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AIP conference proceedings

Entropy Type Complexity in Quantum Systems

Noboru Watanabe

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
10.1063/1.3567449

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Entropy Type Complexity in Quantum Systems

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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.

AIP conference proceedings•Jan 1, 2011•0 citations

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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.

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arXiv:1109.2197•Journal of Physics A Mathematical and Theoretical•Dec 23, 2011•14 citations
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.
We consider properties of quantum channels with the use of unified entropies. Extremal unravelings of quantum channel with respect to these entropies are examined. The concept of map entropy is extended in terms of the unified entropies. The map ( q , s )-entropy is naturally defined as the unified ( q , s )-entropy of a rescaled dynamical matrix of given quantum channel. Inequalities of Fannes type are obtained for introduced entropies in terms of both the trace and Frobenius norms of difference between corresponding dynamical matrices. Additivity properties of introduced map entropies are discussed. The known inequality of Lindblad with the entropy exchange is generalized to many of the unified entropies. For the tensor product of a pair of quantum channels, we derive a two-sided estimate on the output entropy of a maximally entangled input state.