rightarrow{\Cal B}(\Cal H)$
there exist a *-representation $\{{\Cal K}, \pi\}$ of $\Cal A$
and a bounded linear map $V : {\Cal H}\rightarrow{\Cal K}$ such that
$$\Lambda \left( A \right)=V^*\pi \left( A \right)V\quad \quad \forall A
\in {\Cal A}\eqno(5.1)$$
\medskip
In particular, if ${\Cal A}={\Cal B}(\Cal H)$ in the above Theorem 5.1,
then we have the following lemma due to Kraus.
 
\bigskip
{\bf Lemma 5.2} [9] :
Let $\Cal H$ be a Hilbert space.
A linear map $\Lambda : {\Cal B}(\Cal H)\rightarrow{\Cal B}(\Cal H)$
is a completely positive if and only if it has the form
$$\Lambda \left( A \right)=\sum\limits_{i=1}^n {V_i^*AV_i}\quad \quad
 A\in {\Cal A}\eqno(5.2)$$
where $V_i : {\Cal H}\rightarrow{\Cal H}$ are partial isometries and
$n\le \dim \left( {\Cal H} \right)$.
\bigskip
We shall give an concrete
expression for partial isometries $V_i$ in our model.
 
From the form (2.4) of the Hamiltonian $H$ of the total system
${\Cal H}_0\otimes{\Cal H}_1$, it is easy to see that
$$\left[ {H_0+H_1,\;H_{01}} \right]=0.\eqno(5.3)$$
Hence the time evolution of the system comes from only the interaction
Hamiltonian $H_{01}$:
$$U_t=\exp \left( {-itH_{01}} \right).\eqno(5.4)$$
 
Clearly the spectra of $H_0+H_1$ are natural numbers, so that
let ${\Cal K}_n$ be the eigenspace of $H_0+H_1$ for an eigenvalue
$n\in{\bold N}$ , then ${\Cal K}_n$ is a subspace of ${\Cal
H}_0\otimes{\Cal H}_1$
spanned by $\left\{ {\left| {j\otimes n-j} \right\rangle \;;\;j=0,\cdots ,n}
\right\}$ and $H_{01}{\Cal K}_n\subseteq {\Cal K}_n$. Therefore we define
$H_{01}^{(n)}$ by :
$$H_{01}^{(n)}\equiv H_{01}\upharpoonright K_n\eqno(5.5)$$
$H_{01}^{(n)}$ is a finite dimensional self-adjoint operator
${\bold C}^{n+1}\to {\bold C}^{n+1}$ and has $(n+1)$ eigenvalues
$\lambda_0^{(n)},\cdots ,\lambda_n^{(n)}$
and the corresponding eigenvectors $\psi _k^{(n)}$
associated to $\lambda _k^{(n)}\,( k=0,\cdots$\par
\noindent $,n )$; namely, 
$$H_{01}^{(n)}\psi _j^{(n)}=\lambda _j^{(n)}\psi _j^{(n)}
\quad \left( {j=0,\cdots ,n} \right)\eqno(5.6)$$
$$\psi _j^{(n)}=\sum\limits_{\alpha =0}^n {C_\alpha ^{(n,j)}
\left| {\alpha \otimes n-\alpha } \right\rangle }\;\in \;{\Cal K}_n\eqno(5.7)$$
\noindent
where $C_\alpha ^{(n,j)}$ is the coefficient 
determined by $\left\langle {\psi _i^m,\,\psi _j^n} \right\rangle=
\delta _{ij}\delta _{mn}$ and the completeness of 
$\left\{ {\psi_j^{(n)}} \right\}$.
(5.6) implies
$$H_{01}^{(n)}=\sum\limits_{j=0}^n {\lambda _j^{\left( n \right)}
\left| {\psi _j^{\left( n \right)}} \right\rangle \left\langle
 {\psi _j^{\left( n \right)}} \right|}\eqno(5.8)$$
$$H_{01}=\sum\limits_{n=0}^\infty  {H_{01}^{(n)}}=
\sum\limits_{n=0}^\infty  {\sum\limits_{j=0}^n 
{\lambda _j^{\left( n \right)}\left| {\psi _j^{\left( n \right)}} \right
\rangle \left\langle {\psi _j^{\left( n \right)}} \right|}}\eqno(5.9)$$
Using the above expressions, a unitary operator
$U_t=\exp \left( {-itH} \right)$ on ${\Cal H}_0\otimes {\Cal H}_1$
can be written as  
$$\eqalignno{U_t&=\exp \left( {-itH} \right)=\exp \left( {-it\left( {H_0+H_1}
 \right)} \right)\exp \left( {-i\varepsilon tH_{01}} \right)\cr
&=\sum\limits_{m=0}^\infty  e^{-itm} \left| m \right\rangle \left\langle m
\right|
 \sum\limits_{n=0}^\infty  {\sum\limits_{j=0}^n
 {e^{-i\varepsilon t\lambda _j^{\left( n \right)}}\left|
 {\psi _j^{\left( n \right)}} \right\rangle \left\langle 
{\psi _j^{\left( n \right)}} \right|}}\cr
&=\sum\limits_{n=0}^\infty  {\sum\limits_{j=0}^n
{e^{-it\left( {n+\varepsilon \lambda _j^{(n)}} \right)}\left|
{\psi _j^{(n)}} \right\rangle\left\langle {\psi _j^{(n)}} \right|}}\cr
&=\sum\limits_{n=0}^\infty  {\sum\limits_{j=0}^n 
{\sum\limits_{\alpha ,\beta =0}^n {e^{-it\left( {n+\varepsilon
 \lambda _j^{\left( n \right)}} \right)}}C_\alpha ^{\left( {n,j} \right)}
\overline {C_\beta ^{\left( {n,j} \right)}}\left| \alpha  
\right\rangle \left\langle \beta  \right|}}\otimes \left| {n-\alpha }
 \right\rangle \left\langle {n-\beta } \right|&(5.10)\cr}$$
Therefore the channel $\Lambda^*\rho \in {\frak S}_0$
is written by
$$\eqalignno{\Lambda ^*\rho &=tr_{{\Cal H}_1}U_t\left( {\rho \otimes \omega
} \right)U_t^*\cr
&=\sum\limits_{m,n=0}^\infty  {\sum\limits_{i=0}^m {\sum\limits_{j=0}^n
 {\sum\limits_{\alpha ,\beta =0}^n {\sum\limits_{\alpha ',\beta '=0}^m
 {e^{-it\left( {n+\varepsilon \lambda _j^{\left( n \right)}} \right)}
e^{it\left( {n+\varepsilon \lambda _j^{\left( m \right)}} \right)}
C_\alpha ^{\left( {n,j} \right)}\overline {C_\beta ^{\left( {n,j} \right)}}
\overline {C_{\alpha '}^{\left( {m,i} \right)}}C_{\beta '}^{\left( {m,i}
 \right)}}}}}}\cr
&\times Tr_2\left| \alpha  \right\rangle \left\langle \beta  \right|\rho
 \left| {\beta '} \right\rangle \left\langle {\alpha '} \right|\otimes
 \left| {n-\alpha } \right\rangle \left\langle {n-\beta } \right|\,\omega 
\,\left| {m-\beta '} \right\rangle \left\langle {m-\alpha '} \right|\cr
&=\sum\limits_{m,n=0}^\infty  {\sum\limits_{\alpha ,\beta =0}^n
 {\sum\limits_{\alpha ',\beta '=0}^m {\left( {\sum\limits_{j=0}^n
 {e^{-it\varepsilon \lambda _j^{\left( n \right)}}C_\alpha ^{\left( {n,j}
 \right)}\overline {C_\beta ^{\left( {n,j} \right)}}}} \right)
\left( {\sum\limits_{i=o}^m {e^{it\varepsilon \lambda _i^{\left( m \right)}}
\overline {C_{\alpha '}^{\left( {m,i} \right)}}C_{\beta '}^{\left( {m,i}
 \right)}}} \right)}}}\cr
&\times \left| \alpha  \right\rangle \left\langle \beta  \right|
\rho \left| {\beta '} \right\rangle \left\langle {\alpha '} \right|\;
\left\langle {n-\beta } \right|\,\omega \,\left| {m-\beta '} \right\rangle
 \;\,\delta _{m-\alpha ',n-\alpha }\cr
&=\sum\limits_{m,n=0}^\infty  {\sum\limits_{\alpha ,\beta =0}^n 
{\sum\limits_{\alpha ',\beta '=0}^m {d_{\alpha ,\beta }^{(n)}\overline 
{d_{\alpha ',\beta '}^{(m)}}}}}\left\langle {n-\beta } \right|\,\omega 
\,\left| {m-\beta '} \right\rangle \left\langle \beta  \right|\rho 
\left| {\beta '} \right\rangle \;\delta _{m-\alpha ',n-\alpha }\left| 
\alpha  \right\rangle \left\langle {\alpha '} \right|\cr
&&(5.11)\cr}$$
where
$$d_{\alpha ,\beta }^{(n)}\equiv \sum\limits_{j=0}^n 
{e^{-it\varepsilon \lambda _j^{\left( n \right)}}C_\alpha ^{\left( {n,j}
 \right)}\overline {C_\beta ^{\left( {n,j} \right)}}}\eqno(5.12)$$
 
Let the spectral decomposition of the reservoir state
$\omega \in {\frak S}_1$ be :
$$\omega =\sum\limits_\gamma  {p_\gamma^\omega \left| {\xi_\gamma
 } \right\rangle \left\langle {\xi_\gamma } \right|}\eqno(5.13)$$
Then the state $\Lambda^*\rho \in {\frak S}_0$ becomes
$$\eqalignno{\Lambda ^*\rho &=\sum\limits_\gamma  {p_\gamma ^\omega }
\sum\limits_{m,n=0}^\infty  {\sum\limits_{\alpha ,\beta =0}^n 
{\sum\limits_{\alpha ',\beta '=0}^m {d_{\alpha ,\beta }^{(n)}\overline 
{d_{\alpha ',\beta '}^{(m)}}}}}\left\langle {n-\beta } \right|\,\left.
 {\xi _\gamma } \right\rangle \left\langle {\xi _\gamma } \right.\,
\left| {m-\beta '} \right\rangle \left\langle \beta  \right|\rho \left|
 {\beta '} \right\rangle \;\delta _{m-\alpha ',n-\alpha }\left| \alpha 
 \right\rangle \left\langle {\alpha '} \right|\;\cr
&&(5.14)\cr}$$
Now we define new channel $\Lambda^*_{\xi_\gamma }$:
$$\eqalignno{\Lambda ^*_{\xi _\gamma }\rho &\equiv
 \sum\limits_{m,n=0}^\infty  {\sum\limits_{\beta =0}^n 
{\sum\limits_{\beta '=0}^m {\sum\limits_{\nu =0}^{m\wedge n} 
{\left| {n-\nu } \right\rangle \left\langle {m-\nu } \right|}
d_{n-\nu ,\beta }^{(n)}\overline {d_{m-\nu ,\beta '}^{(m)}}}}}
\left\langle {n-\beta } \right|\,\left. {\xi _\gamma } \right\rangle
 \left\langle {\xi _\gamma } \right.\,\left| {m-\beta '} \right\rangle
 \left\langle \beta  \right|\rho \left| {\beta '} \right\rangle \cr
&=\sum\limits_{\nu =0}^\infty  {\sum\limits_{m,n=\nu }^\infty 
 {\sum\limits_{\beta =0}^n {\sum\limits_{\beta '=0}^m 
{\left| {n-\nu } \right\rangle \left\langle {m-\nu } \right|}
d_{n-\nu ,\beta }^{(n)}\overline {d_{m-\nu ,\beta '}^{(m)}}}}}
\left\langle {n-\beta } \right|\,\left. {\xi _\gamma } 
\right\rangle \left\langle {\xi _\gamma } \right.\,\left|
 {m-\beta '} \right\rangle \left\langle \beta  \right|\rho 
\left| {\beta '} \right\rangle\cr
&&(5.15)\cr}$$
Then the final state $\Lambda^*\rho $ becomes :
$$\Lambda ^*\rho =\sum\limits_\gamma  {p_\gamma ^\omega }
\Lambda ^*_{\xi _\gamma }\rho \eqno(5.16)$$
If $\rho =\left| \eta  \right\rangle \left\langle \eta  \right|$
 (pure state), then $\Lambda^*_{\xi_\gamma }\rho $ is
$$\eqalignno{&\Lambda ^*_{\xi _\gamma }\left| \eta  \right\rangle
 \left\langle \eta  \right|=\sum\limits_{\nu =0}^\infty  
{\sum\limits_{m,n=\nu }^\infty  {\sum\limits_{\beta =0}^n 
{\sum\limits_{\beta '=0}^m {\left| {n-\nu } \right\rangle \left\langle 
{m-\nu } \right|}d_{n-\nu ,\beta }^{(n)}\overline 
{d_{m-\nu ,\beta '}^{(m)}}}}}\left\langle {n-\beta } \right|\,
\left. {\xi _\gamma } \right\rangle \left\langle {\xi _\gamma } \right.\,
\left| {m-\beta '} \right\rangle \left\langle \beta  \right|\,\left. 
\eta  \right\rangle \left\langle \eta  \right.\,\left| {\beta '}
 \right\rangle \cr
&\quad=\sum\limits_{\nu =0}^\infty  {\sum\limits_{m,n=\nu }^\infty
  {\left( {\sum\limits_{\beta =0}^n {d_{n-\nu ,\beta }^{(n)}\left\langle 
\beta  \right|\,\left. \eta  \right\rangle \left\langle {n-\beta } 
\right|\,\left. {\xi _\gamma } \right\rangle \left| {n-\nu } \right\rangle }}
 \right)\left( {\sum\limits_{\beta '=0}^m {d_{m-\nu ,\beta '}^{(m)}
\left\langle {\beta '} \right|\,\left. \eta  \right\rangle 
\left\langle {m-\beta '} \right|\,\left. {\xi _\gamma }
 \right\rangle \left| {m-\nu } \right\rangle }} \right)^*}}\cr
&&(5.17)\cr}$$
Take $\varphi_{n-\nu }^{(n)}$ as
$$\varphi _{n-\nu }^{(n)}\equiv \sum\limits_{\beta =0}^n 
{d_{n-\nu ,\beta }^{(n)}\left\langle \beta  \right|\,\otimes
 \left\langle {n-\beta } \right|\,}\in {\Cal K}_n\eqno(5.18)$$
Then we have
$$\eqalignno{\Lambda ^*_{\xi _\gamma }\left| \eta  \right\rangle \left\langle 
\eta  \right|&=\sum\limits_{\nu =0}^\infty  {\sum\limits_{m,n=\nu }^\infty 
 {\left( {\left\langle {\varphi _{n-\nu }^{(n)}} \right|\,\,\left.
 {\eta \otimes \xi _\gamma } \right\rangle \left| {n-\nu } \right\rangle }
 \right)\left( {\left\langle {m-\nu } \right|\overline
 {\left\langle {\varphi _{m-\nu }^{(m)}} \right|\,\,\left.
 {\eta \otimes \xi _\gamma } \right\rangle }} \right)}}\cr
&=\sum\limits_{\nu =0}^\infty  {T_\nu \left( {\left| \eta  \right\rangle
 \left\langle \eta  \right|\otimes \left| {\xi _\gamma } 
\right\rangle \left\langle {\xi _\gamma } \right|} \right)\,}T_\nu ^*\cr
&&(5.19)\cr}$$
where
$$T_\nu \equiv \sum\limits_{n=\nu }^\infty  {\left| {n-\nu } \right\rangle 
\left\langle {\varphi _{n-\nu }^{(n)}} \right|}\eqno(5.20)$$
Thus we can get  the following formula :
$$\Lambda ^*\rho =\sum\limits_{\nu =0}^\infty  {T_\nu \left( {\rho \otimes 
\omega } \right)\,}T_\nu^*\eqno(5.21)$$
The following lemma shows that $\left\{ {\left| {\varphi _{n-\nu }^{(n)}}
\right\rangle \;;\;n\ge \nu ,\;n,\nu \in N} \right\}$ is an othogonal system,
so that the above $T_\nu$ are partial isometries.
\bigskip
{\bf Lemma 5.1 :}
$$\left\langle {{\varphi _{n-\nu }^{(n)}}} \mathrel{\left | 
{\vphantom {{\varphi _{n-\nu }^{(n)}} {\varphi _{n'-\nu '}^{(n')}}}} \right.
 \kern-\nulldelimiterspace} {{\varphi _{n'-\nu '}^{(n')}}} \right\rangle
 =\delta _{n,n'}\delta _{\nu ,\nu '}\eqno(5.22)$$
{\bf Proof :}
Since $\Lambda^*\rho $ of (5.21) is a state on ${\Cal H}_1$,
$tr\Lambda^*\rho=1$ which implies $\sum\limits_{\nu =0}^\infty 
 {T_\nu ^*T_\nu =1}$.
$$tr\Lambda ^*\rho =1=\sum\limits_{\nu =0}^\infty  {T_\nu ^*T_\nu }
\;=\sum\limits_{\nu \ge 0} {\sum\limits_{n\ge \nu } 
{\left| {\varphi _{n-\nu }^{(n)}} \right\rangle \left\langle
 {\varphi _{n-\nu }^{(n)}} \right|}}\eqno(5.23)$$
From the definition of $\varphi _{n-\nu }^{(n)}$ by (5.18),
if $n\ne n'$. Let us prove $\left\langle {{\varphi _{n-\nu }^{(n)}}}
 \mathrel{\left | {\vphantom {{\varphi _{n-\nu }^{(n)}}
 {\varphi _{n-\nu '}^{(n)}}}} \right. \kern-\nulldelimiterspace} 
{{\varphi _{n-\nu '}^{(n)}}} \right\rangle =\delta _{\nu ,\nu '}$.
$$\eqalignno{\left\langle {{\varphi _{n-\nu }^{(n)}}} \mathrel{\left | 
{\vphantom {{\varphi _{n-\nu }^{(n)}} {\varphi _{n-\nu '}^{(n)}}}} \right. 
\kern-\nulldelimiterspace} {{\varphi _{n-\nu '}^{(n)}}} \right\rangle 
&=\sum\limits_{\beta ,\beta '\le n} {d_{n-\nu ,\beta }^{(n)}\overline 
{d_{n-\nu ',\beta '}^{(n)}}\left\langle {\beta } \mathrel{\left | 
{\vphantom {\beta  {\beta '}}} \right. \kern-\nulldelimiterspace} 
{{\beta '}} \right\rangle \left\langle {{n-\beta }} \mathrel{\left | 
{\vphantom {{n-\beta } {n-\beta '}}} \right. \kern-\nulldelimiterspace} 
{{n-\beta '}} \right\rangle }\cr
&=\sum\limits_{\beta \le n} {d_{n-\nu ,\beta }^{(n)}\overline 
{d_{n-\nu ',\beta }^{(n)}}}\cr
&=\sum\limits_{\beta \le n} {\left( {\sum\limits_{j_1=0}^n 
{e^{-i\varepsilon t\lambda _{j_1}^{(n)}}C_{n-\nu }^{(n,j_1)}\overline 
{C_\beta ^{(n,j_1)}}}} \right)\left( {\sum\limits_{j_2=0}^n 
{e^{i\varepsilon t\lambda _{j_2}^{(n)}}\overline {C_{n-\nu '}^{(n,j_2)}}
C_\beta ^{(n,j_2)}}} \right)}\cr
&=\sum\limits_{j_1,j_2=0}^n {e^{-i\varepsilon t\left( {\lambda _{j_1}^{(n)}
-\lambda _{j_2}^{(n)}} \right)}C_{n-\nu }^{(n,j_1)}\overline 
{C_{n-\nu '}^{(n,j_2)}}}\sum\limits_{\beta =0}^n {\overline 
{C_\beta ^{(n,j_1)}}C_\beta ^{(n,j_2)}}\cr
&=\sum\limits_{j=0}^n {C_{n-\nu }^{(n,j)}\overline {C_{n-\nu '}^{(n,j)}}}
&(5.24)\cr}$$
where we applied the last equality come from
$$\psi _{j_1}^{(n)}\bot \psi _{j_2}^{(n)}\;\left( {j_1\ne j_2} \right)
\Rightarrow \left\langle {{\psi _{j_1}^{(n)}}} 
\mathrel{\left | {\vphantom {{\psi _{j_1}^{(n)}} {\psi _{j_2}^{(n)}}}}
 \right. \kern-\nulldelimiterspace} {{\psi _{j_2}^{(n)}}} 
\right\rangle =\sum\limits_{\alpha =0}^n {\overline {C_\alpha ^{(n,j_1)}}
C_\alpha ^{(n,j_2)}}=\delta _{j_1,j_2}. \eqno(5.25)$$
The matrix
$$\left( {u_{j,\alpha }} \right)=\left( {C_\alpha ^{(n,j)}} \right)$$
is unitary, which a transition matrix between the two ONS 
$\left( {\psi _j^{(n)}} \right)$ and $\left| \alpha  \right\rangle \otimes
 \left| {n-\alpha } \right\rangle $.
Therefore we get
$$\sum\limits_{j=0}^n {u_{j,\alpha }\overline {u_{j,\beta }}}=
\delta _{\alpha ,\beta }\;\Rightarrow \;\sum\limits_{j=0}^n 
{C_{n-\nu }^{(n,j)}\overline {C_{n-\nu '}^{(n,j)}}}=\delta _{n-\nu ,n-\nu '}
=\delta _{\nu ,\nu '}\eqno(5.26)$$
Thus, from (5.24) and (5.26), we obtain
$$\left\langle {{\varphi _{n-\nu }^{(n)}}} \mathrel{\left | 
{\vphantom {{\varphi _{n-\nu }^{(n)}} {\varphi _{n-\nu '}^{(n)}}}}
 \right. \kern-\nulldelimiterspace} {{\varphi _{n-\nu '}^{(n)}}}
 \right\rangle =\delta _{\nu ,\nu '}$$
$$\therefore \left\langle {{\varphi _{n-\nu }^{(n)}}}
 \mathrel{\left | {\vphantom {{\varphi _{n-\nu }^{(n)}} 
{\varphi _{n'-\nu '}^{(n')}}}} \right. \kern-\nulldelimiterspace} 
{{\varphi _{n'-\nu '}^{(n')}}} \right\rangle =\delta _{n,n'}
\delta _{\nu ,\nu '}\eqno\qed$$
\bigskip
Therefore (5.20) and (5.21) give a concrete  Stinespring
representation of $\Lambda^*$.
This representation is useful for the computation of several entropies.
In the following section, we rigorously compute the mutual entropy.
 
\Bigskip
\leftline{\fontB 6 Computation of the Mutual Entropy in our model}
\bigskip
From (5.20) and (5.21), we have
$$\Lambda ^*\rho =\sum\limits_\nu  {\sum\limits_{m,n\ge \nu }
 {\left\langle {\varphi _{n-\nu }^{(n)}} \right|\rho \otimes \omega
 \,\left| {\varphi _{m-\nu }^{(m)}} \right\rangle \left| {n-\nu } 
\right\rangle \left\langle {m-\nu } \right|}}\eqno(6.1)$$
Here, in the calculation below, we assume the following Schatten
decompositions :
$$\rho =\sum\limits_l {\lambda_l^\rho \left| l \right\rangle \left\langle l
\right|}
  \omega =\sum\limits_\gamma  {\mu_\gamma^\omega \left| \gamma  \right\rangle
 \left\langle \gamma  \right|}.\eqno(6.2)$$
Then (6.2) implies
$$\eqalign{&\Lambda ^*\rho =\sum\limits_\nu  {\sum\limits_{m,n\ge \nu } 
{\left\langle {\varphi _{n-\nu }^{(n)}} \right|\sum\limits_{l,\gamma } 
{\lambda _l^\rho \mu _\gamma ^\omega \left| l \right\rangle \left\langle l 
\right|\otimes \left| \gamma  \right\rangle \left\langle \gamma  \right|}\,
\left| {\varphi _{m-\nu }^{(m)}} \right\rangle \left| {n-\nu } \right\rangle 
\left\langle {m-\nu } \right|}}\cr
&\quad \;\;=\sum\limits_{l,\gamma } {\lambda _l^\rho \mu _\gamma ^\omega }
\sum\limits_\nu  {\sum\limits_{m,n\ge \nu } {\left\langle 
{{\varphi _{n-\nu }^{(n)}}} \mathrel{\left | {\vphantom 
{{\varphi _{n-\nu }^{(n)}} {l\otimes \gamma }}} \right. 
\kern-\nulldelimiterspace} {{l\otimes \gamma }} \right\rangle 
\left\langle {{l\otimes \gamma }} \mathrel{\left | 
{\vphantom {{l\otimes \gamma } {\varphi _{m-\nu }^{(m)}}}} \right.
 \kern-\nulldelimiterspace} {{\varphi _{m-\nu }^{(m)}}} 
\right\rangle \left| {n-\nu } \right\rangle \left\langle {m-\nu } \right|}}\cr
&\quad \;\;=\sum\limits_{l,\gamma } {\lambda _l^\rho \mu _\gamma ^\omega }
\sum\limits_\nu  {\left( {\sum\limits_{n\ge \nu } {\left\langle
 {{\varphi _{n-\nu }^{(n)}}} \mathrel{\left | {\vphantom 
{{\varphi _{n-\nu }^{(n)}} {l\otimes \gamma }}} \right.
 \kern-\nulldelimiterspace} {{l\otimes \gamma }} \right\rangle
 \left| {n-\nu } \right\rangle }} \right)\left( {\sum\limits_{m\ge \nu }
 {\overline {\left\langle {{\varphi _{m-\nu }^{(m)}}} \mathrel{\left | 
{\vphantom {{\varphi _{m-\nu }^{(m)}} {l\otimes \gamma }}} \right. 
\kern-\nulldelimiterspace} {{l\otimes \gamma }} \right\rangle }\left\langle
 {m-\nu } \right|}} \right)}\cr}$$
Put $\left| {\Phi _{l,\gamma }^\nu } \right\rangle $ as
$$\left| {\Phi _{l,\gamma }^\nu } \right\rangle \equiv 
\sum\limits_{n\ge \nu } {\left\langle {{\varphi _{n-\nu }^{(n)}}} 
\mathrel{\left | {\vphantom {{\varphi _{n-\nu }^{(n)}} {l\otimes \gamma }}} 
\right. \kern-\nulldelimiterspace} {{l\otimes \gamma }} \right\rangle
 \left| {n-\nu } \right\rangle }\eqno(6.3)$$
Then we have :
$$\Lambda ^*\rho =\sum\limits_{l,\gamma } 
{\lambda _l^\rho \mu _\gamma ^\omega }\sum\limits_\nu  
{\left| {\Phi _{l,\gamma }^\nu } \right\rangle \left\langle 
{\Phi _{l,\gamma }^\nu } \right|}\eqno(6.4)$$
Because of $\varphi _{n-\nu }^{(n)}\in {\Cal K}_n$, (6.3) becomes
$$\left| {\Phi _{l,\gamma }^\nu } \right\rangle \equiv 
\sum\limits_{n\ge \nu } {\left\langle {{\varphi _{n-\nu }^{(n)}}} 
\mathrel{\left | {\vphantom {{\varphi _{n-\nu }^{(n)}} {l\otimes \gamma }}}
 \right. \kern-\nulldelimiterspace} {{l\otimes \gamma }} \right\rangle
 \left| {n-\nu } \right\rangle }=\left\langle
 {{\varphi _{l+\gamma -\nu }^{(l+\gamma )}}} 
\mathrel{\left | {\vphantom {{\varphi _{l+\gamma -\nu }^{(l+\gamma )}}
 {l\otimes \gamma }}} \right. \kern-\nulldelimiterspace} {{l\otimes \gamma }} 
\right\rangle \left| {l+\gamma -\nu } \right\rangle \eqno(6.5)$$
Therefore (6.4) implies
$$\eqalignno{\Lambda ^*\rho &=\sum\limits_{l,\gamma } 
{\lambda _l^\rho \mu _\gamma ^\omega }\sum\limits_\nu  
{\left| {\left\langle {{\varphi _{l+\gamma -\nu }^{(l+\gamma )}}} 
\mathrel{\left | {\vphantom {{\varphi _{l+\gamma -\nu }^{(l+\gamma )}} 
{l\otimes \gamma }}} \right. \kern-\nulldelimiterspace} {{l\otimes \gamma }}
 \right\rangle } \right|^2}\left| {l+\gamma -\nu } \right\rangle \left\langle
 {l+\gamma -\nu } \right|\cr
&=\sum\limits_{l,\gamma } {\lambda _l^\rho \mu _\gamma ^\omega }
\sum\limits_\nu  {\left| {d_{l+\gamma -\nu ,l}^{(l+\gamma )}} 
\right|^2}\left| {l+\gamma -\nu } \right\rangle \left\langle 
{l+\gamma -\nu } \right|\cr
&=\sum\limits_{l,\gamma ,\nu } {\lambda _l^\rho \mu _\gamma ^\omega \left| 
{d_{l+\gamma -\nu ,l}^{(l+\gamma )}} \right|^2\left| {l+\gamma -\nu }
 \right\rangle \left\langle {l+\gamma -\nu } \right|}\cr
&=\sum\limits_i {\left( {\sum\limits_{\scriptstyle {\quad \;\;l,\nu }
\hfill\atop\scriptstyle {\left( {\nu -l} \right)+i\ge 0}\hfill}
 {\lambda _l^\rho \mu _{\left( {\nu -l} \right)+i}^\omega \left|
 {d_{i,l}^{(\nu +i)}} \right|^2}} \right)}
\left| i \right\rangle \left\langle i \right|\cr
&=\sum\limits_i {q_i\left| i \right\rangle \left\langle i \right|}&(6.6)\cr
}$$
where
$$q_i=\sum\limits_{\scriptstyle {\quad \;\;l,\nu }\hfill\atop
  \scriptstyle {\left( {\nu -l} \right)+i\ge 0}\hfill} 
{\lambda _l^\rho \mu _{\left( {\nu -l} \right)+i}^\omega 
\left| {d_{i,l}^{(\nu +i)}} \right|^2}\eqno(6.7)$$
Clearly (6.6) is also a Schatten decomposition.
Suppose that the decomposition (6.2) of $\rho$ is unique.
 
From (6.6) we find
$$\Lambda ^*E_k=\sum\limits_i {q_i^{\left( k \right)}\left| i 
\right\rangle \left\langle i \right|}\eqno(6.8)$$
where
$$q_i^{\left( k \right)}=\sum\limits_{\scriptstyle {\quad \;\;\nu }\hfill\atop
\scriptstyle {\left( {\nu -k} \right)+i\ge 0}\hfill} {\mu _{\left( {\nu -k}
 \right)+i}^\omega \left| {d_{i,k}^{(\nu +i)}}
 \right|^2}\eqno(6.9)$$
Then, in order to compute the mutual entropy in (3.6),
$S\left( {\Lambda ^*E_k\left| {\Lambda ^*\rho } \right.} \right)$
with nondegenerate eigenvalues becomes
$$\eqalignno{S\left( {\Lambda ^*E_k, {\Lambda ^*\rho } } \right)
&=\sum\limits_{s,t} {\left\langle {s} \mathrel{\left | {\vphantom 
{s {\Lambda ^*E_kt}}} \right. \kern-\nulldelimiterspace} {{\Lambda ^*E_kt}}
 \right\rangle \log {{\left\langle {t} \mathrel{\left | 
{\vphantom {t {\Lambda ^*E_ks}}} \right. \kern-\nulldelimiterspace} 
{{\Lambda ^*E_ks}} \right\rangle } \over {\left\langle {t} \mathrel{\left | 
{\vphantom {t {\Lambda ^*\rho s}}} \right. \kern-\nulldelimiterspace}
 {{\Lambda ^*\rho s}} \right\rangle }}}\cr
&=\sum\limits_t {q_t^{\left( k \right)}\log {{q_t^{\left( k \right)}}
 \over {q_t}}}&(6.10)\cr}$$
because of
$$\left\{ {\matrix{{\left\langle {s} \mathrel{\left | {\vphantom {s 
{\Lambda ^*E_kt}}} \right. \kern-\nulldelimiterspace} {{\Lambda ^*E_kt}} 
\right\rangle =\sum\limits_i {q_i^{\left( k \right)}\left\langle {t} 
\mathrel{\left | {\vphantom {t i}} \right. \kern-\nulldelimiterspace} {i}
 \right\rangle \left\langle {i} \mathrel{\left | {\vphantom {i s}} \right.
 \kern-\nulldelimiterspace} {s} \right\rangle }=q_t^{\left( k \right)}}}\cr
{\left\langle {s} \mathrel{\left | {\vphantom {s {\Lambda ^*\rho t}}} \right.
 \kern-\nulldelimiterspace} {{\Lambda ^*\rho t}} \right\rangle =\sum\limits_i
 {q_i\left\langle {t} \mathrel{\left | {\vphantom {t i}} \right. 
\kern-\nulldelimiterspace} {i} \right\rangle \left\langle {i} 
\mathrel{\left | {\vphantom {i s}} \right. \kern-\nulldelimiterspace} {s}
 \right\rangle }=q_t}\cr}}} \right.$$
Therefore the relative entropy is
$$I\left( \rho ; \Lambda^*  \right)=
\sum\limits_{k,t} {\lambda _k^\rho \sum\limits_{\scriptstyle 
{\quad \quad \nu }\hfill\atop
  \scriptstyle {\left( {\nu -k} \right)+t\ge 0}\hfill} 
{\mu _{\left( {\nu -k} \right)+t}^\omega \left| 
{d_{t,k}^{\left( {\nu +t} \right)}} 
\right|^2\log {{\sum\limits_{\scriptstyle {\quad \quad \nu _1}\hfill\atop
  \scriptstyle {\left( {\nu _1-k} \right)+t\ge 0}\hfill} 
{\mu _{\left( {\nu _1-k} \right)+t}^\omega \left| {d_{t,k}^{\left( 
{\nu _1+t} \right)}} \right|^2}} \over 
{\sum\limits_{\scriptstyle {\quad \quad l,\nu }\hfill\atop
  \scriptstyle {\left( {\nu _2-l} \right)+t\ge 0}\hfill} 
{\lambda _l^\rho \mu _{\left( {\nu _2-l} \right)+t}^\omega \left| 
{d_{t,l}^{\left( {\nu _2+t} \right)}}
 \right|^2}}}}}\eqno(6.11)$$
 
The typical situations are shown in the followings :
\item{(i)}
The state $\omega \in {\frak S}_1$ is a vacuum state : 
$\omega =\left| 0 \right\rangle \left\langle 0 \right|$. Then
$$I\left( \rho ; \Lambda^*  \right)
=\sum\limits_{k,t} {\lambda _k^\rho \left| {d_{t,k}^{\left( k \right)}}
 \right|^2\log {{\left| {d_{t,k}^{\left( k \right)}} \right|^2}
 \over {\sum\limits_l {\lambda _l^\rho \left| {d_{t,l}^{\left( l \right)}}
 \right|^2}}}}\eqno(6.12)$$
\item{(ii)}
The state $\omega \in {\frak S}_1$ is a Gibbs state : 
$$\omega _\beta ={{e^{-\beta H_1}} \over {tre^{-\beta H_1}}}
=\sum\limits_n {{{e^{-\beta n}} \over {tre^{-\beta H_1}}}\left|
 n \right\rangle \left\langle n \right|}=\left( {1-e^{-\beta }}
 \right)\sum\limits_n {e^{-\beta n}\left| n \right\rangle
 \left\langle n \right|}$$
Then
$$I\left( \rho ; \Lambda^*  \right)=\left( {1-e^{-\beta }}
 \right)\sum\limits_{k,t} {\lambda _k^\rho D_{k,t}^\beta
 \log {{D_{k,t}^\beta } \over {\sum\limits_l
 {\lambda _l^\rho D_{l,t}^\beta }}}}\eqno(6.13)$$
where
$$D_{k,t}^\beta =\sum\limits_{\scriptstyle {\quad \quad \nu }\hfill\atop
  \scriptstyle {\left( {\nu -k} \right)+t\ge 0}\hfill} {e^{-\beta
 \left( {\left( {\nu -k} \right)+t} \right)}\left| {d_{t,
k}^{\left( {\nu +t} \right)}} \right|^2}\eqno(6.14)$$
%\vfill\eject
\Bigskip
\leftline{\fontB 7 Computer Experiments}
\bigskip
Under the above computations we show some numerical results of
 the mutual entropy $I\left( \rho ; \Lambda^*  \right)$ 
w.r.t. $T$, the temperture of Gibbs state in the reservoir, and 
$I\left( \rho ; \Lambda^{*n}  \right)$ w.r.t. $n$, the times of 
the channel, respectively.
\Bigskip
\finder
\Bigskip
As expected, the mutual entropy decreases with respect to $n$, and
the effect of the Gibbs state as a noise is obviously
larger than that of the vacuum state. These numerical experiments
show that our model of open system gives a dissipative change of a state,
although the interaction Hamiltonian is symmetric in the two modes $a$ and $b$.
Clearly this symmetry has been broken by taking partial trace in (3.1).
%\vfill\eject
\Bigskip
\leftline{\fontB 8 Generalized Mutual Entropy}
\bigskip
 
Finally we mention a new definition of the mutual entropy for two liftings
${\Cal E}^*$ and ${\Cal L}^*:{\frak S}(\Cal A)\rightarrow
{\frak S}({\Cal A}\otimes{\Cal B})$ satisfying the marginal conditions
$$
\eqalignno{
{\Cal E}^*(\varphi)\upharpoonright{\Cal A}&={\Cal L}^*(\varphi)
\upharpoonright{\Cal A}&(8.1)\cr
{\Cal E}^*(\varphi)\upharpoonright{\Cal B}&={\Cal L}^*(\varphi)
\upharpoonright{\Cal B}&(8.2)\cr}
$$
for any state $\varphi\in{\frak S}(\Cal A)$.
 
Let $\varphi =\sum\limits_k {\lambda _k\varphi _k}$ be a fixed certain
decomposition. Then the generalized mutual entropy is given by
$$I\left( {\varphi \;;\;{\Cal E}^*,{\Cal L}^*} \right)=S\left( {\sum\limits_k
 {\lambda _k{\Cal E}^*\left( {\varphi _k} \right)}\left|
 {{\Cal L}^*\left( \varphi \right)} \right.} \right)\eqno(8.3)$$
If the decomposition is not fixed, then we have to take the supremum
on a possible decompositions needed for information transmission.
A typical example is the case that ${\frak S}(\Cal A)$ is the set of all
density operators on $\Cal H$ and so is ${\frak S}({\Cal A}\otimes{\Cal B})$
on ${\Cal H}\otimes{\Cal K}$.
In this case, when a density operator $\rho\in{\frak S}(\Cal A)$ is given,
the marginal conditions become
$$
\eqalignno{
tr_{\Cal K}{\Cal E}^*(\rho)&=tr_{\Cal K}{\Cal L}^*(\rho)&(8.4)\cr
tr_{\Cal H}{\Cal E}^*(\rho)&=tr_{\Cal H}{\Cal L}^*(\rho)&(8.5)\cr}
$$
and the generalized mutual entropy is
$$I\left( \rho \;;\;{\Cal E}^*,{\Cal L}^* \right)=
\mathop{\sup}\limits_E\left\{S\left( \sum\limits_k{\lambda_k{\Cal E}^*
\left( {E_k} \right)}\vert{{\Cal L}^*\left( \rho \right)}\right)
\right\}\eqno(8.6)$$
when the supremum is taken over all Schatten decompositions
$$\rho =\sum\limits_k {\lambda _kE_k},\quad
 E_k\in ex{\frak S}\left(\Cal A \right)$$
In particular, when
$$\eqalignno{
&\sum\limits_k {\lambda _k{\Cal E}^*\left( {\rho _k} \right)}
=\sum\limits_k {\lambda _kE_k\otimes \Lambda ^*E_k}&(8.7)\cr
&{\Cal L}^*\left( \rho  \right)=\rho \otimes \Lambda ^*\rho &(8.8)\cr}
$$
for a channel ${\Lambda ^*:{\frak S}\left( {\Cal A} \right)\to 
{\frak S}\left( {\Cal B} \right)}$, the generalized mutual entropy is
equivalent to the mutual entropy :
$$I\left( {\rho \;;\;{\Cal E}^*,{\Cal L}^*} \right)=
I\left( {\rho \;;\;\Lambda ^*} \right)\eqno(8.9)$$
The generalized mutual entropy so defined cannot be interpreted as the 
information transmitted a state to another.
The use of this generalized mutual entropy will be discussed in other
occasion.
 
\Bigskip
\leftline{\fontB 9 Conclusion}
\bigskip
In this paper we have constructed the Stinespring expression of a dissipative
evolution of an open system model, and we have rigorously computed
 the mutual entropy of the associated QMC.
 
Once we have the concrete expression of the mutual entropy,
it will be useful to investigate the dynamical change of the mutual entropy
in analysis of communication processes and nonequilibrium behavior
in statistical mechanics.
The detailed discussion of the change of mutual entropy
will be reported in the forthcoming paper [6]. 
 
\Bigskip
\leftline{\fontB References}
\bigskip
 
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\bye

