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AIP conference proceedings

On General Quantum Mutual Entropy and Capacity

Noborn Watanabe

Published Jan 1, 2005• 0 citations
Publisher

Abstract

In communication processes, a channel has an activity to communicate information of input system to output system. The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983, and he extended it to general quantum systems by means of the relative entropy of Araki and Uhlmann. The capacity shows the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of Ohya mutual entropy with respect to a certain subset of the initial state space. One of the most important theorems in quantum communication theory is coding theorem. The quantum coding theorems are discussed by using the mutual entropy — type measures introduced in the studies of quantum information.In this paper, we compare with these mutual entropy‐type measures in order to obtain most suitable one for discussing the quantum coding theorem. We concretely calculate the quantum capacity for quantum channels.

DOI
10.1063/1.1874588

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In communication processes, a channel has an activity to communicate information of input system to output system. The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983, and he extended it to general quantum systems by means of the relative entropy of Araki and Uhlmann. The capacity shows the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of Ohya mutual entropy with respect to a certain subset of the initial state space. One of the most important theorems in quantum communication theory is coding theorem. The quantum coding theorems are discussed by using the mutual entropy — type measures introduced in the studies of quantum information.In this paper, we compare with these mutual entropy‐type measures in order to obtain most suitable one for discussing the quantum coding theorem. We concretely calculate the quantum capacity for quantum channels.
The quantum entropy was introduced by von Neumann around 1932, which describes the amont of information of the quantum state itself. It was extended by Ohya for C*‐systems. The quantum relative entropy was first defined by Umegaki for σ ‐finite von Neumann algebras, which denotes a certain difference between two states. It was extended by Araki and Uhlmann for general von Neumann algebras and *‐algebras, respectively. The development of communication theory is closely connected with study of entropy theory. In order to discuss the commuinication processes, we need at least two systems, such as input and output systems. The signal of the input system is carried through a physical device, which is called a channel. The mathematical representation of the channel is a mapping from the input state space to the output state space. The mutual entropy (information) defined by using the relative entropy between the product state consist of the input state and the output state and the compound state given by the communication process, expresses the amount of information correctly transmitted through the channel from the input system to the output system. In classical communication theory, the mutual entropy was formulated by using the joint probability distribution between the input system and the output system. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. In fully quantum system, there does not exist the joint probability distribution in general. Instead of the joint probability distribution, Ohya invented the quantum (Ohya) compound state, and he introduced the fully quantum mutual entropy (information), which is called Ohya mutual entropy, for quantum input and output systems, describes the amount of information correctly sent from the quantum input system to the quantum output system through the quantum channel. It was extended by Ohya for general quantum systems. Recently Shor and Bennet et al took the coherent entropy and defined the mutual type entropy to discuss a sort of coding theorem for communication processes.In this paper, we concretely calculate these mutual entropy‐type measures for the attenuation channel in order to assay these abilities.
The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel.The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers.The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983.In this paper, we co\mathrm{c}oco mpare with mutual entroΨ\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}_{\Psi}entroΨ​ -type measures and show some resuls for quantum capacity.
The mutual entroy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel.The (semi-classical) mutual entropies for classical input and quantum output were defined by several researchers.The funy quantum mutual entropy, which is called Ohya mutual entroy, for quantum input and output by using the relative entropy was defined by Ohya in 1983.In this paper, we ∞mpue\infty mpue∞mpue With mutual entropy−WPeentropy-WPeentropy−WPe measures and show some resuls for quantum capacity.
The quantum entropy introduced by von Neumann around 1932 describes the amount of information of the quantum state itself. It was extended by Ohya for C*-systems before Conne-Narnhoffer-Thirring (CNT) entropy. The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras and it was subsequently extended by Araki and Uhlmann for general von Neumann algebras and *-algebras, respectively. By introducing a new notion, the so-called compound state, in 1983 Ohya succeeded to construct the mutual entropy in a complete quantum mechanical system (i.e., input state, output state and channel are all quantum mechanical) describing the amount of information correctly transmitted through the quantum channel. In this paper, we briefly review Ohya's S-mixing entropy and the quantum mutual entropy for general quantum systems. Based on a concept of structure equivalent, we apply the general framework of quantum communication to the Gaussian communication processes.
von Neumann entropy represents the amount of information in the quantum state, and this was extended by Ohya for general quantum systems [10]. Umegaki first defined the quantum relative entropy for -finite von Neumann algebras, which was extended by Araki, and Uhlmann, for general von Neumann algebras and *-algebras, respectively. In 1983 Ohya introduced the quantum mutual entropy by using compound states; this describes the amount of information correctly transmitted through the quantum channel, which was also extended by Ohya for general quantum systems. In this paper, we briefly explain Ohya?s -mixing entropy and the quantum mutual entropy for general quantum systems. By using structure equivalent class, we will introduce entropy type functionals based on quantum information theory to improve treatment for the Gaussian communication process.
The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and he extended it to general quantum systems by the relative entropy of Araki and Uhlmann. When the author introduced the quantu m mutual entropy, he did not indicate that it contains other definitions of the mutual entropy including classical one, so that there exist several misu nderstandings for the use of the mutual entropy (information) to compute the capacity of quantum channels. Therefore in this note we point out that our quantum mutual entropy generalizes others and where the m isuse occurs.