数理解析研究所講究録
第 1906 巻 2014 年 162-171

162

Operator Algebraic Shannon’s Interpretation
for Entropy-preserving Stochastic Averages
Marie Choda
Abstract

We study various relations of and from the view point of the
von Neumann entropy. Here and are a state and a unital positiveof $n\cross n$ complex
Tr-preserving linear map on the algebra
arisingmatrices respectively. For the state and the new stateas the composition, we show among the others that these two states
have the same value of the von Neumann entropy if and only if.
behaves for as some automorphism of$\Phi$$\rho$$\Phi$$\rho$$M_{n}(\mathbb{C})$$\rho\circ\Phi$$\rho$$\Phi$$M_{n}(\mathbb{C})$$\rho$

1

Introduction

Shannon([8, p.3954 denotes as the followings: If we perform any”averagof the forming “ operation on the$\{p_{i}\}_{i=1,\cdots,n}$$p_{i}’= \sum_{j}a_{ij}p_{j}$where $p_{i} \geq 0, \sum_{i}p_{i}=1 and a_{ij}\geq 0, \sum_{i}a_{ij}=\sum_{j}a_{ij}=1)$ , the entropy $H$
increases (except in the special case where this transformation amounts to
no more than a permutation of the with $H$ of course remaining the same).
of a probability vector
This means the followings: The entropy
for aof the probability vector
and the entropy$b=[b_{ij}]$
are always in the relation thatbistachastic matrix
and the two values are equal if and only if the bistochastic matrix behaves.
just as a permutation , i.e.
(resp. a bistachastic matrix b) toReplacing a probability vector(resp. a unital positive Tr-preserving linear map on
a state of$($$p_{i}$$H(\lambda)$$\lambda b$$H(\lambda b)$$\lambda=(\lambda_{1}, \cdots, \lambda_{n})$$H(\lambda)\leq H(\lambda b)$$b$$\sigma$$\lambda b=(\lambda_{\sigma(1)}, \cdots, \lambda_{\sigma(n)})$$\lambda\in \mathbb{R}^{n}$$\rho$$M_{n}(\mathbb{C})$$\Phi$

163, we show, among the others, that the von Neumann entropy
increases by performing any on (except in the special case where thistransformation amounts to no more than an automorphism of state withof course remaining the same).
$S$$M_{n}(\mathbb{C}))$$\Phi$$\rho$$\alpha$$\rho$$S$

2

Notations, terminologies and basic facts

The main tool is the entropy function$\eta(t)=-t\log t$$\eta$

defined on the interval$(0<t\leq 1)$and, 1 by$]$$\eta(0)=0.$

The is strictly concave, i.e. for two -tuples of real numbers$t_{i}>0,$that, it holds that$k$$\eta$$[0$$\{s_{i}\},$$\{t_{i}\}$such$\sum_{i=1}^{k}t_{i}=1$$s_{i}\geq 0,$

$k$

$k$

$\sum t_{i}\eta(s_{i})\leq\eta(\sum t_{i}s_{i})$,$i=1 i=1$
and the equality holds if and only if
for all $j.$
Moreover, is strictly operator-concave, i.e. the similar relations hold byreplacingto any bounded self-adjoint operatorswith spectra in[0,1], i.e.
$i,$$s_{i}=s_{j}$$\eta$$\{s_{i}\}_{i}$$\{x_{i}\}_{i}$$\sum_{i=1}^{k}t_{i}\eta(x_{i})\leq\eta(\sum_{i=1}^{k}t_{i}x_{i})$and the equality implies that $x_{i}=x_{j}$ for all $i,j$ . (see for example 4, $B],$
[5, 6
$[$

Let
and

be a probability vector in, i.e.
. The Shannon entropy
for is given as$\lambda=(\lambda_{1}, \cdots, \lambda_{n})$$\sum_{i}\lambda_{i}=1$$\mathbb{R}^{n}$$H(\lambda)$$\lambda_{i}\geq 0$

for all$i$$\lambda$$H(\lambda)=\eta(\lambda_{1})+\cdots+\eta(\lambda_{n})$.

It holds always that $H(\lambda)\leq\log n$ and $H(\lambda)=\log n$ if and only if
for all $i=1,$
Throughout this note, let $H$ be an -dimensional Hilbert space. We denote by $M$ the algebra $B(H)$ of linear operators on $H$ so that $M$ is isomorphic, i.e. the -algebra oftomatrices over the complex field . By
Tr we mean the standard trace of $M$ such that $Tr(e)=1$ for every minimal
projection in $M.$$\lambda_{i}=1/n$$\cdots,$ $n.$$n$$C^{*}$$M_{n}(\mathbb{C})$$e$$n\cross n$$\mathbb{C}$164
Every positive linear functional on $M$ is of the form $\phi(x)=Tr(D_{\phi}x)$ ,$(x\in M)$ for a unique positive element $D_{\phi}\in M$ which is called the density
operator or density matrix of . If is a state of $M$ , then the density matrix
is a positive operator in $M$ such that $R(D_{\rho})=1.$
of , the von Neumann entropy
By using the eigenvalue list
are defined byfor andand$\phi$$\phi$$\rho$$D_{\rho}$$\{\lambda_{1}, \cdots, \lambda_{n}\}$$S(\rho)$$S(D_{\rho})$$D_{\rho}$$D_{\rho}$$\rho$$S( \rho)=S(D_{\rho})=\sum_{i=1}^{n}\eta(\lambda_{i})$.

The von Neumann entropy and stochastic
averages3

Our purpose of this note is to give a generalized version of Shannon’s interpretation for entropy-preserving stochastic averages of probability vectors to.
the framework of von Neumann entropy for states on
In this section, we discuss the Shannon’s interpretation in the framworkof the von Neumann entropy as follows:, a bistachastic
Replace a probability vectors to a state of, and theonmatrix to a unital positive trace preserving map
then a permutation changes
Shannon entropy $H$ to the von Neumann, i.e., $S(\rho\circ\Phi)=S(\rho)$ if and only if
into an automorphism offor some automorphism$M_{n}(\mathbb{C})$$\lambda$$\rho$$M_{n}(\mathbb{C})$$\Phi$$b$$M_{n}(\mathbb{C})$$S$$M_{n}(\mathbb{C})$$\alpha$$\rho 0\Phi=\rho 0\alpha$The pair

3.1
Let$D_{\rho}$$\alpha.$$\{\rho, \Phi\}$of state$\rho$and positive map$\Phi.$

the density matrix of , i.e.,. We denote by
be a state ofwhich satisfies thatis a positive operator in$M_{n}(\mathbb{C})$$\rho$$D_{\rho}$$\rho$$M_{n}(\mathbb{C})$$h(D_{\rho})=1$

and$\rho(x)=R(D_{\rho}x)$

for all$x\in M_{n}(\mathbb{C})$.

be a positive unital Tr preserving map. Then:
and $h(\Phi(D_{\rho}))=1.$
is a operator in, we need the
In order to see the state whose density matrix is: The inner productsystem of the Hilbert-Schmidt inner product ofLet$\Phi(D_{\rho})$$\Phi$$M_{n}(\mathbb{C})arrow M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$$\Phi(D_{\rho})$$M_{n}(\mathbb{C})$165
and the norm are given by$<x,$ $y>=$

Tr

and

$(y^{*}x)$$\Vert x\Vert_{2}=(Tr(x^{*}x))^{1/2}$

The *-preserving map induces the adjoint map$\cdot>by$with respect to this$\Phi$

for$x,$ $y\in M_{n}(\mathbb{C})$$\Phi^{*}$.

:$M_{n}(\mathbb{C})arrow M_{n}(\mathbb{C})$

.

(3.1)

$<\cdot,$$Tr(y\Phi^{*}(x))=Tr(\Phi(y)x) x, y\in M_{n}(\mathbb{C})$

Since is positive, it follows thatis positive, andis a state by$\Phi=Tr.$
property
thethat TrThe
is the density matrix of this statebecause$\Phi$$\Phi^{*}$$\rho\circ\Phi^{*}$$\Phi(D_{\rho})$$\rho 0\Phi^{*}$$\rho\circ\Phi^{*}(x)=Tr(D_{\rho}\Phi^{*}(x))=Tr(\Phi(D_{\rho})x) , (x\in M_{n}(\mathbb{C}))$We let the set of eigenvalues of$\lambda=$$(\lambda_{1}, \cdots , \lambda_{n})$$D_{p}$

andand$\Phi(D_{\rho})$.

be$\mu=(\mu_{1}, \cdots , \mu_{n})$

(3.2)

,

respectively. Here we arrange them always in a decereasing order, i.e.,$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$and$\mu_{1}\geq\mu_{2}\geq\cdots\geq\mu_{n}$

.

(3.3)

resp.
be mutually orthogonal minimal
projections, which gives the spectral decomposition of(resp.
):We let$\{e_{1}, \cdots, e_{n}\}$$\{p_{1}, \cdots,p_{n}\})$$($$\Phi(D_{\rho})$$D_{\rho}$$D_{\rho}= \sum_{i=1}^{n}\lambda_{i}e_{i}$

$($

resp.

$\Phi(D_{\rho})=\sum_{j=1}^{n}\mu_{j}p_{j})$.

(resp.
We denote by) the maximal abelian subalgebra ofwhich is generated by the projections
resp.
$A$

$B$

$\{e_{1}, \cdots, e_{n}\}$

3.1.1

The unitary$u_{(\rho,\Phi)}$In these setting, a unitary$($arising from the pair$u_{(\rho,\Phi)}$(3.4)$M_{n}(\mathbb{C})$$\{p_{1}, \cdots,p_{n}\})$.

$\{\rho, \Phi\}.$appears and satisfies the following relation:$u_{(\rho,\Phi)}e_{i}=p_{i}u_{(\rho,\Phi)}$,for all$i=1,$ $\cdots,$ $n$

(3.5)

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Bistochastic matrix

3.1.2

for the pair$b_{\rho}(\Phi)$

Definition 3.1. We define a matrix$b_{\rho}(\Phi)$$\{\rho, \Phi\}$by the formula$b_{\rho}(\Phi)_{ij}=Tr(\Phi(e_{i})p_{j}) , (1\leq i\leq n, 1\leq j\leq n)$

.

(3.6)

, and letbe a unital positive TrLemma 3.2. Let be a state of. Let and be the probability vectors of thepreserving map on
respectively. Then the followings hold:andeigenvalues of(1) Theis a bistoch,astic
is transposed to the probability vector(2) The probability vector:
by the matrix$\Phi$$M_{n}(\mathbb{C})$$\rho$$\lambda$$M_{n}(\mathbb{C})$$\mu$$\Phi(D_{\rho})$$D_{\rho}$$matr\cdot ix.$$b_{\rho}(\Phi)$$\lambda\in \mathbb{R}^{n}$$b_{\rho}(\Phi)$$\mu\in \mathbb{R}^{n}$$\lambda b_{\rho}(\Phi)=\mu.$Definition 3.3. For each , we set$j$$I_{j}=\{i:b_{\rho}(\Phi)_{ij}\neq 0\}.$

Lemma 3.4. Let
preser ving map on, we have that$\rho$, and letbe a unital positive Trbe a state of. Assume that $S(\Phi(D_{\rho}))=S(D_{\rho})$ . Then, for each$\Phi$$M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$$j$$\lambda_{i}=\lambda_{k}$for all$i,$ $k\in I_{j}.$Under the assumption that $S(\Phi(D_{\rho}))=S(D_{\rho})$ , we denote the constant. Remark that each is a nonin the above Lemma by
foris a bistochastic matrix, andempty set because$\lambda_{i}$$\lambda^{(j)}$$i\in I_{j}$$I_{j}$$b_{\rho}(\Phi)$

for all$\lambda^{(j)}=\frac{\sum_{i\in I_{j}}\lambda_{i}}{|I_{j}|}=\lambda_{k}$$k\in I_{j}.$and let :Theorem 3.5. Let be aofunital positive Tr preserving map. Then the followings are equivalent:(i) $S(\rho\circ\Phi^{*})=S(\rho)$ , i.e. $S(\Phi(D_{\rho}))=S(D_{\rho})$ .
, i.e.
(ii)
denotes the transpose.
where$\mathcal{S}late$$\rho$$\lambda=\mu b_{\rho}(\Phi)^{T}$$M_{n}(\mathbb{C})$$\Phi$$M_{n}(\mathbb{C})arrow M_{n}(\mathbb{C})$$\lambda=\lambda b_{\rho}(\Phi)b_{\rho}(\Phi)^{T}$$\{\}^{T}$(iii)$\lambda_{i}=\mu_{i}$(iv) Thefor all $i=1,$
satisfies that

$unitar^{v}yu_{(\rho,\Phi)}$$\cdots,$$n.$$\Phi(D_{\rho})=u_{(\rho,\Phi)}D_{\rho}u_{(\rho,\Phi)}^{*}.$

be a

167
Remark 3.6. If a state is the normalized trace $Tr/n$ , then the density
matrix is
so that the all statements in the above theorem are trivial.
$\rho$$I_{n}/n$

for a bistochastic matrix
Remark 3.7. In the case of $n=2$ , ifthen is the nontrivial permutation.
In fact, let. Every
bistochastic matrix $b=(b_{ij})$ iswritten as $b_{11}=b_{22}=b_{1}$ for some $0\leq b_{1}\leq 1$ and $b_{12}=b_{21}=b_{2}=1-b_{1}.$, thenIf
andThis implies that $\lambda_{1}b_{1}(2b_{1}-2)+b_{1}(1-b_{1})=0$ . Hence ifthen $b_{1}=0$
or $b_{1}=1$ , which means that is permutation matrix. Assume that
, Thenor
We may omit the case $b_{1}=1$ and so we assume$b_{1}=$ O.
As we omit that is the trivial case so that $b_{1}=0$ , i.e. is the
non-trivial permutation.
$\lambda bb^{T}=\lambda$

$b,$

$b$

$2\cross 2$$\lambda=(\lambda_{1}, \lambda_{2})$$\lambda bb^{T}=\lambda$$\lambda_{1}=\lambda_{1}(b_{1}^{2}+b_{2}^{2})+2\lambda_{2}b_{1}b_{2}$$\lambda_{2}=\lambda_{2}(b_{1}^{2}+b_{2}^{2})+2\lambda_{1}b_{1}b_{2}.$$\lambda_{1}=0$$b$$\lambda_{1}\neq 0.$$b_{1}\neq 1$$\lambda_{1}=1/2$$b$$\lambda$

Corollary 3.8.
that $S(\Phi(D_{\rho}))=S(\rho)$ holds for the pairstate ofand a unital positive TJ preserving map on$A_{\mathcal{S}}sume$$\Phi$$M_{n}(\mathbb{C})$$\rho$$<\Phi(D_{\rho})$, $\Phi(e_{k})>=<D_{\rho},$ $e_{k}>$ for all$\{\rho, \Phi\}$$M_{n}(\mathbb{C})$of a. Then$k.$(the tensor
A linear map on
said to be 2-positive if
product of and the identity map on) on
is positive. It
is well known that if is 2-positive, thenis 2-positive and the so-called
Kadison-Schwartz inequality holds [2], (cf. [4, 5, 6$\Phi$$\Phi\otimes id$$M_{n}(\mathbb{C})is$$\Phi$$M_{2}(\mathbb{C})$$\Phi$$M\otimes M_{2}(\mathbb{C})$$\Phi^{*}$$\Phi^{*}(x^{*})\Phi^{*}(x)\leq\Phi^{*}(x^{*}x) , (x\in M)$.

Corollary 3.9. Let be a state of, and letbe a unital positive.
Tr preserving map on
thenconditions are equivalent:
If is$S(\Phi(D_{\rho}))=S(D_{\rho})$(i’)(iv) $\Phi(D_{\rho})=uD_{\rho}u*for$ some unitary $u.$(v)
$M_{n}(\mathbb{C})$$\rho$$\Phi$$M_{n}(\mathbb{C})$$\Phi$$A^{6}-positive_{f}$$tl\iota efollowir\iota g$$\Phi^{*}\Phi(D_{\rho})=D_{\rho}$

Related results are obtained in [7] and [3].
Example 3.10. The conditional expectation conditioned by $Tr/n$ is a mosttypical example of unital completely posive (so that 2-positive) $H$-preserving

168
linear map of
a -subalgebra$M_{n}(\mathbb{C})$$C^{*}$

. Let

$E$

be such a conditional expectation of $A=M$ towith $1_{A}=1_{B}$ . Then$B$

if and only if$S(E(D_{\rho}))=S(D_{\rho})$$D_{\rho}\in B.$

In fact, the conditional expectation $E$ satisfies that $E^{*}E=E$ . By combining this fact with Corollary 3.7, we have that $S(E(D_{\rho}))=S(\rho)$ if andonly if $D_{\rho}=E^{*}E(D_{\rho})=E(D_{\rho})$ which means that
$D_{\rho}\in B.$

3.2

Relations among various entropies

The weighted entropy
andwith respect to a probability vector
the following forms:$H^{\lambda}(b)$$H_{\lambda}(b)$In the case where$\lambda_{i}=1/n$$\lambda=(\lambda_{1}, \cdots, \lambda_{n})$and$H^{\lambda}(b)= \sum_{j=1}^{n}\lambda_{j}\sum_{k=1}^{n}\eta(b_{jk})$

for a bistochastic matrix $b=[b_{ij}]$
are defined in [10] by$H_{\lambda}(b)= \sum_{k=1}^{n}\lambda_{k}\sum_{j=1}^{n}\eta(b_{jk})$.

for all , these are denoted by $H(b)$ simply:$i$$H(b)= \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\eta(b_{ij})$.

Definition 3.11. We let$J_{\lambda}=\{k;\lambda_{k}\neq 0\}.$is a density matrix whichis positive and $R$-preserving, each$\rho_{i}(x)=Tr(\Phi(e_{i})x)$.
induces the state given by
for all

Since$\Phi$$\Phi(e_{i})$$x\in M_{n}(\mathbb{C})$$\rho_{i}$which is a
Definition 3.12. Now we pick up the following constant$\{S(\Phi(e_{i}));i=1,\cdots,n\}$convex combination of the entropies
with respect tothe eigenvalues of the density matrix :$S_{\rho}(\Phi)$$D_{\rho}$$S_{\rho}( \Phi)=\sum_{i=1}^{n}\lambda_{i}S(\Phi(e_{i}))=\sum_{i=1}^{n}\lambda_{i}S(\rho_{i})$.

The algebra $B$ is a typical von Neumann subalgebra of the -factor
from
onto
and there exists always a positive linear map$N$ such that $aE(x)b=E(axb)$ for all $x\in M$ and$b\in B$ which is called
conditional expectation of
onto $B.$$I_{n}$$E_{B}$$M_{n}(\mathbb{C})$$a,$$M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$169
Lemma 3.13. Let

be the conditional expectation$E_{B}$$ofM_{n}(\mathbb{C})$for each$E_{B}( \Phi(e_{i}))=\sum_{j=1}^{n}b_{\rho}(\Phi)_{ij}p_{j}$onto B. Then$i.$so that$H^{\lambda}(b_{\rho}( \Phi))=\sum_{i=1}^{n}\lambda_{i}S(E_{B}(\Phi(e_{i})))$.

, and letTheorem 3.14. Let be a state of
be a unital positive$R$ -preserving map on.
Then the following relations hold for the weighted entropies of the bistochastic matrix
with respect to the eigenvalue listofand the eigenvalue list:of(1)$\Phi$$M_{n}(\mathbb{C})$$\rho$$M_{n}(\mathbb{C})$$\lambda=(\lambda_{1}, \cdots, \lambda_{n})$$b_{\rho}(\Phi)$$D_{\rho},$$\mu=(\mu_{1}, \cdots, \mu_{n})$$S_{\rho}(\Phi)$$\Phi(D_{\rho})$$S_{\rho}(\Phi)\leq H^{\lambda}(b_{\rho}(\Phi))\leq S(\rho\circ\Phi^{*})\leq S(\rho)+S_{\rho}(\Phi)$

(2)
(3)

$S_{\rho}(\Phi)=H^{\lambda}(b_{\rho}(\Phi))$

if and only if
if and only if$\Phi(e_{i})\in B$$i\in J_{\lambda}.$$H^{\lambda}(b_{\rho}(\Phi))=S(\rho\circ\Phi^{*})$$(\mu_{1}, \cdots , \mu_{n})=(b_{\rho}(\Phi)_{i1}, \cdots , b_{\rho}(\Phi)_{in})$

(4)
(5)

for all.

$S_{\rho}(\Phi)=S(\rho\circ\Phi^{*})$

for allfor every

if and only if
if and only if the$i\in J_{\lambda}.$$\Phi(D_{\rho})=\Phi(e_{i})$$S(\rho\circ\Phi^{*})=S(\rho)+S_{\rho}(\Phi)$

Remark 3.15. The above statement (3) says that
if and only if
has the following form:$\rho$

is a pure state.

$H^{\lambda}(b_{\rho}(\Phi))=S(\Phi(D_{\rho}))$$b_{\rho}(\Phi)$$b_{\rho}(\Phi)=\{\begin{array}{lll}\mu_{1} \mu_{2} \mu_{n}\mu_{l} \mu_{2} \mu_{n}\mu_{1} \mu_{2} \mu_{n}b_{\rho}(\Phi)_{k1} b_{\rho}(\Phi)_{k2} b_{\rho}(\Phi)_{kn}b_{\rho}(\Phi)_{n1} b_{\rho}(\Phi)_{n2} b_{\rho}(\Phi)_{nn}\end{array}\}$Here $k=|J_{\lambda}|+1$ for the cardinality$|J_{\lambda}|$of$J_{\lambda}.$$i\in J_{\lambda}.$170
Corollary 3.16. If$\lambda_{i}\neq 0$$H^{\lambda}(b_{\rho}(\Phi))=S(\rho 0\Phi^{*})$

for all $i=1,$$\rho 0\Phi^{*}$and if$j=1,$$\cdots,$$\Phi$

satisfies that, then$\mu_{j}=b_{\rho}(\Phi)_{ij}=\frac{1}{n}$so that$\cdots,$ $n$

for all

$i,$

$n,$

is the nor nalized trace $Tr/n$ and $S(\rho\circ\Phi^{*})=\log n.$

Remark 3.17. (A connection with Hadamard matrix). A bistochastic
matrix is said to be unistochastic if it is induced from some unitary matrix
Aby thatunitary matrix isfor all $i,j=1,$$|u_{i,j}|=1/\sqrt{n}$called a Hadamard matrix if
for all $i,j=1,$
The above corollary means that if
has only non-zero eigenvalues, (i.e.,
for all i) and if
thenis a unistochastic
matrix induced from a Hadamard matrix.
$b$$b_{i,j}=|u_{i,j}|^{2}$$u$$n\cross n$$\cdots,$ $n.$$u$$\cdots,$ $n.$$D_{\rho}$$H^{\lambda}(b_{\rho}(\Phi))=S(\Phi(D_{\rho}))$$\lambda_{i}\neq 0$$b_{\rho}(\Phi)$

Example 3.18. Here, we give some examples.
(1) If is a pure state, then the four kinds constants satisfy that$\rho$$S(\rho)=0$

and$S_{\rho}(\Phi)=H^{\lambda}(b_{\rho}(\Phi))=S(\rho\circ\Phi^{*})$

for all positive unital Tr-preserving map(2) If$\Phi$

is$a^{*}$$\Phi.$-isomorphism, then for all stateand$S_{\rho}(\Phi)=0$$\rho$

the followings hold:$S(\rho\circ\Phi^{*})=S(\Phi(D_{\rho}))=S(D_{\rho})=S(\rho)$.

In fact, if ia-isomorphism, thenis a minimal projection for a$S_{\rho}(\Phi)=\sum_{i=1}^{n}\lambda_{i}S(\Phi(e_{i}))=0$
minimal projection , so thatand of course$\Phi$$a^{*}$$\Phi(e)$$e$$S(\Phi(D_{\rho}))=S(D_{\rho})$.

(3) Ifis a unital positive Tr-preserving map to the center, then$\Phi$$\mathbb{C}1_{M}$of$M_{n}(\mathbb{C})$$S_{\rho}(\Phi)=H^{\lambda}(b_{\rho}(\Phi))=S(\rho\circ\Phi^{*})=\log n$

In fact, for each , put

forfor all state$\rho.$, then $1=rb(e_{i})=$$Tr(\Phi(e_{i}))=\alpha_{i}H(\Phi(1_{M}))=\alpha_{i}n$ so that $\Phi(e_{i})=1_{M}/n$ .
This implies that
$S_{\rho}( \Phi)=\sum_{i}\lambda_{i}S(1_{M}/n)=Tr(\eta(1_{M}/n))=\log n$ . Remember that in general
for all state . Hence we have theequality.
$i$$\Phi(e_{i})=\alpha_{i}1_{M}$$S_{\rho}(\Phi)\leq H^{\lambda}(b_{\rho}(\Phi))\leq S(\rho\circ\Phi^{*})\leq\logn$$\alpha_{i}\in \mathbb{C}$$\rho$

171

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Osaka Kyoiku University
Asahigaoka, Kashiwara 582-8582
JAPANmarie@cc.osaka-kyoiku.ac.jp