A Construction of Dynamical Entropy on CAR Algebras

TL;DR

This paper develops a new definition of dynamical entropy specifically for CAR algebras, which are used to describe quantum spin systems, extending the AOW entropy framework from von Neumann algebras.

Contribution

It formulates a novel dynamical entropy on CAR algebras based on the AOW entropy construction, enabling analysis of quantum spin systems.

Findings

Entropy computed for 2x2 matrix algebra case

Provides a tool for quantifying quantum information and uncertainty in spin systems

Links dynamical entropy to quantum spin system models

Abstract

The dynamical entropy on von Neumann algebras defined by Accardi, Ohya and Watanabe (AOW entropy) is a natural noncommutative extension of the classical dynamical entropy. On the other hand, quantum spin lattice systems currently used in quantum computing and communication processes are mathematically described by $C^*$-algebras called CAR algebras. Therefore, in order to obtain the average amount of quantum information and to calculate the uncertainty of the dynamics of quantum spin systems, it is necessary to define dynamical entropy on CAR algebras. In this paper, we formulate dynamical entropy on CAR algebras based on the construction of the AOW entropy. Moreover, we compute the introduced entropy for a $2 \times 2$ matrix algebra case which has relation to the quantum spin system.

Equations

$$\sum _j{\gamma }_j=1_{\mathcal{A}},{\gamma }_i{\gamma }_j={\delta }_{ij}{\gamma }_j,$$

$$E_e(\sum _{i,j}{e}_{ij}\otimes A_{ij})=\sum _i{A}_{ii},$$

$${\mathcal{E}}_{\gamma ,\theta }(a\otimes b):=\theta \circ E_e(p_{\gamma ,e}^{\ast }(a\otimes b)p_{\gamma ,e}),a\otimes b\in M_d\otimes \mathcal{A},$$

$$\psi :=\{\phi ,{\mathcal{E}}_{\gamma ,\theta }\}\in \mathfrak{S}({\otimes }^{\mathbb{N}}{M}_d),$$

$$\psi (j_1(a_1)j_2(a_2)\cdots j_n(a_n))=\phi ({\mathcal{E}}_{\gamma ,\theta }(a_1\otimes {\mathcal{E}}_{\gamma ,\theta }(a_2\otimes \cdots \otimes {\mathcal{E}}_{\gamma ,\theta }(a_n\otimes 1_{\mathcal{A}})\cdots ))),n\in \mathbb{N},a_i\in M_d,$$

$$j_k(a):=1\otimes \cdots \otimes 1\otimes \stackrel{k-th}{a}\otimes 1\otimes \cdots$$

$$\phi (A)=\mathrm{Tr}\rho A,\forall A\in \mathcal{A}.$$

$$\psi (j_1(a_1)j_2(a_2)\cdots j_n(a_n))=\phi ({\mathcal{E}}_{\gamma ,\theta }(a_1\otimes {\mathcal{E}}_{\gamma ,\theta }(a_2\otimes \cdots \otimes {\mathcal{E}}_{\gamma ,\theta }(a_n\otimes 1_{\mathcal{A}}\cdots ))))$$

$$=$$

$${\rho }_{[0,n]}$$

$${\rho }_n$$

$${\Gamma }_{i_n\cdots i_1}:={\theta }^{n-1}({\gamma }_{i_n})\cdots \theta ({\gamma }_{i_2}){\gamma }_{i_1}$$

$$P_{i_1,\cdots i_n}:={\mathrm{Tr}}_{\mathcal{A}}{\Gamma }_{i_n\cdots i_1}\rho {\Gamma }_{i_n\cdots i_1}^{\ast }={\mathrm{Tr}}_{\mathcal{A}}|{\Gamma }_{i_n\cdots i_1}{|}^2\rho ,$$

$$S_n(\gamma ,\theta ):=-\mathrm{Tr}{\rho }_n\log {\rho }_n=-\sum _{i_1,\cdots ,i_n}{P}_{i_1,\cdots i_n}\log P_{i_1,\cdots i_n}.$$

$$\tilde{S}(\gamma ;\theta )$$

$${\mathcal{H}}_N:=\underset{N}{⨂}\mathcal{H}.$$

$${\mathcal{P}}_N:=\{\sigma ;\{1,\cdots ,N\}\to \{1,\cdots ,N\},\sigma \mathrm{is}\mathrm{injective}\}.$$

$$P_{\sigma }(x_1\otimes \cdots \otimes x_N)=x_{\sigma (1)}\otimes \cdots \otimes x_{\sigma (N)},x_j\in \mathcal{H},j\in \{1,\cdots ,N\}$$

$$P_{\sigma }{P}_{\tau }=P_{\tau \sigma },\sigma ,\tau \in {\mathcal{P}}_N.$$

$$\stackrel{N}{\bigwedge }(\mathcal{H}):=\{x\in {\mathcal{H}}_N;x\mathrm{is}\mathrm{anti}-\mathrm{symmetric}\}.$$

$$x_1\bigwedge x_2\bigwedge \cdots \bigwedge x_N:=\frac{1}{\sqrt{N!}}\sum _{\sigma \in {\mathcal{P}}_N}\mathrm{sgn}(\sigma )x_{\sigma (1)}\otimes \cdots \otimes x_{\sigma (N)}$$

$$x_1\bigwedge \cdots \bigwedge x_i\bigwedge \cdots \bigwedge x_j\bigwedge \cdots \bigwedge x_N=-x_1\bigwedge \cdots \bigwedge x_j\bigwedge \cdots \bigwedge x_i\bigwedge \cdots \bigwedge x_N$$

$$x_1\bigwedge x_2\bigwedge \cdots \bigwedge x_N=0.$$

$$\mathcal{F}(\mathcal{H}):=\underset{N=0}{\overset{\infty }{⨁}}{\mathcal{H}}_N$$

$$=\left\{x=\{x^{(N)}{\}}_{N=0}^{\infty };x^{(N)}\in {\mathcal{H}}_N,N\ge 0,\sum _{N=0}^{\infty }\Vert x^{(N)}{\Vert }_{{\mathcal{H}}_N}^2<\infty \right\}$$

$$⟨x,y⟩:=\sum _{N=0}^{\infty }⟨x^{(N)},y^{(N)}{⟩}_{{\mathcal{H}}_N}.$$

$${\mathcal{F}}_f(\mathcal{H}):=\underset{N=0}{\overset{\infty }{⨁}}\stackrel{N}{\bigwedge }(\mathcal{H})$$

$$=\left\{x=\{x^{(N)}{\}}_{N=0}^{\infty };x^{(N)}\in \stackrel{N}{\bigwedge }(\mathcal{H}),N\ge 0,\sum _{N=0}^{\infty }\Vert x^{(N)}{\Vert }^2<\infty \right\}$$

$${\mathcal{F}}_{f,0}(\mathcal{H}):=\{x\in {\mathcal{F}}_f(\mathcal{H});\mathrm{If}\mathrm{there}\mathrm{exists}{N}_0\mathrm{such}\mathrm{that}N\ge N_0,x^{(N)}=0\}$$

$${\mathcal{F}}_{f,fin}(\mathcal{D}):=\{x\in {\mathcal{F}}_f(\mathcal{H});x^{(N)}\in A_N({\hat{\otimes }}^N\mathcal{D}),N\ge 0,$$

$$\mathrm{If}\mathrm{there}\mathrm{exists}{N}_0\mathrm{such}\mathrm{that}N\ge N_0,x^{(N)}=0\}$$

Full text

1. 1 Introduction

2. 2 AOW entropy

3. 3 CAR Algebras

4. 3.1 Fermion Fock spaces

5. 3.2 CAR Algebras

6. 4 Markov States and Chains on CAR Algebras

7. 5 A Construction of Dynamical Entropy on CAR Algebras

8. 6 Model Computation

9. 7 Conclusion

1 Introduction

The classical dynamical entropy for a measure-preserving invertible transformation of a Lebesgue space has important roles in both pure mathematics and classical information theory. In mathematical side, it shows that two classical dynamical systems are isomorphic. In classical information theory, the dynamical entropy gives the average of infomation amount of an information source [9], [11].

 There are several ways to define the dynamical entropy on noncommutative algebras [5], [13], [15]. In [6], using quantum Markov chains on matrix algebras [1], [7], Accardi, Ohya and Watanabe formulated the dynamical entropy on von Neumann algebras. The entropy is called AOW entropy and is a natural noncommutative (or quantum) extension of the classical dynamical entropy. Moreover, due to the simplicity of its formulation, one can easily obtain the average amount of quantum information for quantum dynamical systems using the AOW entropy.

 Incidentally, in recent years, quantum spin lattice systems are used in quantum computing and quantum communication processes. Since the observables of spin systems are expressed by the elements of CAR algebras, it is necessary to define dynamical entropy on this algebra to discuss the dynamics and the average information of spin systems strictly.

 Therefore, in this paper, we define dynamical entropy on CAR algebras based on the construction method of AOW entropy and the definition of Markov chains on CAR algebras given by Accardi, Fidaleo and Mukhamedov [2].

 We organize the paper as follows. In section 2 we recall the definition of the AOW entropy, namely the dynamical entropy through a quantum Markov chain on matrix algebras. In section 3, we briefly review the definitions of the CAR algebra and the Fermion Fock space which gives the algebra. Section 4 is devoted to the notions of Markov states and chains on CAR algebras. In section 5, we formulate dynamical entropy on CAR algebras based on the construction of the AOW entropy, the defintion of Markov chains on CAR algebras, and using an Umegaki conditional expectation on CAR algebras. In section 6 we calculate our dynamical entropy for a simple model associated with the two-state system.

2 AOW entropy

In this section, we construct dynamical entropy on von Neumann algebras using quantum Markov chains on matrix algebras and a noncommutative extension of measurable partitions of a metric space.

Let - a Hilbert space: $\mathcal{H}$,

• a von Neumann algebra with an identity operator $1_{\mathcal{A}}$ acting on $\mathcal{H}$: $\mathcal{A}$,

• the set of all normal states on $\mathcal{A}$: $\mathfrak{S}(\mathcal{A})$,

• a *-automorphism on $\mathcal{A}$: $\theta$,

• the set of all $d\times d$ matrices on $\mathbb{C}$: $M_{d}$,

• the tensor product of $\mathbb{N}$ copies of $M_{d}$: $\otimes_{\mathbb{N}}M_{d}$.

Then the triplet $(\mathcal{A},\mathfrak{S}(\mathcal{A}),\theta)$ describes the dynamics of a quantum system.

Furthermore, let $\gamma:=\{\gamma_{j}\}$ be a finite orthogonal partition of $1_{\mathcal{A}}\in\mathcal{A}$, i.e.

[TABLE]

and $E_{e}$ be a completely positive map from $M_{d}\otimes\mathcal{A}$ to $\mathcal{A}$:

[TABLE]

with the matrix unit $e_{ij}\in M_{d}$. Then a transition expectation [1], [3] $\mathcal{E}_{\gamma,\theta}:M_{d}\otimes\mathcal{A}\to\mathcal{A}$ with respect to $\theta$ is given by

[TABLE]

where $p_{\gamma,e}:=\sum_{j}e_{jj}\otimes\gamma_{j}$. In the above notations, a quantum Markov chain on $\otimes_{\mathbb{N}}M_{d}$ is defined by

[TABLE]

where $\varphi$ is called the initial distribution of $\psi$. Then $\psi=\{\varphi,\mathcal{E}_{\gamma,\theta}\}$ is given by

[TABLE]

where $j_{k}$ is an embedding from $a\in M_{d}$ into the $k$-th factor of $\otimes_{\mathbb{N}}M_{d}$, i.e.

[TABLE]

Let $\varphi$ be a stationary state. Then there exists unique density operator $\rho$ such that

[TABLE]

For any $a_{1}\otimes\cdots\otimes a_{n}\otimes 1_{\mathcal{A}}\in M_{d}\otimes\cdots\otimes M_{d}\otimes\mathcal{A}$, we have

[TABLE]

[TABLE]

where

[TABLE]

Hence we obtain the density operator $\rho_{n}$ of $\varphi_{n}$ on $\otimes^{n}_{1}M_{d}$. Denoting

[TABLE]

and

[TABLE]

Accardi, Ohya and Watanabe defined the entropy with respect to $\gamma,\theta$ and $n$ as

[TABLE]

Definition 1

The dynamical entropy through a quantum Markov chain with respect to $\gamma$ and $\theta$ is given by

[TABLE]

The above entropy is called AOW entropy.

Remark 1

Then $P_{i_{1},\cdots,i_{n}}$ (7) is the time ordered correlation kernel [3], [4] over $\mathcal{A}$.

Remark 2

The AOW entropy was extended to dynamical mutual entropy and is used to study quantum communication processes [14], [17].

3 CAR Algebras

The CAR algebra is a $C^{*}$-algebra generated by the observables of Fermion systems. In this section, we recall the basic mathematical definitions and physical backgrounds of this algebra [10].

3.1 Fermion Fock spaces

First, we fix $N\in\mathbb{N}$. Let $\mathcal{H}$ be a Hilbert space of the state of 1-particle. Then the Hilbert space of states of many body systems is give by

[TABLE]

Then $\mathcal{H}_{N}$ is the Hilbert space that $N$-particles can be distinguished from each other. Moreover, $\mathcal{P}_{N}$ denotes a $N$-dimensional group of permutations, i.e.

[TABLE]

If $\sigma$ is even (resp. odd), we put ${\rm sgn}(\sigma)=1$ (resp. ${\rm sgn}(\sigma)=-1$) where sgn is sign of $\sigma$.

For each $\sigma\in\mathcal{P}_{N}$, there exists a unique unitary operator $P_{\sigma}$ on $\mathcal{H}_{N}$ satisfying

[TABLE]

and

[TABLE]

$P_{\sigma}$ is called a permutation operator with respect to a permutation $\sigma$. Using $P_{\sigma}$, we can classify the vectors of $\mathcal{H}_{N}$.

Definition 2

For any $\sigma\in\mathcal{P}_{N}$,

if $P_{\sigma}x=x$ holds, $x$ is called symmetric. 2. 2.

if $P_{\sigma}x={\rm sgn}(\sigma)x$ holds, $x$ is called anti-symmetric.

Put

[TABLE]

The elements of $\bigwedge^{N}(\mathcal{H})$ are called Fermions.

Remark 3

For any $x_{1},\cdots,x_{N}\in\mathcal{H}$,

[TABLE]

is an anti-symmetric vector. For any $i,j\in\{1,\cdots,N\}$ $(i\neq j)$,

[TABLE]

holds. Especially, if $x_{i}=x_{j}$, one can see that

[TABLE]

(15) implies that two fermions can not exist in the same 1-particle state. The property is called the Pauli exclusion principle. We will see the algebraic representation of this principle below.

In general, the number of particles in many body systems can change. Therefore one has to consider the generalized Hilbert space of $\mathcal{H}_{N}$:

[TABLE]

[TABLE]

where $\mathcal{H}_{0}=\mathbb{C}$. Then the scalar product of $\mathcal{F}(\mathcal{H})$ is defined by

[TABLE]

Hilbert space $\mathcal{F}(\mathcal{H})$ is called a full Fock space.

Definition 3

Let $\bigwedge^{0}(\mathcal{H}):=\mathbb{C}$.

[TABLE]

[TABLE]

is called a Fermion Fock space.

$\mathcal{F}_{f}(\mathcal{H})$ is a closed subspace of the full Fock space $\mathcal{F}(\mathcal{H})$.

Definition 4

A subspace of $\mathcal{F}_{f}(\mathcal{H})$:

[TABLE]

is called a finite particles subspace and is dense in $\mathcal{F}_{f}(\mathcal{H})$.

3.2 CAR Algebras

Let $\mathcal{D}$ be a dense subspace of $\mathcal{H}$ and let

[TABLE]

[TABLE]

where $A_{N}:=\frac{1}{N!}\sum_{\sigma\in\mathcal{P}_{N}}P_{\sigma}$. Then $\mathcal{F}_{f,fin}(\mathcal{D})$ is called a finite particles subspace on $\mathcal{D}$ and is dense in $\mathcal{F}_{f}(\mathcal{H})$.

For each $x\in\mathcal{H}$, we define the operator $a_{+}(f)$ on $\mathcal{F}_{f}(\mathcal{H})$ as follows:

[TABLE]

[TABLE]

One can know that $b_{+}(f)$ is closed. Since $\mathcal{F}_{f,fin}(\mathcal{D})\subset\mathrm{Dom}(a_{+}(f))$ holds, $a_{+}(f)$ is densely defined. Therefore the adjoint operator

[TABLE]

exists. Furthermore, since $a_{+}(f)$ is closed,

[TABLE]

$a(f)^{*}$ is called a Fermion creation oprator and $a(f)$ is called a Fermion annihilation operator respectively.

Then the following poposition is satisfied. Let $\Omega$ be a vacuum state.

Proposition 1

For any $f_{j}\in\mathcal{H}\ (j=1,\cdots,n)$,

[TABLE]

[TABLE]

The Fermion operators satisfy the following algebraic relations.

Theorem 1

For each $f\in\mathcal{H}$, $a(f),\ a(f)^{*}\in\mathbf{B}(\mathcal{F}_{f}(\mathcal{H}))$ and the following relation hold:

[TABLE]

[TABLE]

where $\{\cdot,\cdot\}$ is the anti commutator $\{A,B\}:=AB+BA$.

In (30), if $f=g$, there hold

[TABLE]

These are the algebraic representations of the Pauli exclusion principle (15).

Now we state the definition of the CAR algebra.

Definition 5

A $C^{*}$-algebra generated by bounded operators $\{a(f),a(f)^{*}\ ;\ f\in\mathcal{H}\}$ is called a CAR algebra.

The CAR algebra is a $C^{*}$-algebra of observables of Fermion systems.

4 Markov States and Chains on CAR Algebras

In section 3, we showed that the CAR algebra describes the Fermi systems. In this section, we construct Markov chains on CAR algebras following [2].

Since the Markov chain is a stochastic process, it is natural to consider on $\mathbb{Z}_{+}:=\mathbb{N}\cup\{0\}$ as follows.

Let $\mathcal{A}_{\mathbb{Z}_{+}}$ be the CAR algebra generated by $\{a_{i},a_{i}^{*}\ ;\ i\in\mathbb{Z}_{+}\}$. Namely, $\{a_{i},a_{i}^{*}\ ;\ i\in\mathbb{Z}_{+}\}$ satisfy:

[TABLE]

[TABLE]

where $\{\cdot,\cdot\}$ is the anti-commutator. For any $I\subset\mathbb{Z}_{+}$, $\mathcal{A}_{I}$ denotes the subalgebra generated by $\{a_{i},a_{i}^{*}\ ;\ i\in I\}$. In particular,

[TABLE]

Remark 4

*In general, the -algebra generated by a subset $\mathcal{M}\subset\mathcal{A}$ is given by

[TABLE]

[TABLE]

*Especially, in CAR case, the -algebra generated by $\{a_{i},a_{i}^{*}\ ;\ i\in\{k\}\}$ is written as

[TABLE]

Due to the relations (29)-(30), $\mathcal{A}_{k}$ becomes

[TABLE]

Therefore, we only consider the elements $\{a_{k},a_{k}^{*},a_{k}^{*}a_{k},a_{k}a_{k}^{*}\}$ in $I=\{k\},\ k\in\mathbb{Z}_{+}$ case.

Put $\mathcal{A}:=\mathcal{A}_{\mathbb{Z}_{+}}$.

Definition 6

We denote $\Theta_{I}$ as the unique automorphism on $\mathcal{A}$ which satisfy

[TABLE]

[TABLE]

$\Theta_{I}$* is called a parity automorphism on $\mathcal{A}$.*

Let $\Theta:=\Theta_{\mathbb{Z}_{+}}$. Then $\Theta$ induces even and odd parts of $\mathcal{A}$:

[TABLE]

Definition 7

Let $\mathcal{A},\ \mathcal{B}$ be CAR algebras and $E$ be a map from $\mathcal{A}$ to $\mathcal{B}$. If

[TABLE]

holds, $E$ is called even.

If $E$ is even,

[TABLE]

holds for each $a\in\mathcal{A}_{-}$.

Definition 8

Let $\varphi$ be a state on $\mathcal{A}$. If there exists a quasi-conditional expectation $E_{n}$ w.r.t. $\mathcal{A}_{n-1]}\subset\mathcal{A}_{n]}\subset\mathcal{A}_{n+1]}$ which satisfy

[TABLE]

[TABLE]

$\varphi$* is called a Markov state.*

Let $\{\mathcal{E}_{n}\}_{n\in\mathbb{N}}$ be a sequence of completely positive identity-preserving maps satisfying for each $n\in\mathbb{N}$:

[TABLE]

[TABLE]

[TABLE]

where $\alpha$ is the one-step right shift. One can see that (43) gives the following diagram.

[TABLE]

Let $\rho$ be a density operator on $\mathcal{A}_{0}$ which satisfies the stationarity in the sense of

[TABLE]

In the above notations one obtain the sequence of states $\{\varphi_{n}\}_{n\in\mathbb{N}}$:

[TABLE]

Due to (42), (43) and (45), one can extend $\varphi_{n}$ to a shift-invariant state $\varphi$ on $\mathcal{A}_{\mathbb{N}}$. The shift-invariant state is called the quantum Markov chain generated by $(\rho,\{\mathcal{E}_{n}\}_{n\in\mathbb{N}})$.

5 A Construction of Dynamical Entropy on CAR Algebras

In this section, giving an Umegaki conditional expectation [8], [16], we formulate dynamical entropy on CAR algebras.

Let $\mathcal{A}_{0}$ be a CAR algebra generated by $\{a_{i},a_{i}^{*}\ ;\ i\in\{0\}\}$ and $\varphi_{0}$ a faithful normal state on $\mathcal{A}_{0}$. $\mathcal{E}_{\gamma,\theta}$ denotes an Umegaki conditional expectation from $\mathcal{A}_{\{0,n\}}$ to $\mathcal{A}_{0}$ as

[TABLE]

where $1_{0}$ is the identity of $\mathcal{A}_{0}$, $\theta$ is a *-automorphism on $\mathcal{A}_{0}$, and $\sum_{i=1}\gamma_{i}=1_{0}$ is an operator partition of $1_{0}$. Since $\theta(1_{0})=1_{0}$, $\sum\theta(\gamma_{i})$ is again operator partition of $1_{0}$.

Now we construct Markov chain with 1-time evolution:

[TABLE]

[TABLE]

[TABLE]

Since

[TABLE]

we have

[TABLE]

respectively. Therefore, we obtain a state through a Markov chain on $\mathcal{A}_{[0,1]}$ as

[TABLE]

Moreover, we consider the case of $n$-time evolution:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence a state thorough a Markov chain on $\mathcal{A}_{[0,n]}$ is given by

[TABLE]

Therfore, we get an $n$-time developped state on $\mathcal{A}_{[1,n]}$ as

[TABLE]

[TABLE]

Now we focus on the constant of $\rho_{n}$. Denote

[TABLE]

Obviously, then there holds

[TABLE]

Hence, in the above notations, we can define dynamical entropy on the CAR algebra as following.

Definition 9

For a quadruple $(\mathcal{A},\varphi_{0},\gamma,\theta)$, the dynamical entropy on the CAR algebra is given by

[TABLE]

6 Model Computation

In this section, we calculate the complexity of a time-evolution of a simple model associated with the quantum spin system using the introduced dynamical entropy (52).

Let

[TABLE]

be orthonormal system on $\mathbb{C}^{2}$ and let

[TABLE]

$e_{0}$ and $e_{1}$ describe the models of spin-down and spin-up of a quantum spin system respectively. Furthermore, by the operations:

[TABLE]

one can see that $a_{0}^{*}$ is the creation operator and $a_{0}$ is the annihilation operator of this system. Since

[TABLE]

$\mathcal{A}_{0}:=M(2,\mathbb{C})$ is generated by $\{a_{0}^{*},a_{0}\}$. Now we investigate the complexity of a time development on this system using our dynamical entropy. According to

[TABLE]

we put:

An operatior partition of the identity $1\in\mathcal{A}_{0}$:

[TABLE]

a *-automorphism:

[TABLE]

[TABLE]

Moreover, we denote a density matrix by

[TABLE]

Due to

[TABLE]

[TABLE]

[TABLE]

(51) becomes

[TABLE]

[TABLE]

Now we mention the following result.

Lemma 1

Under the above notations,

[TABLE]

**Proof **

According to the anti-commutation relation (30),

[TABLE]

The above equation gives

[TABLE]

Therefore, if $(i_{1},i_{2},\cdots,i_{n})=(1,1,\cdots,1)$, there holds

[TABLE]

[TABLE]

Moreover,

[TABLE]

This implies that

[TABLE]

Finally, the equation:

[TABLE]

induces

[TABLE]

Hence we obtain

[TABLE]

From the above lemma, we then get the dynamical entropy (52):

[TABLE]

Remark 5

In general, the complexities of unitary operators on a unital $C^{*}$-algebra are 0 [13]. Therefore, this result also tells us that our dynamical entropy is defined correctly.

7 Conclusion

In this paper, based on the construction method of the AOW entropy, we have defined a new type of dynamical entropy on CAR algebras using an Umegaki conditional expectation from $\mathcal{A}_{\{0,n\}}$ to $\mathcal{A}_{0}$ with a *-automorphisms on a local algebra $\mathcal{A}_{0}$. Moreover, we have computed the introduced entropy for a $2\times 2$ matrix algebra case.

 Incidentally, if one investigate the complexities of the lattice translation or the Bogoliubov automorphism using dynamical entropy through a Markov chain, we think that it is necessary to reformulate our dynamical entropy with completely positive identity preserving map from $\mathcal{A}_{[0,n+1]}$ to $\mathcal{A}_{[0,n]}$ associated with a *-automorphism on $\mathcal{A}_{\mathbb{Z_{+}}}$.

Acknowledgements The authors would like to thank Prof. Accardi for his useful suggestions and warm encouragements.