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International Symposium on Information Theory and its Applications

Property of Classical Reliability Function for BPSK Signals Coded by Simplex Codes with SRM

Ken Masaki, Tiancheng Wang, Shogo Usami, Souichi Takahira, Tsuyoshi Sasaki Usuda

Published Nov 10, 2024• 0 citations

Abstract

There is a gap between properties of quantum and classical reliability functions. That is, the upper bound of the quantum reliability function diverges, whereas that of classical one does not, even if the same coherent signal is used. We are interested in the origin of this gap. In this paper, we investigate the upper bound of the reliability function for quantum signals coded by simplex codes when using square-root measurement (SRM). The reliability function in this case has a much steeper slope with respect to the rate than in the uncoded case. This result suggests the divergence of the upper bound of the quantum reliability function in the previous study.

DOI: 10.23919/isita60732.2024.10858245

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There is a gap between properties of quantum and classical reliability functions. That is, the upper bound of the quantum reliability function diverges, whereas that of classical one does not, even if the same coherent signal is used. We are interested in the origin of this gap. In this paper, we investigate the upper bound of the reliability function for quantum signals coded by simplex codes when using square-root measurement (SRM). The reliability function in this case has a much steeper slope with respect to the rate than in the uncoded case. This result suggests the divergence of the upper bound of the quantum reliability function in the previous study.

In information theory, the reliability function and its bounds, describing the exponential behavior of the error probability, are important quantitative characteristics of the channel performance. From a more general point of view, these bounds provide certain measures of distinguishability of a given set of classical states. In a previous paper, quantum analogs of the random coding and expurgation lower bounds for the case of pure signal states were introduced. Here we discuss the case of general quantum states, in particular, we prove the expurgation bound conjectured previously and find the quantum cutoff rate for arbitrary mixed signal states.

In information theory the reliability function and its bounds, describing the exponential behavior of the error probability, are the most important quantitative characteristics of the channel performance. From a general point of view, these bounds provide certain measures of distinguishability of a given set of states. In an earlier paper we introduced quantum analogs of the random coding and the expurgation lower bounds for the case of pure signal states. Here we discuss the general case, in particular, we prove the previously conjectured expurgation bound and find the quantum cutoff rate in the case of arbitrary mixed signal states.

To derive an optimal quantum detection that minimizes the average probability of detection error is an important problem in quantum information theory. Especially, a study of mixed-state signals is important not only for reliable communications but also for secure communications. To date, the Square-Root Measurement (SRM) has been received much attention because it is the optimal or asymptotically-optimal detection for quantum state signals. The main purpose of this correspondence is to consider the Pseudo-Classicality of the SRM.

We study the reliability function of general classical-quantum channels, which describes the optimal exponent of the decay of decoding error when the communication rate is below the capacity. As the main result, we prove a lower bound, in terms of the quantum Rényi information in Petz's form, for the reliability function. This resolves Holevo's conjecture proposed in 2000, a long-standing open problem in quantum information theory. It turns out that the obtained lower bound matches the upper bound derived by Dalai in 2013, when the communication rate is above a critical value. Thus, we have determined the reliability function in this high-rate case. Our approach relies on Renes' breakthrough made in 2022, which relates classical-quantum channel coding to that of privacy amplification, as well as our new characterization of the channel Rényi information.

In quantum discrimination, the value of the minimum error probability and the set of measurement operators which achieve this minimum are often difficult to derive. Here we present a comparison of the performance obtained by the optimal solution and by the available bounds, namely the square root measurement (SRM) and the Chernoff bound. Applied to some Gaussian states, namely to coherent states with thermal noise, it is shown that the SRM provides a much tighter bound with respect to the Chernoff bound, with a comparable numerical complexity.