数理解析研究所講究録
第 1819 巻 2012 年 9-21

9Entropy via partitions of unity
大阪教育大学1長田

まりゑ (Marie Choda)

Introduction

In this paper, we take up the notions of “finite partion of unity“. The firstis a finite partion of 1 in real numbers. The second is it in the ergodic
theory. The third is those in operator algebras, two kinds of”finite partionof unity” One is finite partions of unity via positive operator. The other isfinite operationsl partion of unity.
We state relations between “mutual orthogonality for two subalgebras
and $B$ ” of thecomplex matrix algebra and the entropy for “finite
partion of unity” induced from unitary operators which are related to thepair $\{A, B\}$ . Generally speaking, isomorphic two subalgebras are mutuallyorthogonal if and only if the related entropy takes the maximum value, andthe value is the logarithm of the dimension of the subalgebra.
This relation reminde us that the area of a rhombus takes the maximal
value if and only if two sides intersect at right angles, and the maximal value
is the square of the length of the side.
$A$$n\cross n$

Furthermore, by applying the notion of finite opemtionsl partion of unity
to unital completely positive (called the UCP for short) maps, we extend
the notion of von Neumann entropy for states to that for UCP maps, andshow computations of entropy for UCP maps (in special Cuntz’s canonicalendomorphisms).
In this paper we denote by
the algebra ofcomplex matrices,and bythe standard trace, that is, the sum of all diagonal components.
A matrixis called a density matrix if $D$ is a positive operator
with $Tr_{n}(D)=1$ (cf. [11] [12]).
$M_{n}(\mathbb{C})$$R_{n}$$D\in M_{n}(\mathbb{C})$$n\cross n$

10
The notation
tion defined by$\eta$

is called the entropy function in usual, and it is the func$\eta(t)=\{\begin{array}{ll}-t\log t, (0<t\leq 1)0, t=0\end{array}$2

Several kinds of”finite partitions of unity”

Here, we discuss on several kinds of notions which we may call a finite partition of unity, and we apply the entropy function to those finite partition$\eta$of unity.

2.1

Entropy for finite 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
finite partition of 1. Let$\lambda_{i}\geq 0$

with$\sum_{i}\lambda_{i}=1$. We say that the set$H(\lambda)=\eta(\lambda_{1})+\cdots+\eta(\lambda_{n})$and we say that

2.2

$H(\lambda)$$\lambda$

is a

(2.1)

,

is the entropy for the finite partiton$\lambda$

of 1.

Entropy for finite measurable partitions

Let us remember the definition of the entropy for finite measurable partitions
is discussed under the following
in the ergodic theory. The notation
setting: Let $(X, \mu)$ be a Lebesque space. Let$H(\lambda)$$\xi=\{X_{1}, \cdots, X_{n}\}$be a finite measurable partition of $X$ . Then
1 as follows:$\xi$

induces the finite partition of$\lambda_{\mu}(\xi)=\{\mu(X_{1}), \cdots, \mu(X_{n})\}$.

The entropy $H(\xi)$ for the finite partition of $X$ is nothing else butthat is,$H(\xi)=\eta(\mu(X_{1}))+\cdots+\eta(\mu(X_{n}))$ .
$\xi$

(2.2)
$H(\lambda_{\mu}(\xi))$

,

11
2.3

Finite partitions via projections

We can discuss the notion of finite measurable partitions in the abelian algebra. Let
be the characteristic function of. Then $\chi=$is a partition of unity in:$L^{\infty}(X, \mu)$$X_{i}$$\chi_{i}$$\{\chi_{1}, \cdots, \chi_{n}\}$$L^{\infty}(X, \mu)$$\chi_{1}+\cdots+\chi_{n}=1_{L^{\infty}(X,\mu)}$Letbe the multiplicative representation of
on the Hilbert space$H=L^{2}(X, \mu)$ . We denote by $M$ the von Neumann algebra,
and by the projection
for $i=1,$. Then $p_{1}+\cdots+p_{n}=1_{M}$ . Let$L^{\infty}(X, \mu)$$\pi$$\pi(L^{\infty}(X, \mu))$$\pi(\chi_{i})$$p_{i}$$\cdots,$ $n$$p=\{p_{1}, \cdots,p_{n}\}.$

Thus the set is a finite partition via projections of unity 1 in $M$ , and its
entropywith respect to the measure is considered as$p$$H_{\mu}(p)$$\mu$$H_{\mu}(p)=\eta(\mu(p_{1}))+\cdots+\eta(\mu(p_{n}))$

2.4

.

(2.3)

Finite partitions via positive operators

extended the notion of “finite partition via projectionsof unity” in the abelian von Neumann algebra $\pi(L^{\infty}(X, \mu))$ to the case of afinite von Neumann algebra $M.$
Let be a tracial state of $M.$ $A$ set $x=\{x_{1}, \cdots, x_{n}\}\subset M$ is said tobe a finite partition of unity in $M$ if each $x_{i}\in M$ is a positive operator for$i=1,$
and $\sum_{i-1}^{n}x_{i}=1_{M}$ . The entropy for (finite partition of unity
in $M)$ with respect to is given by$Connes-St\emptyset mer([8])$$\tau$$\cdots,$ $n$$x$$\tau$$H_{\tau}(x)=\eta(\tau(x_{1}))+\cdots+\eta(\tau(x_{n}))$

2.5

(2.4)

.

Finite operational partition of unity

The terminology, a finite opemtional partition of unity, was first given by
Lindblad ([10]) and after then it is used by Alicki-Fannes([l]).
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}.$12
Let be a state of , and let $x=\{x_{1}, \ldots, x_{k}\}$ be an operational partitionthematrix whose $\{i,j\}$ componentof unity in . We denote byis given by$A$$\varphi$$A$$\rho_{\varphi}[x]$$k\cross k$$\rho_{\varphi}[x](i,j)$$\rho_{\varphi}[x](i,j)=\varphi(x_{j}^{*}x_{i}) , (i,j=1, \cdots, k)$.

. We callanis a positive operator inThen
the density matrix associate with andbe the eigenvalues of the density matrix
Letis a positive op. Then
is a finite partition of 1 because. Hence we have the entropy
erator inandin (3.1).
Letbe the von Neumann entropy ([11, 12]) for the density op, that is,. Thenis nothing else but
erator$M_{k}(\mathbb{C})$$\rho_{\varphi}[x]$$x$$d^{r}R_{k}(\rho_{\varphi}[x])=1$$\rho_{\varphi}[x]$$\varphi.$$\lambda(\rho_{\varphi}[x])=\{\lambda_{1}, \lambda_{2}, \cdots, \lambda_{k}\}$$\rho_{\varphi}[x]$$\lambda(\rho_{\varphi}[x])$$\rho_{\varphi}[x]$$Tr_{k}(\rho_{\varphi}[x])=1$$M_{k}(\mathbb{C})$$H(\lambda(\rho_{\varphi}[x]))$$S(\rho_{\varphi}[x])$$H(\lambda(\rho_{\varphi}[x]))$$S(\rho_{\varphi}[x])$$\rho_{\varphi}[x]$$S( \rho_{\varphi}[x])=Tr_{k}(\eta(\rho[x]))=H(\lambda(\rho_{\varphi}[x]))=\sum_{i}\eta(\lambda_{i})$

2.6

(2.5)

.

Unitary matrix and patition of 1

Let $u=((u(i,j))_{ij}$ be an$n\cross n$

unitary matrix. Then$\sum_{i=1}^{n}|u(i,j)|^{2}=1=\sum_{j=1}^{n}|u(i,j)|^{2}$,for all $i,j.$Let$\lambda(u)=\{\frac{|u(1,1)|^{2}}{n}, \frac{|u(1,2)|^{2}}{n}, \cdots, \frac{|u(n,n)|^{2}}{n}\}.$

is a finite partition of 1, and we callthe finite partition of 1
Thensatisfies the following equality
induced from . The entropy $H(\lambda(u))$ ofby (3.1):$\lambda(u)$$\lambda(u)$$\lambda(u)$

$u$

.

(2.6)

$H( \lambda(u))=\frac{1}{n}\sum_{i,j}\eta(|u(i,j)|^{2})+\log n$

2.7

Entropy for unistochastic matrices

A matrix$b\in M_{n}(\mathbb{C})$$b(i,j)\geq 0$ ,is said to be bistochastic if
and$\sum_{i=1}^{n}b(i,j)=\sum_{j=1}^{n}b(i,j)=1,$$(\forall i,j)$.

13
Zyczkowski-Ku\’{s}-Slomczy\’{n}ski-Sommers defined in [17] the entropy $H(b)$for a bistochastic matrix by$b$$H(b)= \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\eta(b(i,j))$.

We pick up a bist chastic matrix which is called a unistochastic matrix. Let
us consider a unitary matrix, and let$0$$u\in M_{n}(\mathbb{C})$$(b(u))(i,j)=|u(i,j)|^{2} \forall i,j.$

The $b(u)$ is a bistochastic matrix which is called unistochastic matrix inducedfrom
We have the following relation between two kinds of entropy $H(\lambda(u))$ and$H(b(u))$ :$H(\lambda(u))=H(b(u))+\log n$ .
(2.7)
$u.$

where $H(\lambda(u))$ is the entropy for the finite partitionof 1 induced from
$H(b(u))$andis the entropy for the unistochastic matrix induced from$\lambda(u)$

$u$

$u.$

We show in the next section the meaning of the value $H(b(u))$ from theview point of the theory of operator algebras.

3

Characterization of Orthogonality

In this section, we give a characterization for the mutual orthogonality of
two subalgebras by the maximal value of related entropy to the pair of subalgebras.

3.1

Mutually orthogonal subalgebras

Let $M$ be a finite von Neumann algebra and let be a fixed normal faithful
tracial state. If $M=M_{n}(\mathbb{C})$ , then $\tau(x)=E_{n}(x)/n$ . The inner product$<x,$ $y>$ for $x,$ $y\in M$ is given by $<x,$ $y>=\tau(y^{*}x)$ . Let
and $B$ be von
Neumann subalgebras of $M$ such that $1_{M}\in A$ and $1_{M}\in B$ . Then andare said to be mutually orthogonal by Popa([15]) if$\tau$$A$$A$$\tau(a)=0=\tau(b)\Rightarrow\tau(ab)=0$

$B$

(3.1)

14
for all $a\in A$ and $b\in B$ . This means that$(A\ominus \mathbb{C}1_{A})\perp(B\ominus \mathbb{C}1_{B})$with respect to the above inner product.
We remark that in some papers (for example [14]) mutual orthogonality
for subalgebras is called complementarity.

Conditional relative entropy $h(A|B)$

3.2

Let $M$ be a finite von Neumann algebra, and let and be two von Neumann subalgebras of $M$ . Let be a tracial state of $M$ . Then there exists the: $Marrow B$ conditiones by: $Marrow A$ andconditional expectations
due to Connes andWe modify the relative entropy $H(A|B)$ of andas follows:
St
$B$

$A$

$\tau$$E_{B}$$E_{A}$$A$$\tau.$$B$$\emptyset mer([8])$Definition 3.2. ([2]). Let(3.2)$h(A|B)= \sup_{(x.)}\sum_{i}(\tau\eta E_{B}(x_{i})-\tau\eta x_{i})$is a finite partition of unity in
wherepositive operators contained in and$(x_{i})$

$A$

$A$

, that is,$(x_{i})$is a finite set of$1_{A}= \sum_{i}x_{i}.$We remark that in the definition of $H(A|B)$ , Connes and St
as a finite partition of unity in $M.$
take$\emptyset mer([8])$$(x_{i})$

3.3

Mutual orthogonality is maximality of entropy

We have the following relation between the entroppy
and the entroppy $H(b(u))$ :Theorem 3.3. ([2]). Let, and letin
$M_{n}(\mathbb{C})$$h(D_{n}(\mathbb{C})|uD_{n}(\mathbb{C})u^{*})$

be the algebm of the diagonal matrices
be a unitary matrix. Then

$D=D_{n}(\mathbb{C})$$u\in M_{n}(\mathbb{C})$$h(D|uDu^{*})= \frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\eta(|u(i,j)|^{2})=H(b(u))$.

so that$\{h(D|uDu^{*})$ : $u\in M_{n}(\mathbb{C})$unitary $\}=[0, \log n].$

(3.3)

15

Remark 3.3. We remark that the above relations do not hold in the caseof Connes and St
relative entropy $H(A|B)$ . See for example [14].
$\emptyset mer$

Corollary 3.3. ([2]). The following conditions are equivalent:

1. $D$ and $uDu^{*}$ are mutually orthogonal,
2. $h(D|uDu^{*})=\log n,$
3. $|u(i,j)|=1/\sqrt{n}$Let$\forall i,j.$andbe maximal abelian subalgebras (called MASA for short)in. Then there exists a unitary $u(A, B)\in M_{n}(\mathbb{C})$ andis thetypical MASA of
so that we have the following:
$A$

$B$

$M_{n}(\mathbb{C})$$D_{n}(\mathbb{C})$$M_{n}(\mathbb{C})$Conclusion 3.3. ([2]). Letandbe maximal abelian subalgebras in. Then following conditions are equivalent:$A_{0}$$B_{0}$$M_{n}(\mathbb{C})$1.

$A_{0}$

and$B_{0}$are mutually orthogonal,

2. $h(A_{0}|B_{0})= \max\{h(A|B)$ : $A,$

$B\subset M_{n}(\mathbb{C})$3. $H(b(A_{0}, B_{0}))= \max\{H(b(A, B))$ : $A,$

, MASA$B\subset M_{n}(\mathbb{C})$$\},$, MASA

$\}$

4. $h(A_{0}|B_{0})=H(b(A_{0}, B_{0}))=\log n$
Extended version tofactor. In the paper [3], the above equivalent
three relations {1, 2, 4} were extended in a connection to Jones index theory, to some kinds of subfactors infactors, where the notion of mutual“
orthogonality is replaced by the commuting square condition”$II_{1}$$II_{1}$

3.4

Unitary and finite operational partition

Let be a finite von Neumann algebra (the most simple case $L=M_{k}(\mathbb{C})$for some integer ), and let
be a fixed normal faithful tracial state of $L.$$M$
Let
be the tensor product, and letbe the tracial state of$M$ which is given as thetensor product. Let $\{e_{ij};i,j=1, \cdots, n\}$
$L$

$k$

$\tau_{L}$$M_{n}(\mathbb{C})\otimes L$$R_{n}/n\otimes\tau_{L}$$\tau_{M}$16be a system of a matrix units ofunique form

$M_{n}(\mathbb{C})$. Then each $u\in M$ is written as the$u= \sum_{i,j=1}^{n}e_{ij}\otimes u_{ij}, (u_{ij}\in L)$

and the$u$,

is unitary if and only if$\sum_{j=1}^{n}u_{ij}u_{kj}^{*}=\delta_{ik}1_{L}, \sum_{i=1}^{n}u_{ij}^{*}u_{ik}=\delta_{jk}1_{L}.$

Assume that$u\in M_{n}(\mathbb{C})\otimes L$

is unitary. Let$U= \{\frac{1}{\sqrt{n}}u_{11)}\frac{1}{\sqrt{n}}u_{12}, \cdots, \frac{1}{\sqrt{n}}u_{nn}\}.$

Then is a finite operational partition of unity in the von Neumann algebraof size$U$$n^{2}.$$L$

Definition 3.4. ([4]). Let$u$be a uunitary in$M_{n}(\mathbb{C})\otimes L$. By putting$\rho[U](ij, kl)=\tau(u_{kl}^{*}u_{ij}) , i,j, k, l=1, \cdots, n^{2}$

(3.4)

,

which associates with the unitary . Here,matrix$\rho[U](ij, kl)$ means the double indexed $(ij, kl)$ component of the matrix

we define a$n^{2}\cross n^{2}$$\rho[U]$$u$$\rho[U].$which satisfies
is a positive operator in
It is obvious thatis a density matrix. Hence we have the vonthat $R_{n^{2}}(\rho[U])=1$ , so that
Neumann entropy $S(\rho[U])$ of the density operator$M_{n^{2}}(\mathbb{C})$$\rho[U]$$\rho[U]$$\rho[U]..$

Remark 3.4. We remark that the following holds in general:$0\leq S(\rho[U])\leq 2\log n.$

be a finite von Neumann algebm and letnormalized tmce of L. We let $M=M_{n}(\mathbb{C})\otimes L$ and $\tau=$ Tr
and that is a unitary opemtor in $M.$
that

Theorem 3.4. ([4]). Let$L$$/n\otimes\tau_{L}.$$N=M_{n}(\mathbb{C})\otimes 1_{L}$Then the following$u$$\omega$nditions are equivalent:

1. $N$ and $uNu^{*}$ are mutually orthogonal,$\cdot$2.

$n^{2}\rho[U]$is the$n^{2}\cross n^{2}$

identity matrix,$\cdot$$\tau_{L}$be a$A_{\mathcal{S}sume}$

17
3. $S(\rho[U])=2\log n=$ log dim $N.$
Here is the finite opemtional partition of unity induced by
Below, we show some operator algebraic role of $S(\rho[U])$ .
$U$

3.5
Let
and

$u.$

Subfactors of matrix algebras

and $B$ be subalgebras ofand assume that both subalgebras$B$
are isomorphic to. Then $k=nm$ , and we can assume thatand
. By the assumption, there
exists a unitary matrix
such that $B=uAu^{*}$We denote by $u(A, B)$ this unitary. Now, by letting be, we apply
the discussion in the section 4.4 to $u(A, B)$ . Then we have the density matrix$\rho[U(A, B)]$ induced by $u(A, B)$ . We denote by
the set of subalgebras ofwhich are isomorphic to
.
$A$

$A$

$M_{k}(\mathbb{C})$$M_{n}(\mathbb{C})$$M_{k}(\mathbb{C})=M_{n}(\mathbb{C})\otimes M_{m}(\mathbb{C})$$A=M_{n}(\mathbb{C})\otimes \mathbb{C}1$$u\in M_{k}(\mathbb{C})=M_{n}(\mathbb{C})\otimes M_{m}(\mathbb{C})$$L$$M_{m}(\mathbb{C})$$\mathfrak{F}_{k,n}$$M_{k}(\mathbb{C})$$M_{n}(\mathbb{C})$Conclusion 3.5. ([5]). Letandbe subalgebras which are containedin. Then following conditions are equivalent:$A_{0}$$B_{0}$$\mathfrak{F}_{k,n}$1.

$A_{0}$

and$B_{0}$are mutually orthogonal,

2. $S( \rho[U(A_{0}, B_{0})])=\max\{S(\rho[U(A, B)]) : A, B\in \mathfrak{F}_{k,n}\},$The maximal value is $2\log n$ which is the logarithm of the dimension of thesubalgebra$A_{0}.$

4
Let

Entropy for UCP maps

andbe unital-algebras and let : $Aarrow B$ be a linear map. If:
is positive for all positive integer $n,$then is said to be completely positive. If $\Phi(1_{A})=1_{B}$ , then is said tobe unital. We call a unital completely positive map UCP map, and develope
our entropy for UCP maps via finite operational partition of unity. ([5])$A$$\Phi\otimes id_{n}$$\Phi$$B$$C^{*}$$\Phi$$A\otimes M_{n}(\mathbb{C})arrow B\otimes M_{n}(\mathbb{C})$$\Phi$

18

Finite operational partition and UCP map

4.1-algebra, and let $v=\{v_{1}, \cdots, v_{n}\}\subset A$ be a finite
be a unitalLet
operational partitions of unity. Let$A$$C^{*}$(4.1)$\Phi(x)=\sum_{i=1}^{n}v_{i}^{*}xv_{i}, x\in A$

Since is a finite operational partitions of unity $( i.e. \sum_{i=1}^{n}v_{i}^{*}v_{i}=1)$ , itfollows that is a UCP map of . We call this the $UCP$ map associated
$v$

$A$

$\Phi$$\Phi$

with the finite opemtional partitions of unity . In the theory of -algebras,
we have an interesting example of finite operational partitions of unity.
$C^{*}$$v$

Cuntz relation and finite operational partitions

4.2
Let

$\{S_{1}, S_{2}, \cdots, S_{n}\}$

be

$n$

isometries on some Hilbert space such that:$S_{1}S_{1}^{*}+S_{2}S_{2}^{*}+\cdots S_{n}S_{n}^{*}=1$

(4.2)

.

. Then theThis (5.2) is called the Cuntz relation. Let
Cuntz relation implies that the set is a finite operational partitions of unity.
-algebra generated by the set
be the universal
Let-algebras called the$(n\geq 2)$ is a unital, simple, purely infinite
Then
Cuntz algebra.
$s=\{S_{1}^{*}, S_{2}^{*}, \cdots, S_{n}^{*}\}$$s$$\{S_{1}, S_{2}, \cdots, S_{n}\}.$$C^{*}$$0_{n}$$C^{*}$$O_{n},$

Entropy for UCP via finite operational partition

4.3-algebra, and let $v=\{v_{1}, \cdots, v_{n}\}\subset A$ be a finite
be a unitalLet
operational partitions of unity. If is a state of , then we have the density
forand also we have the von Neumann entropy
matrixas in the section 3.5.
$A$$C^{*}$$A$$\varphi$$S(\rho_{\varphi}[v])$$\rho_{\varphi}[v]$$\rho_{\varphi}[v]$be the UCP map associated with the finite
Definition 4.3. ([5]). Let
operational partitions of unity , and let$\Phi$$v$(4.3)$S(\Phi)=S(\rho_{\varphi}[v])$We call the $S(\Phi)$ the entropy for$\Phi$

associated with$v.$

Remark 4.3. ([5]). The following two facts give us the ground that wethe entropy forconsider$S(\Phi)$$\Phi.$

19
1. Let: $B(K)arrow B(H)$ be a UCP map, where $H$ and $K$ are finite
dimensional. Then there exists a unique (up to unitary matrix) $v=$
, which is a finite operational partition of unity associatedwith . This implies that the value
does not depend on the$S(\Phi)$choice of . Hence the value
is determined by only$\Phi$$\{v_{1}, \cdots, v_{n}\}$$\Phi$$S(\rho_{\varphi}[v])$$v$$\Phi.$

2. The most typical UCP map
way: Let be a state of. In this case,entropyfor$\varphi$$x\in M_{n}(\mathbb{C})$is induced from a state by the following, and letfor all
is nothing else but the von Neumann$\phi$$M_{n}(\mathbb{C})$$\Phi(x)=\varphi(x)1_{M_{n}(\mathbb{C})}$$S(\rho_{\varphi}[v])$$S(\varphi)$

4.4$\Phi$$\varphi.$Cuntz canonical shift

The Cuntz canonical endomorphism$\Phi_{n}$$\Phi_{n}$is defined by$\Phi_{n}(x)=\sum_{i=1}^{n}S_{i}xS_{i}^{*}, x\in O_{n}.$That is,is the UCP map associated with thestandard left inverseofis defined by$\Phi_{n}$$\hat{\Phi}_{n}$$s$in the section 5.2. The$\Phi_{n}$$\hat{\Phi}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}S_{i}^{*}xS_{i}$and so-called canonical state. Then we have$\varphi$of$O_{n}$,

for all$x\in O_{n},$is induced from the standard left inverse$\hat{\Phi}_{n}$Theorem 4.4. ([5]).
$S(\Phi_{n})=\log n$

.

(4.4)

Remark 4.4. We compare this value with those which we showed before:
1. The Voiclescu topological entropy$ht(\Phi_{n})$2. The Connes-Narnhofer-Thirring entropy$\log n([7])$.

is $\log n$ ([6]).
$h_{\varphi}(\Phi_{n})$with respect to$\varphi$is20

References[1] R. Alicki and M. Fannes, Defining Quantum Dynamical entropy, Lett.
Math. Phys., 32 (1994)[2] M. Choda, Relative entropy for maximal abelian subalgebms of matrices and the entropy of unistochastic matrices, Internatinal J. Math., 19(2008), 767-776.
[3] M. Choda, Conjugate Pairs of Subfactors and Entropy for Automorphisms, to appear in Internatinal J. Math., 22(2011).
[4] M. Choda, von Neumann entropy and relative position between subalgebms, arXiv:1007.1037.
[5] M. Choda, Entropy for unital completely$p_{0\mathcal{S}}$itive maps, Preparation[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 algebras, Acta

[9] A. Connes, H.Narnhofer, W. Thrring Dynamical entropy of -algebms
and von Neumann algebras, Comm. Math.. Phys., 112(1987), 691-719.
$C^{*}$[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. St
Springer-Verlag, Berlin(2006).
$\emptyset rmer$[12] M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, 2004.

[13] D. Petz, Algebmic complementarity in quantum theory, J. Math. Phys.
51 (2010),[14] D. Petz, A. Sz\’ant\’o and M. Weiner, Complementari ty and the algebmic
structure of 4-level quantum systems, Infin. Dimens. Anal. Quantum
Probab. Relat. Top., 12 (2009), 99-116.

21[15] S. Popa, Orthogonal pairs o$f^{*}-subalgebm\mathcal{S}$ in finite von Neumann algebms, J. Operator Theory, 9 (1983), 253-268.
[16] D. Voiculescu, Dynamical approximation entropies and topological entropy in opemtor algebms, Commun. Math. Phys., 170 (1995), 249-281.

[17] K. Zyczkowski, M. Ku\’{s}, W. Slomczy\’{n}ski and H.-J. Sommers, Random
unistochastic matrices, J. Phys., A36 (2003), 3425-3450.