## Abstract

Quantum entropy and channel are fundamental concepts for quantum information theory progressed recently in various directions. We will review the fundamental aspects of mean entropy and mean mutual entropy and calculate them for open system dynamics.

## 1. Introduction

We consider a dynamical change of states in open systems, which is not isolated, connected with the environment. In order to treat the open system phenomenologically, one can make use of several concepts and methods such as probability, entropy (information), stochastic process and statistical inference, etc. Those methods are particularly important for discussing complex systems. One of these modalities was formulated by Shannon [1]. He created a mathematical foundation of information theory based on the idea that ‘information obtained from a system with a large vagueness has been highly profitable’. He introduced the entropy measuring the amount of information of the state of system and the mutual entropy (information) expressing the amount of information correctly carried from the initial system to the final system through a channel. This theory can deal with the efficiency of the information transmission in a commutative signal space described by the electric wave or the electric current. The information theory based on the Shannon's entropy theory is completely formulated mathematically. The information theory being able to treat quantum effects is the so-called quantum information theory, which has been developed with quantum entropy theory and quantum probability. In quantum information theory, one of the important problems is to study how much information is exactly transmitted to the output system from the input system through a quantum channel.

The amount of information of the quantum input system is described by the quantum entropy defined by von Neumann [2] in 1932. Furthermore, it had been desired to extend Shannon's mutual entropy (information) of classical information theory to that in the quantum one, but it was a very difficult task. It was shown [3] that there does not generally exist a joint probability distribution in quantum systems, describing a correlation between the input and output systems, to formulate the mutual entropy in quantum systems. The semi-classical mutual entropy was introduced by Holevo [4], Levitin [5], Ingarden [6], for classical input and quantum output passing through a semi-classical channel. By introducing a new notion, the so-called compound state, Ohya defined the quantum mutual entropy (information) in complete quantum systems (i.e. input state, output state and channel are dealt with in quantum systems) in 1983 [7]. By calculating this mutual entropy, one can investigate the efficiency of the information transmission in quantum communication processes, which allows the detailed analysis of optical communication [8–11].

The quantum relative entropy was introduced by Umegaki [12], and it is extended to general quantum systems by Araki [13], Uhlmann [14] and Donald [15]. Moreover, the study of the channel for the quantum communication processes was discussed in [7]. The characterization of quantum processes based on the transition expectation introduced by Accardi *et al.* [16], in which quantum Markov process [17] was studied, is discussed in [18], and beam splitting was rigorously studied by Fichtner *et al.* [19]. By using quantum communication processes, the noisy optical channel was introduced in [10]. The channel capacities are studied in [20–24]. Recently, these methods started to be explored in applications to biology and cognition [25–27].

The C*-mixing entropy was introduced in [28] and its property was studied in [29,30]. The Umegaki's relative entropy was extended for general state space by Araki [13,31] in von Neumann algebra and by Uhlmann [14] in *-algebra. The mutual entropy in C*-algebra was defined by Ohya [32]. These quantum entropies are expounded in [33].

Kolmogorov–Sinai entropy S(T) [34] for a measure preserving transformation T was defined on a message space through finite partitions of the measurable space. It shows the maximal mean entropy with respect to finite partitions. Based on the mean dynamical entropy and the mean dynamical mutual entropy, the classical coding theorems of Shannon are formulated to analyse the aspect of communication processes. The quantum dynamical entropy (QDE) was studied by Connes & Størmer [35], Emch [36], Connes *et al.* [37], Alicki & Fannes [38] and others [39–43], which were formulated on observable spaces. The quantum dynamical entropy and the quantum dynamical mutual entropy were studied by Ohya [32,44], which were defined on the state spaces. The quantum dynamical entropy through the quantum Markov chain (QMC) was defined in [45]. The dynamical entropy for a completely positive (CP) map was introduced in [46]. These relations are discussed in [47]. Some calculations according to the generalized AOW entropy given by KOW entropy for several channels were discussed in [11,48].

The mean dynamical entropy expresses the amount of information per one letter of a signal sequence carried from the input source, and the mean dynamical mutual entropy denotes the amount of information per one letter of the signal obtained in the output system.

In this paper, we will discuss the efficiency of information transmission of the open system dynamics to calculate the mean mutual entropy with respect to the input states and the quantum channel for the open system dynamics.

## 2. Quantum channels

Let and be the input and output quantum systems, respectively, where is the set of all bounded linear operators on a separable complex Hilbert space and is the set of all density operators on (*k*=1,2). A quantum channel is defined by a mapping *Λ** from the input quantum state space to the output quantum state space . A quantum channel *Λ** satisfying an affine property such as
for any and any λ∈[0,1] is said to be a linear. Almost physical processes can be expressed by a CP map *Λ*, which is a linear map from to holding
for any *n*∈**N**, any and any . The dual map *Λ** of *Λ* defined by *trΛ**(*ρ*)*B*=*trρΛ*(*B*) for any and any is called a CP channel [9,30,33,49,50]. Here we explain the examples of quantum channels.

### (a) Open system dynamics

Let *S*_{1} be a system described by a Hilbert space and *S*_{2} be an external system described by another Hilbert space , interacting with *S*_{1}. Let *H* be the total Hamiltonian of *S*_{1} and *S*_{2}. We denote the initial states of *S*_{1} and *S*_{2} by *ρ* and *σ*, respectively. The time evolution of the interacted state *ρ*⊗*σ* at time *t* according to the unitary operator is given by the combined state *θ*_{t}
A CP channel of the open system dynamics [30,33,50] is obtained by taking the partial trace of *θ*_{t} with respect to such as
where is defined by
with a complete orthogonal system .

### (b) Quantum communication processes

Based on [7], we briefly explain a quantum communication process in consideration of the influence of noise and loss.

Let and be two complex Hilbert spaces relating to noise and loss systems, respectively. A quantum communication channel *Λ** from to is defined by the composition of three CP channels *a**, *Π**, *γ**
where *γ** is a CP channel from to defined by
*a** is a CP channel from to given by
and *Π** is a CP channel from to given by physical properties of the communication processes.

### (c) Noisy optical channel

One of the examples of the quantum communication channel is a noisy optical channel. The noisy optical channel *Λ** with a fixed noise state *ζ*=|*m*_{1}〉〈*m*_{1}|, which is the *m*_{1} photon number state in , was defined in [10], such as
where *V* is a linear mapping from to given by
and |*n*_{1}〉 is the *n*_{1} photon number state vector in , and , and *α*, *β* are complex numbers satisfying |*α*|^{2}+|*β*|^{2}=1. In the noisy optical channel *Λ**, the above CP channel *Π** from to is given by
for any states . For the coherent input and noise states , the output and loss states of the CP channel *Π** is obtained by
where *Π** is called a generalized beam splitting because it means that two coherent states are split to two coherent states by passing through the CP channel *Π**. On generalized Fock spaces, the mathematical formulations of beam splitting are studied in [19] based on liftings in the sense of Accardi & Ohya [51] is denoted by
where is a mapping from to .

### (d) Attenuation channel

The noisy optical channel with a vacuum noise is called the attenuation channel introduced in [7] by
where |0〉〈0| is the vacuum state in and *V* _{0} is a linear mapping from to given by

### (e) Connected channel

Here we explain a connected channel. For *n*∈**N**, a *n*-connected channel with a fixed noise state *ζ* was defined by
for any , where *Π*^{*n} is *n*-folds composition of the CP channels *Π**, *V* ^{n} and *V* ^{*n} are also *n*-folds composition of *V* and *V* *, respectively. Then *n*-connected channel can be denoted as the following representation [11].

### Theorem 2.1

*The n-connected channel* *with noise state* |*m*〉〈*m*| *is denoted by*
*where* {|*y*_{l}〉} *and* {|*z*_{k}〉} *are CONS in* *and* *respectively*. {|*i*〉} *is the set of number states in* .

For a *n*-connected channel with a fixed noise state *ζ*, we have

### Theorem 2.2

*For the n-connected channel* *with* *if n is given by* *then Π*^{*n} *is the identity channel id, that is*

We obtain the following theorem in [11]

### Theorem 2.3

*For the n-connected channel* *with* *if n is given by* *then Π*^{*n} *is the exchanged channel, that is*
*is held for any* *. If n is given by* *one can obtain the output and loss states of n-folds composition of the CP channels Π** *such as the maximal entangled state*
*for the input state ρ*=|1〉〈1| *and the noise state ζ*=|0〉〈0|. *It means that Π**^{n} *generates the maximal entangled state.*

## 3. Quantum entropy and quantum mutual entropy

For a density operator , von Neumann [2] defined the quantum entropy by The properties of entropy are explained in [9,33,50].

The mutual entropy for purely quantum systems denoted by *I*(*ρ*;*Λ**) depends on an input quantum state *ρ* and a quantum channel *Λ** should hold the following three conditions:

(1) The quantum mutual entropy with the identity channel

*Λ**=id is equal to the von Neumann entropy:*I*(*ρ*;id)=*S*(*ρ*).(2) The quantum mutual entropy w.r.t. the classical systems reduces to classical one.

(3) Shannon's type fundamental inequalities 0≤

*I*(*ρ*;*Λ**)≤*S*(*ρ*) is satisfied.

Instead of the joint state in classical systems, Ohya defined the compound state *σ*_{E} of the input state *ρ* and the output state *Λ***ρ* by
where *E* represents a one-dimensional orthogonal base {*E*_{k}} of the Schatten decomposition [52] of *ρ*, which is not always unique unless every eigenvalue of *ρ* is not degenerated. The compound state depends on how we distinguish the state *ρ* into basic states. By using two compound states *σ*_{E} and *σ*_{0}=*ρ*⊗*Λ***ρ*, Ohya introduced [7,53] the quantum mutual entropy (information) as
where the supremum is taken over all Schatten decompositions of *ρ*, and *S*(*σ*_{E},*σ*_{0}) is the Umegaki's relative entropy [12] defined by
where *s*(*σ*_{E})≪*s*(*σ*_{0}) means the support projection *s*(*σ*_{0}) of *σ*_{0} is greater than the support projection *s*(*σ*_{E}) of *σ*_{E}. The above conditions (1)∼(2) are held for the quantum mutual entropy. Moreover condition (3) follows from the monotonicity of relative entropy [7]:

### Theorem 3.1

For a linear channel, one has the following form [7]:

### Theorem 3.2

*The quantum mutual entropy is denoted as*

When the input system is classical, an input state *ρ* is given by a probability distribution or a probability measure. In either case, the Schatten decomposition of *ρ* is unique, namely, for the case of probability distribution; *ρ*={*μ*_{k}},
where *δ*_{k} is the delta measure, that is
Therefore for any channel *Λ**, the mutual entropy becomes
which equals to the following usual expression when one of the two terms is finite for an infinite dimensional Hilbert space
The above equality has been taken by Levitin [5] and Holevo [4] associated with classical-quantum channels. The quantum mutual entropy by Ohya contains their semi-classical mutual entropies as a special one.

## 4. Mean entropy and mean mutual entropy

We here review the quantum mean entropy and quantum mean mutual entropy defined by Ohya [32,54].

Let be a unital *C**-algebra, be the set of all states over , be an automorphism of . A *C**-triple with a stationary state with respect to ; that is, is held, represents a stationary information source in quantum information theory.

We denote an output *C**-dynamical system by the *C**-triple . Let be a covariant channel, that is its predual map is a CP unital map satisfying . Let and (*m*=1,…,*M*, *n*=1,…,*N*) be finite dimensional unital *C**-algebras, and and be C unital maps. We express a pair of finite sequences of {*α*_{m}},{*β*_{n}} by *α*^{M}=(*α*_{1},*α*_{2},…,*α*_{M}), *β*^{N}=(*β*_{1},*β*_{2},…,*β*_{N}).

We briefly explain functionals , , and introduced in [32,54].

Let be a weak * convex subset of the state space of . Based on Choquet's theorem, for any state , there exits regular Borel probability maximal measures representing extremal decomposition of *φ*
Let be the set of such measures *μ* on . For a given and a given extremal decomposition measure *μ*∈ of *φ*, the compound state of on the tensor product algebra is defined by [32,54]
and is a compound state of and with respect to *α*^{M}∪*β*^{N}≡(*α*_{1},*α*_{2},…,*α*_{M},*β*_{1},*β*_{2},…,*β*_{N}) given by
For a given pair of *α*^{M}=(*α*_{1},…,*α*_{M}), *β*^{N}=(*β*_{1},…,*β*_{N}), the functionals and are defined in [32,54] by taking the supremum for all possible extremal decomposition measure *μ* of *φ* in such as
where and are given by
for any pair (*α*^{M},*β*^{N}) of finite sequences *α*^{M}=(*α*_{1},…,*α*_{M}) and *β*^{N}=(*β*_{1},…,*β*_{N}) of CP unital maps *α*_{m}, *β*_{n} and any extremal decomposition measure *μ* of *φ*, and *S*(⋅,⋅) is the relative entropy.

Let (resp. ) be a unital *C**-algebra with a fixed automorphism (resp. ), *α* (resp. *β*) be CP unital maps from a finite dimensional unital *C**-algebra (resp. ) to (resp. ), and *Λ* be a covariant CP unital map from to , and *φ* be an invariant state over , i.e. . Let *α*^{N} and be finite sequences of , given by
For each CP unital map *α* and *β*, the functionals , and are defined by taking the supremum for all possible 's, *α*'s, 's and *β*'s:
where , and are given in [32,54] by
The fundamental inequalities with respect to , and are held in [32].

### Proposition 4.1

Because these functionals are given by using a channel transformation, they contains the dynamical entropy (e.g. usual K-S entropies, i.e. K-S type entropy and K-S type mutual entropy) as a special case [32,54].

### Proposition 4.2

If are abelian *C**-algebras and each *α*_{k} is an embedding, then our functionals coincide with classical K-S entropies:
for any finite partitions of a probability space

In general quantum systems, the following theorems are proved in [32,54].

### Theorem 4.3

*Let α*_{m} *be a sequence of CP maps* *such that there exist CP maps* *satisfying* *in the pointwise topology. Then,*

### Theorem 4.4

*Let α*_{m} *and β*_{m} *be sequences of CP maps* *and* *such that there exist CP maps* *and* *satisfying* *and* *in the pointwise topology. Then one has*

The above theorems show Kolmogorov–Sinai type convergence theorems for the entropy and the mutual entropy [29,32,44,54]. Based on these settings, one can study Shannon's coding theorems in quantum compound systems by using the quantum capacity [8,9,20–23,33,50].

## 5. Computation of mean entropy and mean mutual entropy for open system dynamics

Based on [18], let *H*_{1} be the Hamiltonian of a system *S*_{1} described by a Hilbert space . If a system *S*_{1} interacts with an external system (heat bath) *S*_{2} with the Hamiltonian *H*_{2} described by another Hilbert space and the initial states of *S*_{1} and *S*_{2} are *ρ* and *ξ*, respectively, then the compound state *σ*_{t} of *S*_{1} and *S*_{2} at time *t* after the interaction between two systems is given by
where with the total Hamiltonian *H* of *S*_{1} and *S*_{2}. A channel is obtained by taking the partial trace w.r.t. such as
The total Hamiltonian *H* is described by
For simplify, we assume that the system *Σ*_{2} is given by a single mode
For a given state , the channel *Λ** is denoted by Stinespring–Sudarshan–Kraus representation such as
where *O*_{i} is a partial isometric operator given by
For any *k*, is a subspace spanned by a subset {|*j*〉⊗|*k*−*j*〉;*j*=0,1,2,…,*k*}. Then
holds. Let be a restriction of *H*_{in} into . *H*^{(k)}_{in} is a finite dimensional self-adjoint operator on satisfying
For a stationary initial states and , let and . Then we have two compound states
When *Λ** is the quantum channel given by the above open system dynamics, we get
where *F*_{ji}=|*j*_{i}><*j*_{i}| is the *j*_{i}-photon number state in . Two compound states through the channel *Λ** of open system dynamics are obtained by
5.1and
5.2

### Lemma 5.1

For a stationary initial state we have

By using the above lemma, we have the following theorem.

### Theorem 5.2

(1)

*For a stationary initial state**we have the lower bound of**such as*(2)

*If η*_{0}*=1, η*_{k}*=0 (∀ k≥1) and**of**then**and*

## 6. Conclusion

We have explained the quantum channels associated with the open system dynamics and the quantum communication processes. Some examples of quantum communication channels are discussed. The quantum mutual entropy by Ohya is treated for purely quantum systems, and semi-classical mutual entropy are a special case of the quantum mutual entropy. We briefly reviewed the mean entropy and the mean mutual entropy for general quantum systems. The lower bound of the mean entropy for the open system dynamics is obtained. For a given assumption, the mean entropy and the mean mutual entropy for the open system dynamics are calculated.

## Competing interests

I declare that I have no competing interests.

## Funding

I received no funding for this study.

## Footnotes

One contribution of 14 to a theme issue ‘Quantum foundations: information approach’.

- Accepted January 4, 2016.

- © 2016 The Author(s)