Abstract
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.
Highlights
The first scholar to give much attention to a mathematical treatment of communication processes was Shannon [1]
Various researchers have studied the efficiency of information transmission from the input system to the output system passing through ordinary communication channels based on information theory
We show the construction of the Ohya compound state by using the Jamiolkowski isomorphism, and we review the existence of completely positive channel between the entangled compound state and the Ohya compound state
Summary
The first scholar to give much attention to a mathematical treatment of communication processes was Shannon [1] He created the information theory by introducing measures of information, such as the entropy of the system and the mutual entropy formulated by the relative entropy of the joint probability distribution between input and output determined by the channel and the direct product distribution between input and output. One of the important problem is to examine how accurately information is transmitted when an optical signal is passed through an optical channel To achieve this, it needs to extend the mutual entropy determined in the classical system to the quantum system. We show the quantum mutual entropy type measure defined by using the entangled compound state is not suitable to discuss the efficiency of information transmission from the initial system to the final system
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