On Entropy of Quantum Compound Systems

Noboru Watanabe

In quantum compound systems, joint (or compound) states usually do not exist [26]. The so-called Ohya compound state was defined by using the von Neumann–Schatten decomposition [23] of an input state and a quantum communication channel. Quantum mutual entropy [10] was introduced by using quantum relative entropy [24] between the Ohya compound state and the trivial (product) compound state on the compound (input and output) system. One of the remarkable results related to state entanglement is the Jamiolkowski isomorphism [5]. In this paper, we use the Jamiolkowski isomorphism for the construction of Ohya compound state and we discuss the existence of a completely positive map between compound entangled states and the Ohya compound states. The efficiency of information transmission is investigated by using the mean entropy and the mean mutual entropy [12, 6] of a connected channel.

Göran Lindblad

Olga V. Man’ko

Noboru Watanabe

It has been shown that joint probability distributions of quantum systems generally do not exist, and the key to solving this concern is the compound state invented by Ohya. The Ohya compound state constructed by the Schatten decomposition (i.e., one-dimensional orthogonal projection) of the input state shows the correlation between the states of the input and output systems. In 1983, Ohya formulated the quantum mutual entropy by applying this compound state. Since this mutual entropy satisfies the fundamental inequality, one may say that it represents the amount of information correctly transmitted from the input system through the channel to the output system, and it may play an important role in discussing the efficiency of information transfer in quantum systems. Since the Ohya compound state is separable state, it is important that we must look more carefully into the entangled compound state. This paper is intended as an investigation of the construction of the entangled compound state, and the hybrid entangled compound state is introduced. The purpose of this paper is to consider the validity of the compound states constructing the quantum mutual entropy type complexity. It seems reasonable to suppose that the quantum mutual entropy type complexity defined by using the entangled compound state is not useful to discuss the efficiency of information transmission from the initial system to the final system.

Noboru Watanabe

In Ref. 27, Ohya proposed the information dynamics synthesizing several ways of studying complex systems. In information dynamics, there are two types of complexities, one is a complexity of state representing system itself and another is a transmitted complexity between two systems. Entropies in classical and quantum systems are examples of these complexities of information dynamics. Transmitted complexity is an important tool to analyze the efficiency of information transmission in communication processes. In order to treat a flow of dynamical process, dynamical entropies were introduced in not only classical but also quantum systems. Based on the transition expectation introduced by Accardi to study quantum Markov process, the KOW entropy for completely positive (CP) maps was defined in Ref. 17. The generalized AOW and the AF entropies was constructed by the KOW entropy. The compound states are important tool to define the transmitted entropy 43,44. The transmitted complexity associated with the separable compound states is defined by using the generalized AOW entropy in Refs. 36 and 45. In this paper, we define a transmitted complexity (quantum dynamical mutual entropy) by means of the modified compound states and we prove the fundamental inequalities of the transmitted complexity for the independent quantum dynamical systems.

Vyacheslav Sergeevich Kirchanov