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Kluwer Academic Publishers eBooks

Group Covariant Signals in Quantum Information Theory

Tsuyoshi Sasaki Usuda, Ichi Takumi

Published Dec 19, 2005• 8 citations
Publisher

Abstract

Necessary and sufficient condition for group covariant signals is given. This condition shows what kind of signals correspond to “group covariant signals.” We show by applying this condition that many signals, such as arbitrary binary signals, PPM, FSK, and PSK coherent-state signals, equidistant codes, and binary linear codes, are all group covariant.

DOI
10.1007/0-306-47097-7_5

Related papers

A class of group covariant signal sets and its necessary and sufficient condition

Tsuyoshi Sasaki Usuda, Yoshi-hiro Ishikawa, Keisuke Shiromoto

(G,̂χ)-covariant quantum state signals, which is a generalization of the narrow sense group covariant signals, are defined. Then a necessary and sufficient condition for (G,̂χ)-covariant signals is given and examples of the signals are shown.

AIP conference proceedings•Jan 1, 2014•0 citations

4PSK coherent-state signals can be narrow sense group covariant with respect to F 4 by coding

Minami Tanaka, Asuka Ohashi, Tsuyoshi Sasaki Usuda

The formula of the channel matrix for group co-variant signals for classical-quantum communication has been studied to calculate the channel matrix analytically. However, despite its importance, the formula derived previously cannot be applied to coded q-ary phase shift keying (PSK) signals by linear codes over an extension field F q . To derive the formula for codes over F q , we focus on the 4PSK signals coded by the codes over F 4 . In the present paper, we propose a construction method of a code of length 2n over F 4 from a code of length n. If the original code is a group code, the coded 4PSK signals by the constructed code is narrow sense group covariant.

International Symposium on Information Theory and its Applications•Oct 1, 2016•0 citations

Applying a symmetrization method by coding to non-symmetric mixed-state signals

Souichi Takahira, Tsuyoshi Sasaki Usuda

In quantum communication theory, group covariant signals are important and useful when we consider the minimization problem of the error probability or the maximization problem of the mutual information. However, in practical applications, there are important non-symmetric signals such as amplitude shift keying or quadrature amplitude modulation signals. To manage this situation, we have proposed a symmetrization method of non-symmetric pure-state signals by using coding. In this paper, we consider to apply the symmetrization method to mixed-state signals and give a necessary condition of mixed-state signals to apply the method. Then, we present examples of mixed-state signals to which the symmetrization method can be applied.

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Group-covariant extreme and quasiextreme channels

Laleh Memarzadeh, Barry C. Sanders

Constructing all extreme instances of the set of completely positive trace-preserving (CPTP) maps, i.e., quantum channels, is a challenging and valuable open problem in quantum information theory. Here we introduce a systematic approach that, despite the lack of knowledge about the full parametrization of the set of CPTP maps on arbitrary Hilbert-spaced dimension, enables us to construct exactly those extreme channels that are covariant with respect to a finite discrete group or a compact connected Lie group. Innovative labeling of quantum channels by group representations enables us to identify the subset of group-covariant channels whose elements are group-covariant generalized-extreme channels. Furthermore, we exploit essentials of group representation theory to introduce equivalence classes for the labels and also partition the set of group-covariant channels. As a result, we show that it is enough to construct one representative of each partition. We construct Kraus operators for group-covariant generalized-extreme channels by solving systems of linear and quadratic equations for all candidates satisfying the necessary condition for being group-covariant generalized-extreme channels. Deciding whether these constructed instances are extreme or quasiextreme is accomplished by solving a system of linear equations. Proper labeling and partitioning the set of group-covariant channels leads to a novel systematic, algorithmic approach for constructing the entire subset of group-covariant extreme channels. We formalize the problem of constructing and classifying group-covariant generalized extreme channels, thereby yielding an algorithmic approach to solving, which we express as pseudocode. To illustrate the application and value of our method, we solve for explicit examples of group-covariant extreme channels. With unbounded computational resources to execute our algorithm, our method always delivers a description of an extreme channel for any finite-dimensional Hilbert space and furthermore guarantees a description of a group-covariant extreme channel for any dimension and for any finite-discrete or compact connected Lie group if such an extreme channel exists.

arXiv:2210.03449•Physical Review Research•Sep 13, 2022•8 citations

Formula of channel matrix for coded quantum signals by classical linear codes over Zm

Masaki Ota, Hideyuki Kumazawa, Keisuke Shiromoto +1 more

A formula of channel matrix for linear codes over Zmis presented. m-ary PSK coherent-states are used as letter states and the square-root measurement (SRM) is used as a decoding. First, arbitrary linear codes over Zmare shown to group covariant. Then the formula of channel matrix is derived by generalizing [T.S. Usuda, S. Usami, I. Takumi, and M. Hata, Phys. Lett. A305, pp.125-134, (2002)]. Furthermore, the usefulness of the formula is considered.

International Symposium on Information Theory and its Applications•Oct 1, 2010•0 citations
Necessary and sufficient condition for group covariant signals is given. This condition shows what kind of signals correspond to “group covariant signals.” We show by applying this condition that many signals, such as arbitrary binary signals, PPM, FSK, and PSK coherent-state signals, equidistant codes, and binary linear codes, are all group covariant.
The formula of the channel matrix for group co-variant signals for classical-quantum communication has been studied to calculate the channel matrix analytically. However, despite its importance, the formula derived previously cannot be applied to coded q-ary phase shift keying (PSK) signals by linear codes over an extension field F q . To derive the formula for codes over F q , we focus on the 4PSK signals coded by the codes over F 4 . In the present paper, we propose a construction method of a code of length 2n over F 4 from a code of length n. If the original code is a group code, the coded 4PSK signals by the constructed code is narrow sense group covariant.
In quantum communication theory, group covariant signals are important and useful when we consider the minimization problem of the error probability or the maximization problem of the mutual information. However, in practical applications, there are important non-symmetric signals such as amplitude shift keying or quadrature amplitude modulation signals. To manage this situation, we have proposed a symmetrization method of non-symmetric pure-state signals by using coding. In this paper, we consider to apply the symmetrization method to mixed-state signals and give a necessary condition of mixed-state signals to apply the method. Then, we present examples of mixed-state signals to which the symmetrization method can be applied.
Constructing all extreme instances of the set of completely positive trace-preserving (CPTP) maps, i.e., quantum channels, is a challenging and valuable open problem in quantum information theory. Here we introduce a systematic approach that, despite the lack of knowledge about the full parametrization of the set of CPTP maps on arbitrary Hilbert-spaced dimension, enables us to construct exactly those extreme channels that are covariant with respect to a finite discrete group or a compact connected Lie group. Innovative labeling of quantum channels by group representations enables us to identify the subset of group-covariant channels whose elements are group-covariant generalized-extreme channels. Furthermore, we exploit essentials of group representation theory to introduce equivalence classes for the labels and also partition the set of group-covariant channels. As a result, we show that it is enough to construct one representative of each partition. We construct Kraus operators for group-covariant generalized-extreme channels by solving systems of linear and quadratic equations for all candidates satisfying the necessary condition for being group-covariant generalized-extreme channels. Deciding whether these constructed instances are extreme or quasiextreme is accomplished by solving a system of linear equations. Proper labeling and partitioning the set of group-covariant channels leads to a novel systematic, algorithmic approach for constructing the entire subset of group-covariant extreme channels. We formalize the problem of constructing and classifying group-covariant generalized extreme channels, thereby yielding an algorithmic approach to solving, which we express as pseudocode. To illustrate the application and value of our method, we solve for explicit examples of group-covariant extreme channels. With unbounded computational resources to execute our algorithm, our method always delivers a description of an extreme channel for any finite-dimensional Hilbert space and furthermore guarantees a description of a group-covariant extreme channel for any dimension and for any finite-discrete or compact connected Lie group if such an extreme channel exists.