数理解析研究所講究録
第 1820 巻 2012 年 11-24

11

A representation of unital completely positivemaps
大阪教育大学まりゑ (Marie Choda)

長田AbstractLetbe the algebra ofcomplex mairices, and let be aunital completely positive map of. With the notion
toof the von Neumann entropy for a state in mind, we give a model of-tuppleso that $\Phi(x)=v_{1}^{*}xv_{1}+\cdots+v_{r}^{*}xv_{r},$ $(x\in M_{n}(\mathbb{C}))$ .
The is uniquely determined for and the -tupple is also unique up
to aunitary matrix.
$M_{n}(\mathbb{C})$$n\cross n$$\Phi$$M_{n}(\mathbb{C})$$r$$M_{k}(\mathbb{C})$$\{v_{j}\}_{j=1}^{r}$$\Phi$$r$$r$$r\cross r$

1

Introduction

In the framework of the theory of operator algebras, the notion of entropy for
automorphisms was introduced by Connes-St
in [8], Connes-NarnhoferThirring in [9] and Voiculescu in [14] (which is extended by Brown [4]). The$Connes-St\emptyset mer$ entropy
is defined for-automorphism of finite von$\tau=\tau
0\theta$$M$
Neumann algebrawith, where is a fixed given finite trace$M$of . After then, the Connes-Narnhofer-Thirring entropy
is given as
an extended version of
for a -automorphism of a-algebra by
replacing the trace to a state of , and if is a finite von Neumann
algebra then
. Voiculescu’s topological entropy
is defined$A.$
as an independent version of any state of
We studied these entropies in [6] and [7] for not only *-automorphisms but
also -endomorphisms like so called canonical shifts. As one of interestingsuch *-endomorphisms, we picked up the Cuntz canonical endomorphismon the Cuntz algebra
which has a strong connection to Longo’s canonicalshift (cf. [6]). The
is the -algebra generated by isometries
such that $S_{1}S_{1}^{*}+\cdots+S_{n}S_{n}^{*}=1$ , and theis defined as$\emptyset mer$$a^{*}$$H(\theta)$$\theta$$\tau$$h_{\phi}(\theta)$$*$$H(\theta)$$\phi$$\tau$$A$$A$$C^{*}$$\theta$$A$$h_{\tau}(\theta)=H(\theta)$$ht(\theta)$$*$$\Phi_{n}$$O_{n}$$O_{n}$$C^{*}$$\{S_{1}, \cdots, S_{n}\}$$\Phi_{n}$$\Phi_{n}(x)=S_{1}xS_{1}^{*}+\cdots 十 S_{n}xS_{n}^{*}, (x\in O_{n})$

.

(1.1)

12
Such maps given by the form as the right hand side of (1.1) are unitalarecompletely positive maps, and the above notions
andavailable for unital completely positive maps too.
Conditional expectations are the most typical examples of unital comare considpletely positive maps, ans states of the matrix algebras
ered as the most elementary example of conditional expectations. However,for a conditional expectation , by their definitions it holds always that
$H(E)=h_{\phi}(E)=ht(E)=0$ . On the other hand, in the case of the von, it is possible that $S(\phi)\neq 0.$
for a state of
Neumann entropy
In order to define “entropy”, we need the notion of “finite partition ofunity” (see for example, [11]). The most generalized one of “finite partition of unity” was introduced by Lindblad ([10]), and it is called the “finite
operational partition of unity”
With these facts in mind, here we give a method to induce the finite
operational partition of unity for a given unital completely positive map.
That is, let and be unital -algebras and let be a unital completely
to . We give a method to get a $mo$ del of -tupple
positive map of$v(\Phi)=\{v_{1}, v_{2}, \cdots, v_{r}\}$ such that
$H(\cdot),$$h_{\phi}(\cdot)$$ht(\cdot)$$M_{n}(\mathbb{C})$$E$$S(\phi)$

$B$

$A$

$A$

$M_{n}(\mathbb{C})$$\phi$$C^{*}$$\Phi$$B$$r$$\Phi(x)=v_{1}^{*}xv_{1}+\cdots+v_{r}^{*}xv_{r}, (x\in A)$

.

(1.2)

When and are matrix algebras, such a representation is called Kraus
representation (cf. Appendix in [13]), or obtained as a straightforward application of Stinespring’s theorem (see for example, [1, 3]). We note that thisrepresentation is not unique
Our main purpose in this note is to show, for a given completely positive
map , a unique -tupple $v(\Phi)=\{v_{1}, \cdots, v_{r}\}$ which is suitable to extend
for a state of matrix algebras tothe notion of von Neumann entropy
the entropy $S(\Phi)$ for a unital completely positive map
First, for a given completely positive map from $B(H)$ to $B(K)$ of finite
dimensional Hilbert spaces $H,$ $K$ , we construct the Hilbert spaces
Let $r=\dim(H\otimes_{\Phi}K)$ . Next, we give a -tupple $v(\Phi)=\{v_{1}, v_{2}, \cdots, v_{r}\}$ which
satisfy that $\Phi(x)=v_{1}^{*}xv_{1}+\cdots+v_{r}^{*}xv_{r}$ . The -tupple is unique up to unitaties and induces $S(\Phi)$ . After then, we apply these to the non-commutativewith respect to a state
Bernoulli shift and we define the entropy. These results in this note are in [5].
with
$B$

$A$

$\Phi$$r$$S(\phi)$$\phi$$\Phi.$$\Phi$$H\otimes_{\Phi}K.$

$r$

$r$

$\beta$$\varphi=\varphi\cdot\beta$$S_{\varphi}(\Phi)$$\varphi$132Preliminaries

Here, we denote some notaions and terminologies which we use later.
We denote by
the algebra of $n\cross n$ complex matrices, and by
the standard trace, that is, the sum of all diagonal components.
matrixis called a density matrix if $D$ is a positive operator with
$Tr_{n}(D)=1$ (cf. [11] [12]).
The notation is called the entropy function in usual, and it is the function defined by$M_{n}(\mathbb{C})$$Tr_{n}$$A$$D\in M_{n}(\mathbb{C})$$\eta$$\eta(t)=.\{\begin{array}{ll}-t\log t, (0<t\leq 1)0, t=0\end{array}$

2.1

Finite partitions

The notion $of$ ” finite partition of unity” is the starting point of our study.
$a$2.1. 1Finite partitions of 1

The first one is discuused in the real numbers$\mathbb{R}$. Let$\lambda=\{\lambda_{1}, \cdots, \lambda_{n}\}$be the set of real numbers
is a finite partition of 1.

$\lambda_{i}\geq 0$

with$\sum_{i}\lambda_{i}=1$. We say that the -tupple$n$$\lambda\subset \mathbb{R}$2.1.2

Finite operational partition of unity

The terminology, a finite operational partition of unity, was first given by
Lindblad ([10]) and after then it is used by Alicki-Fannes([2]).
Let be a unital -algebra. Let $x=\{x_{1}, \ldots, x_{k}\}\subset A$ . Then is said tobe a finite operational partition of unity of size if$A$$C^{*}$$x$$k$$\sum_{i}^{k}x_{i}^{*}x_{i}=1_{A}$

.

(2.1)

Such a finite operational partition of unity $x=\{x_{1}, \ldots, x_{k}\}$ in induces an-coefficient in $M_{k}(A)$ , whose $(i,j)$ coefficient $x(j, i)$ is given by the following:
$A$

$A$

$x(j, i)=x_{i}^{*}x_{j}, (1\leq i,j\leq k)$

.

(2.2)

We denote this matrix by . Thenis an -coefficient density matrix in$M_{k}(A)$ , that is,is a positive operator with $Tr([x])= \sum_{i=1}^{k}x(i, i)=1_{A}.$
$[x]$

$[x]$

$[x]$

$A$

14
2.2

Entropy for finite partitions of unity

2.2.1

Entropy for finite partitions of 1

Let a -tupple$n$$\lambda\subset \mathbb{R}$be a a finite partition of 1. Let$H(\lambda)=\eta(\lambda_{1})+\cdots+\eta(\lambda_{n})$Then$H(\lambda)$.

is called the entropy for the finite partiton(2.3)$\lambda$

of 1.

Entropy for Finite operational partition of unity

2.2.2

Let $x=\{x_{1}, \ldots, x_{k}\}$ be an operational partition of unity in a unital
matrix whose
is thealgebra , and let be a state of . The$\{i,j\}$ -component is defined by$C^{*}-$$A$$A$$\varphi$$k\cross k$$\rho_{\varphi}[x]$$\rho_{\varphi}[x](i,j)=\varphi(x_{j}^{*}x_{i}) , (i,j=1, \cdots, k)$

.

(2.4)

the density matrix associateThen
is a density matrix. We call
simply.
bywith and . If is a unique tracial state, we denotebe the eigenvalues of the matrix
Letis a density matrix.
is a finite partition of 1 because
Then.
Hence we have the entropyLetbe the von $Neuma\iota m$ entropy (cf. [11, 12]) for the density, that is,is nothing else but. Thenmatrix$\rho_{\varphi}[x]$$\rho_{\varphi}[x]$$x$$\rho[x]$$\rho_{\varphi}[x]$$\varphi$$\varphi$$\rho_{\varphi}[x].$$\lambda(\rho_{\varphi}[x])=\{\lambda_{1}, \lambda_{2}, \cdots, \lambda_{k}\}$$\rho_{\varphi}[x]$$\lambda(\rho_{\varphi}[x])$$H(\lambda(\rho_{\varphi}[x]))$$S(\rho_{\varphi}[x])$$\rho_{\varphi}[x]$$H(\lambda(\rho_{\varphi}[x]))$$S(\rho_{\varphi}[x])$$S( \rho_{\varphi}[x])=Tr_{k}(\eta(\rho[x]))=H(\lambda(\rho_{\varphi}[x]))=\sum_{i}\eta(\lambda_{i})$

.

(2.5)

Representation of completely posive maps

3

. We
. Put
to
Let be a completely posive map ofgive a method to get a “finite” family $v(\Phi)=\{v_{1}, v_{2}, \cdots, v_{r}\}$ for which
satisfies that $\Phi(x)=v_{1}^{*}xv_{1}+\cdots+v_{r}^{*}xv_{r}$ for all $x\in A$ . We remark that if
is unital, then $v(\Phi)$ is a finite operational partition:$\Phi$$M_{n}(\mathbb{C})$$M_{k}(\mathbb{C})$$A=M_{n}(\mathbb{C})$$\Phi$$\Phi$$\sum_{j=1}^{r}v_{j}^{*}v_{j}=\Phi(1_{A})=1_{B(K)}$

(3.1)

15
3.1

Hilbert space$H\otimes_{\Phi}K$

Let $H$ be an -dimensional Hilbert space, and let: $Aarrow B(K)$ be acompletely positive linear map. Let
be the set of mutuallyorthogonal minimal projections in $B(H)$ withfor all . Let$e_{i}(H)$ be a vector with
$m$ and we extend
for $i=1,$
to an orthonormal basis of $H$ as. Let $\{e_{ij};i,j=1, \cdots, n\}$ be amatrix units of with$m$ . Then eachso thatfor $i=1,$$H$$K$$\zeta\in H$ (the algebraic tensor product$H$ of and ) is written by$n$$\Phi$$\{e_{1}, \cdots, e_{m}\}$$\Phi(e_{i})\neq 0$$\Vert\xi_{i}\Vert=1$$i$$\xi_{i}\in$$\{\xi_{1}, \cdots, \xi_{m}\}$$\cdots,$$\{\xi_{1}, \cdots, \xi_{n}\}$$A$$e_{ij}\xi_{j}=\xi_{i}$$e_{ii}=e_{i}$$\zeta=\sum_{i=1}^{n}\xi_{i}\otimes\mu_{i}$,for some$\cdots,$$\mu_{i}\in K$

.

(3.2)

Definition 3.1.1. We define a sesquilinear form $<.,$$>\Phi$$H$ byon the space$< \sum_{i=1}^{n}\xi_{i}\otimes\mu_{i}, \sum_{j=1}^{n}\xi_{j}\otimes v_{j}>_{\Phi}=\sum_{i,j}<\Phi(e_{ji})\mu_{i}, v_{j}>K$

where$<\cdot,$$\cdot>_{K}$

,

(3.3)

means the inner product of the Hilbert space $K.$

Since
is completely posive, this form $<.,$
turns out positive
semidefinite. The value $<.,$
depends on the choise of the orthonor$H$
mal basis of . However the kernel of this sesquilinear form $<.,$
is$H\otimesK$unique up to unitaries on
as follows:$\Phi$$\cdot>_{\Phi}$$\cdot>_{\Phi}$$\cdot>\Phi$$(resp., \{\xi_{1}’, \cdots, \xi_{n}’\})$ be an
Proposition 3.1.2. Let
orthonormal basis of $H$ , and let
be a unitary on $H\otimes K$ withfor all. Let $Ker(\Phi)$ $(resp., Ker’(\Phi))$ be the kernel of this form via$i=1,$
Then$\{\xi_{1}, \cdots, \xi_{n}\}$$u$$\xi_{i}’=u\xi_{i}$$\cdots,$ $n$$\{\xi_{i}\}_{i}$$(resp., \{\xi_{i}’\}_{i})$$Ker’(\Phi)=(\overline{u}u^{*}\otimes 1)Ker(\Phi)$

where is the unitary matrix on $H$ whose $(i,j)$ -entry is the conjugate complex$n.$number of $u(i,j)$ for all $i,j=1,$$\overline{u}$$\cdots,$

Definition 3.1.3. Now taking the quotient by the space $Ker(\Phi)$ , we havea preHilbert space and complete to get the Hilbert space$H\otimes_{\Phi}K.$

We denote by$( \sum_{i=1}^{n}\xi_{i}\otimes\mu_{i})_{\Phi}$$\sum_{i=1}^{n}\xi_{i}\otimes\mu_{i}\in H$

the element in$H\otimes_{\Phi}K$

corresponding to16
is finite dimensional. We
If $K$ is finite dimensional, of course theto (for an example)
can extend this method to get the Hilbert space
non-commutative Bernoulli shifts, and it induces finite dimensional
even if $H$ and $K$ are infinite dimensional. By Proposition 3.1.2, we have thefollowing:$H\otimes_{\Phi}K$$H\otimes_{\Phi}K$$H\otimes_{\Phi}K$

Proposition 3.1.4. The dimension ofof orthonomal basis of $H.$

$H\otimes_{\Phi}K$

does not depend on the choice

Example 3.1.5.

1. If$\phi$

is a state of$M_{n}(\mathbb{C})$$\dim(\mathbb{C}^{n}\otimes_{\phi}\mathbb{C})=$, thenrank of$\phi,$$(i.e., the rank of the$ density matrx $of \phi)$ .

2. If $E$ is the conditional expectation of, thensubalgebra of$A$$M_{n}(\mathbb{C})$to a maximal abelian$M_{n}(\mathbb{C})$$\dim(\mathbb{C}^{n}\otimes_{E}\mathbb{C}^{n})=n.$

Here, we ramark that

3. Let$B$$M_{n}(\mathbb{C})$$A$

is isomorphic to the diagonal subalgebra

be a subfactor of
to $B$ , then$M_{n}(\mathbb{C})$

. If

$E$

$D_{n}(\mathbb{C})$.

is the conditional expectation$\dim(\mathbb{C}^{n}\otimes_{E}\mathbb{C}^{m})=\frac{n}{m}$Here, we ramark that $B$ is isomorphic to
can be divided.

$M_{m}(\mathbb{C})$for some $m$ by which

$n$

4. If

$\alpha$is an automorphism of$M_{n}(\mathbb{C})$, then$\dim(\mathbb{C}^{n}\otimes_{\alpha}\mathbb{C}^{n})=1.$

The following shows that $\dim(H\otimes_{\Phi}K)$ can be finite, even if $H$ and $K$
are infinite dimensional.
be the non-commutative Bernoulli shift of $A=$
Example 3.1.6. Let. That is, let $A_{i}=M_{n}(\mathbb{C})$ for all $i=1,2,$ , and for each
, let$A(m)=A_{1}\otimes\cdots\otimes A_{m}\otimes 1\otimes\cdots\subset A$ ,(3.4)$\beta$$\cdots$$\otimes_{i=1}^{\infty}M_{n}(\mathbb{C})$$m\in \mathbb{N}$where 1 is the unit of$M_{n}(\mathbb{C})$

.

17
The

$\beta$is given as the shift as the followings:$\beta(x)=1\otimes x\otimes 1\otimes\cdots$Letfor all $i=1,2,$
with $||\Omega||=1$ , and let$H_{i}=\mathbb{C}^{n}$$\cdots$,

for all, and for each$x\in A(m)$ .

$m,$$m\in \mathbb{N}$(3.5). Fix an vector$H(m)=H_{1}\otimes\cdots\otimes H_{m}\otimes\Omega\otimes\cdots\subset H=\otimes_{i=1}^{\infty}H_{i}$,$\Omega\in \mathbb{C}$(3.6)

is a unital completely positive map from $A\subset B(H)$ to $B(H)$ , andthe restrictionof to $A(m)$ is a unital completely positive map from
$A(m)\subset B(H(m))$ to $B(H(m+1))$ . Apply the above method to, wehave always that

The$\beta$$\beta|_{A(m)}$$\beta$$\beta|_{A(m)}$$\dim(H(m)\bigotimes_{\beta 1_{A(m)}}H(m+1))=n$,for all$m$

As a result, we have that$\dim(H\bigotimes_{\beta}H)=n=\dim(H\bigotimes_{E}H)$,

where $E:Aarrow\beta(A)$ is the conditional expectation.
Now, we call the dimension of$H\otimes_{\Phi}K$

the $mnk$ of$\Phi.$A phenomenon As an example, we show a phenomenon of the above
discussion in the case of a state ofwhich indicates how the dimensionofcoincides with the rank of a state of the usual sense.
$M_{2}(\mathbb{C})$$H\otimes_{\phi}K$$\phi$

Example 3.1.7. Let
be an orthonormal basis ofand let $\{e_{ij};i,j=$$1,2\}$ be a matrix units ofwith. We give a vector representation$\{\xi_{i};i=1,2\}$for eachrelative to thisand a matrix representation for
eachrelative to this $\{e_{ij};i,j=1,2\}$ ;$\{\xi_{1}, \xi_{2}\}$$\mathbb{C}^{2}$$\mathbb{C}^{2}$$e_{ij}\xi_{j}=\xi_{i}$$\xi\in \mathbb{C}^{2}$$x\in M_{2}(\mathbb{C})$(3.7)$\xi_{1}=(\begin{array}{l}10\end{array}), \xi_{2}=(\begin{array}{l}01\end{array})$and$e_{11}=\{\begin{array}{ll}1 00 0\end{array}\},$$e_{12}=\{\begin{array}{ll}0 10 0\end{array}\},$$e_{21}=\{\begin{array}{ll}0 01 0\end{array}\},$$e_{22}=\{\begin{array}{ll}0 00 1\end{array}\}$

(3.8)

18
Assume that$\phi$

is a state given by$\phi(x)=\phi(\{\begin{array}{ll}x_{ll} x_{l2}x_{21} x_{22}\end{array}\})= \frac{1}{2}\sum_{i,j=1}^{2}x_{ij}$

.

(3.9)

Then $\phi(e_{i})=1/2$ for $i=1,2$ and the discussion with respect to$\{e_{ij};i,j=1,2\}$ is as follows:
Let be a fixed unit vector in and let$\{\xi_{1}, \xi_{2}\}$and$\mathbb{C}$$\Omega$.

(3.10)

for all $i,j=1,2$ .

(3.11)$\zeta_{i}=\sqrt{2}\xi_{i}\otimes\Omega\in \mathbb{C}^{2}\otimes_{\phi}\mathbb{C}$Then$<\zeta_{i},$This implies that$\zeta_{j}>\phi=2\phi(e_{ij})=1$$\zeta_{i}\in \mathbb{C}^{2}\otimes_{\phi}\mathbb{C}$,

has norm 1 for $i=1,2$ and$\zeta_{1}=\sqrt{2}\xi_{1}\otimes\Omega=\sqrt{2}\xi_{2}\otimes\Omega=\zeta_{2}$.

(3.12)

Next we choose another family of minimal projections with $\phi(p)\neq 0$ andorthonormal basis of . Let$\mathbb{C}^{2}$$e_{11}’= \frac{1}{2}\{\begin{array}{ll}1 11 1\end{array}\}$,(3.13)of minimal projections withthen $\phi(e_{11}’)=1$ so that the set containing. The corresponding orthonormal basis of
is the one point set
and the corresponding matrix units are as follows:$e_{11}’$$\phi(\cdot)\neq 0$$\{e_{11}’\}$$\mathbb{C}^{2}$(3.14)$\xi_{1}’=\frac{1}{\sqrt{2}}(\begin{array}{l}11\end{array}), \xi_{2}’=\frac{1}{\sqrt{2}}(\begin{array}{l}1-1\end{array})$and$e_{11}’,$$e_{12}’= \frac{1}{2}\{\begin{array}{l}-111-1\end{array}\},$$e_{21}’= \frac{1}{2}\{\begin{array}{l}11-1-1\end{array}\},$$e_{22}’= \frac{1}{2}\{\begin{array}{ll}1 -1-1 1\end{array}\}$(3.15)

Let$\zeta_{i}’=\frac{1}{\sqrt{2}}\xi_{i}’\otimes\Omega\in \mathbb{C}^{2}\otimes_{\phi}\mathbb{C}, (i=1,2)$.

(3.16)

Then$< \zeta_{1}’, \zeta_{1}’>\phi=\frac{1}{2}\phi(e_{11}’)=1$,(3.17)

19
and$<\zeta_{2}’,$(3.18)$\zeta’>=\frac{1}{2}\phi(e_{22}’)=0$

This means that$\mathbb{C}^{2}\otimes_{\phi}\mathbb{C}=\mathbb{C}\zeta_{i}’=\xi_{i}’\otimes \mathbb{C}.$We remark thatthe unitary :$\{\xi_{1}, \xi_{2}\}$and$\{\xi_{1}’, \xi_{2}’\}$are combined as$u\xi_{i}=\xi_{i}’,$$(i=1,2)$ by$u$(3.19)$u= \frac{1}{\sqrt{2}}\{\begin{array}{ll}1 11-1 \end{array}\}$Some relation to the Choi matrix. For a completely posive map of, it is given the so called the Choi matrix
In the case of is a state of, we have the following relation$\Phi$$M_{n}(\mathbb{C})$$C_{\Phi}.$$\Phi$$M_{n}(\mathbb{C})$$\phi$

between the sesquilinear form$<\cdot,$$\cdot>\phi$and the coefficient of$C_{\phi}$:(3.20)$<\xi_{i}\otimes\Omega, \xi_{j}\otimes\Omega>\phi=C_{\phi}(j, i)$wherevector in$\{\xi_{1}, \cdots\xi_{n}\}$3.2

is a orthonormal basis of$\mathbb{C}^{n}$and$\Omega$is 1 considered as the$\mathbb{C}.$Operators $\{v_{j};j=1, \cdots , r\}$As the above section, let : $B(H)arrow B(K)$ be a completely positive linear
map. Here, we assume that $H$ and $K$ be finite dimensional, and let $r=$$\dim(H\otimes_{\Phi}K)$ .
$\Phi$

Definition 3.2.1. Let $\{\xi_{i};i=1, \cdots, n\}$ be an orthonormal basis of $H$ , andlet $\{\zeta_{j};j=1, \cdots, r\}$ be an orthonormal basis of. Define : $Karrow H$$H\otimes_{\Phi}K$$v_{j}$

by$v_{j}( \mu)=\sum_{i=1}^{n}<\xi_{i}\otimes\mu, \zeta_{j}>\Phi\xi_{i}, (\mu\in K)$(3.21)

Proposition 3.2.2. Let
be the tupple obtained by $(3.17)$ .
Then(i) They satisfies the desired following property:$\{v_{1}, v_{2}, \cdots, v_{r}\}$$\Phi(x)=v_{1}^{*}xv_{1}+\cdots+v_{r}^{*}xv_{r}, (x\in B(H))$(ii)$||v_{i}||\leq 1$, for all $i=1,$$\cdots,$ $r$;(3.22)

20(iii)$\{v_{1}, v_{2}, \cdots, v_{r}\}$are linearly independent.

Proposition 3.2.3. The tupple
for unital completely possatisfy the following convenient properties to compute the vonitive map
Neumann type entropy $S(\Phi)$ .
:
then
onto
1. If is a conditional expectation of$j\leq r\}$ are mutually orthogonal minimal projections.
$\{v_{1}, v_{2}, \cdots, v_{r}\}$$\Phi$$E$

2. If$\Phi$$D_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$$1\leq$$\{v_{j}$is a -homomorphism, then $\{v_{j}:1\leq j\leq r\}$ are isometries with$*$

$v_{i}v_{j}^{*}=\delta_{ij}1$.

(3.23)We remark that the following results are well known (see for example
for unital completely positive map[1, 13] so that our tupple
is unique up to a unitary matrix:
Proposition 3.2.4. Let $H$ and $K$ be finite dimensional Hilbert spaces. Assume that $v=\{v_{1}, v_{2}, \cdots, v_{r}\}$ and $w=\{w_{1}, \cdots, w_{r}\}$ are two families of$\{v_{1}, v_{2}, \cdots, v_{r}\}$$)$$\Phi$

opemtors in $B(K, H)$ . Then$v_{1}^{*}xv_{1}+\cdots+v_{r}^{*}xv_{r}=w_{1}^{*}xw_{1}+\cdots+w_{r}^{*}xw_{r}$if and only if there is a unitary matrix,

for all$[u(i,j)]\in M_{r}(\mathbb{C})$$x\in B(H)$such that$v_{i}= \sum_{j=1}^{r}u(i,j)w_{j}, i=1, \cdots, r.$

be the non-commutative Bernoulli shift on $A=$
Example 3.2.5. Letof are as follows. We use the same notation. The -tupple
as Example 3.1.6. Let$\beta$$\otimes_{i=1}^{\infty}M_{n}(\mathbb{C})$$\{v_{j}\}_{j}$$n$$\beta$$W_{m}=\{\alpha=(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{m}), \alpha_{j}\in\{1,2, \cdots, n\}\}$.

(3.24)and let(3.25)$\xi_{\alpha}=\xi_{\alpha 1}\otimes\xi_{\alpha_{2}}\otimes\cdots\otimes\xi_{\alpha_{m}}\otimes\Omega\otimes\cdots\in H(m)$Then$\{\xi_{\alpha};\alpha\in W_{m}\}$is an orthonormal basis of $H(m)$ , and(3.26)$\{\xi_{\alpha}\otimes_{\beta}(\xi_{i}\otimes\xi_{\alpha});i=1,2, \cdots, n\}$is an orthonormal basis of $H(m)\otimes_{\beta}H(m+1)$ . Then our tupple for is givenby(3.27).
, for allm,
are isometries satisfying the Cuntz relation.
We remark that$\beta$$v_{j}(\xi_{i}\otimes\xi_{\beta})=\delta_{ij}\xi_{\beta}$$\{v_{j}^{*}\}_{j}$$\beta\in W_{m}$

21

3.3
Let

Relation to Stinespring’s theorem: $A\subset B(H)arrow B(K)$ be a copletely positive map, where $H$ and $K$
are finite dimensional. Let $\{v_{j}:Karrow H\}_{j=1}^{r}$ be the tupple by (3.17). We
denote by the Hilbert space of -direct sum of $H$ , i.e., $H\oplus\cdots\oplus H$ . Let$\Phi$$L$$r$$V(\xi)=(v_{1}\xi, \cdots, v_{r}\xi)\in L, \xi\in K$(3.28).

(3.29)and let$\pi(x)=\{\begin{array}{llll}x 0 \cdots 0 x 0 0 0 \cdots x\end{array}\}, x\in A$

Then we have the followings:

1. The$\pi$

is a representation of the$C^{*}$-algebra$A$

on the Hilbert space

2. The property thatmeans that the operator$K$by (3.20) is an isometry fromto $L.$$\sum_{i=1}^{r}v_{i}^{*}v_{i}=1_{K}$3. The property$\Phi(x)=\sum_{j=1}^{r}v_{j}^{*}xv_{j}$

$V$

$L.$

definedis written as$\Phi(x)=(v_{1}^{*}, \cdots, v_{r}^{*})\{\begin{array}{llll}x 0 \cdots 0 x 0 0 0 \cdots x\end{array}\}(\begin{array}{l}v_{1}v_{r}\end{array})=V^{*}\pi(x)V$(3.30)

4. These imply that $(\pi, V)$ can be considerefd as the pair obtained by the
Stinespring’s representation.

5. The property thatare linearly independent satisfies that $(\pi, V)$
is the minimal pair in the sense of Arveson [1].
$\{v_{j}\}_{j}$4

Entropy for unital completely positive maps

In this section, we denote an application to the notion of entropy. Let be aunital completely positive map of a -algebra $A\subset B(H)$ to $B\subset B(K)$ and
let $v(\Phi)=\{v_{1}, v_{2}, \cdots, v_{r}\}$ be the tupple for obtained by (3.21). Then the
tuppleis a finite operational partition of unity in $B(K)$ . Hence we canapply the discussion in the section 2.2.2 to unital completely positive map.
$\Phi$$C^{*}$$\Phi$$v(\Phi)$22Case of a state

4.1$\phi$

of$M_{n}(\mathbb{C})$,. Let be a state ofFirst, we consider the cae of a state ofand let $v(\phi)=\{v_{j} : 1\leq j\leq r\}$ be the tuple associated with defined by(3.20).
, that is $\tau(x)=Tr(x)/n$ for
Let be the unique tracial state ofis given by (2.4). Letbe. The density matrix
allsuch thatthe mutually orthogonal minimal projections in
Then we see that
(4.1).
$\phi$$M_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$$\phi$$\phi$$M_{n}(\mathbb{C})$$\tau$$\tau[v(\phi)]$$x\in M_{n}(\mathbb{C})$$\{e_{i}\}_{i=1}^{m}$$\phi(e_{i})\neq 0.$$M_{n}(\mathbb{C})$$\tau[v(\phi)](i,j)=\delta_{ij}\sqrt{\phi(e_{i})\phi(e_{j})}$It is clear thatis a diagonal matrix, and the entropy $S(\tau[v(\phi)])$ inof :the section 2.2.2 is nothing else but the von neumann entropy$\tau[v(\phi)]$$S(\phi)$$S( \tau[v(\phi)])=\sum_{j=1}^{r}\eta(\phi(e_{j}))=-\sum_{j=1}^{r}\phi(e_{j})\log\phi(e_{j})=S(\phi)$

4.2.

$\phi$(4.2)

Entropy for unital completely positive maps

On the basis of the fact in the above section 4.1, we denote the $S(\rho[v(\Phi)])$, and in the casefor a unital completely positive map : $Aarrow B$ by
simply.
of the tracial state we use the same notation
Here, we show the case of which has a unique taracial state and we
number the values of typical examples of von Neumann type entropy
for unital completely positive maps
$S(\rho(\Phi))$$\Phi$$S(\Phi)$$\rho$$A$$\tau$$S(\Phi)$$\Phi.$

1. If$\phi$

is a state of$M_{n}(\mathbb{C})$, then$S( \phi)=\sum_{j=1}^{r}\eta(\lambda_{j})$where$\{\lambda_{j}\}$are eigenvalues of(4.3)$\phi.$

2. If $E$ is the conditional expectation ofsubalgebra then$M_{n}(\mathbb{C})$to a maximal abelian$B$$S(E)=\log n$ .

Compare this fact to that $H(E)=ht(E)=0.$

(4.4)

23
3. If

is the conditional expectation of$E$$M_{n}(\mathbb{C})$$S(E)= \log\frac{n}{k}$to a subfactor

$B$

then

.

(4.5)

Here we remark that a subfactor ofis isomorphic tosome , and that is divisible by . Compare tbis fact to that $H(E)=$
$B$

$k$

$n$

$M_{n}(\mathbb{C})$$M_{k}(\mathbb{C})$$k$$ht(E)=0.$

4. If$\alpha$is an automorphism of$M_{n}(\mathbb{C}))$, then(4.6)$S(\alpha)=0$

and this coinsides with the fact that $H(\alpha)=ht(\alpha)=0.$

5. If$\beta$is the non-commutative Bernoulli shift on$\otimes_{i=1}^{\infty}M_{n}(\mathbb{C})$, then(4.7)$S(\beta)=\log n$

and this coinsides with the fact that $H(\beta)=ht(\beta)=\log n.$

6. If$\Phi_{n}$is the Cuntz’s canonical shift on$O_{n}$, then(4.8)$S_{\Psi}(\Phi_{n})=\log n$

and this coinsides with the fact that $h_{\Psi}(\Phi_{n})=ht(\Phi_{n})=\log n$ . Hereis the state ofwhich is given by the left inverse of$\Psi$$\Phi.$$O_{n}$

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[4] N. Brown Topological entropy in exact(1999), 347-367.

$C^{*}-algebra\mathcal{S}$, Math. Ann., 314

24[5] M. Choda, Entropy for unital completely positive finite mnk maps,
Preprint.
[6] M. Choda, Entropy of Cuntz’s canonical endomorphism,
Math.,190(1999), 235-245.

Pacific J.

[7] M. Choda, $AC^{*}$ -dynamical entropy and applications to canonical endomorphisms, J. Funct. Anal., 173 (2000), 453-480.

, Entropy of[8] A. Connes and E. St
Math., 134 (1975), 289-306.
$\emptyset rmer$$II_{1}$von Neumann algebms, Acta

[9] A. Connes, H.Narnhofer, W. Thrring Dynamical entropy of -algebras
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$C^{*}$[10] G. Lindblad, Entropy, information and quantum measurements, Commun. Math. Phys., 33, pp.111-119, 1973.

, Dynamical entropy in opemtor algebms,[11] S. Neshveyev and E.
Springer-Verlag, Berlin(2006).
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[13] D. Petz, Quantum Information Theory and Quantum Statistics,
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marie@cc.osaka-kyoiku.ac.jp