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AIP conference proceedings

Entropy Type Complexity in Quantum Systems

Noboru Watanabe, Andrei Yu. Khrennikov, Gregg S. Jaeger, M. Schlosshauer, Gregor Weihs

Published Jan 1, 2011• 0 citations
Publisher

Abstract

We review a notion of quantum channel and some entropy type complexities for quantum systems. The quantum dynamical entropy for CP map is discussed and numerical computation of this entropy type complexity is accomplished for noisy optical channel.
DOI
10.1063/1.3578710

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Entropy Type Complexity in Quantum Systems

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In 1989, Ohya propose a new concept, so-called Information Dynamics (ID), to investigate complex systems according to two kinds of view points.One is the dynamics of state change and another is measure of complexity.In ID, two complexities C S and T S are introduced.C S is a measure for complexity of system itself, and T S is a measure for dynamical change of states, which is called a transmitted complexity.An example of these complexities of ID is entropy for information transmission processes.The study of complexity is strongly related to the study of entropy theory for classical and quantum systems.The quantum entropy was introduced by von Neumann around 1932, which describes the amount of information of the quantum state itself.It was extended by Ohya for C*-systems before CNT entropy.The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras, which was 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 formulate 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 the entropic complexities for classical and quantum systems.We introduce some complexities by means of entropy functionals in order to treat the transmission processes consistently.We apply the general frames of quantum communication to the Gaussian communication processes.Finally, we discuss about a construction of compound states including quantum correlations.

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In quantum information theory, Emch, Conne, and Stormer were the first who studied the complexity of quantum dynamical processes. After that, Ohya introduced the [Formula: see text]-mixing entropy for general quantum systems and he defined the mean entropy and the mean mutual entropy for quantum dynamical systems based on the [Formula: see text]-mixing entropy. Conne, Narnhoffer and Thirring introduced the dynamical entropy (CNT entropy) and several researchers discussed this concept. Alicki and Fannes defined a different dynamical entropy — AF entropy. In 1995, Voiculescu proposed the dynamical approximation entropy. Accardi, Ohya and Watanabe defined yet another dynamical entropy (AOW entropy) through a quantum Markov process in 1997. In 1999, Kossakowski, Ohya and Watanabe introduced the dynamical entropy (KOW entropy) with respect to completely positive maps. In this paper, we discuss the complexity of quantum dynamical processes to calculate the dynamical entropy for noisy optical channels. In order to discuss the efficiency of information communication processes, a measure of complexity of initial state itself and a measure of transmitted complexity through communication channels are necessary. Quantum entropies were formulated on the basis of the quantum probability theory. In quantum communication systems, von Neumann entropy and Ohya mutual entropy relate to these measures of complexities, respectively. Recently, several mutual entropy type measures (Lindblad-Nielsen entropy and coherent entropy) were defined making use of entropy exchange with respect to a channel and initial state. In this paper, we show which of the measures is the most suitable one for discussing the efficiency of information transmission for quantum processes.
We are interested to study the dynamics of state change or the complexity of state for several systems. Information dynamics (ID) is a new concept introduced by Ohya to construct a theory under the ID's framework by synthesizing these investigating schemes. In ID, a complexity of state describing system itself and a transmitted complexity between two systems are used. One of the example of these complexities are the Shannon's entropy and the mutual entropy (information) in classical entropy theory. In quantum entropy theory, it was known that the von Neumann entropy and the Ohya mutual entropy relate to these complexities. Recently, several mutual entropy type measures (the Lindblad-Nielsen entropy and the Coherent entropy) were proposed by means of the entropy exchange for an input state and a channel. The quantum capacity is defined by taking the supremum for the Ohya mutual entropy. The main purpose of this paper is to compare with these mutual entropy type complexities in order to find most congruous one for studying the quantum communication process. In this paper, we calculate the quantum capacity for some simple models for discussing the information transmission for OOK and PSK modulations.
In 1989, Ohya propose a new concept, so-called Information Dynamics (ID), to investigate complex systems according to two kinds of view points.One is the dynamics of state change and another is measure of complexity.In ID, two complexities C S and T S are introduced.C S is a measure for complexity of system itself, and T S is a measure for dynamical change of states, which is called a transmitted complexity.An example of these complexities of ID is entropy for information transmission processes.The study of complexity is strongly related to the study of entropy theory for classical and quantum systems.The quantum entropy was introduced by von Neumann around 1932, which describes the amount of information of the quantum state itself.It was extended by Ohya for C*-systems before CNT entropy.The quantum relative entropy was first defined by Umegaki for σ-finite von Neumann algebras, which was 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 formulate 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 the entropic complexities for classical and quantum systems.We introduce some complexities by means of entropy functionals in order to treat the transmission processes consistently.We apply the general frames of quantum communication to the Gaussian communication processes.Finally, we discuss about a construction of compound states including quantum correlations.