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Error Exponents of Quantum Communication System with M-ary PSK Coherent State Signal

Kentaro Kato

Published Jan 1, 2011• 2 citations
OpenAlex

Abstract

The optimal distributions on a channel input that yield the random coding exponent, the expurgation exponent, and the zero-rate reliability function for a discrete classical-quantum channel with M-ary Phase Shift Keying (PSK) coherent state signal are analytically derived. In each case, the optimal distributions are given by a uniform distribution over the channel input alphabet. As a result, the expressions of the error exponents for M-ary PSK coherent state signal are simplified. By using this result, typical behavior of the error exponents is sketched in the case of M = 16.

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The optimal distributions on a channel input that yield the random coding exponent, the expurgation exponent, and the zero-rate reliability function for a discrete classical-quantum channel with M-ary Phase Shift Keying (PSK) coherent state signal are analytically derived. In each case, the optimal distributions are given by a uniform distribution over the channel input alphabet. As a result, the expressions of the error exponents for M-ary PSK coherent state signal are simplified. By using this result, typical behavior of the error exponents is sketched in the case of M = 16.
It is well known that the maximal success probability of the binary quantum communication can be improved by using a sub-Poissonian nonstandard coherent state as an information carrier. In the present article, we consider the quantum communication with NNN-ary phase-shift-keying (NNN-PSK) signal for an arbitrary positive integer N>1N>1N>1. By using nonstandard coherent state, we analytically provide the maximal success probability of the quantum communication with NNN-PSK. Unlike the binary case, there is a case that the guessing probability of NNN-PSK quantum communication can be improved by some non-sub-Poissonian, nonstandard coherent state.
In this paper, we investigate the symbol error probability (SEP) of phase-squeezed M-ary phase-shift keying (M-PSK). Since the relevant observable for M-PSK detection is the optical phase, we adopt the adaptive Mark-II receiver which is a physically realizable phase measurement. First, we develop a theoretical analysis based on the phase probability operator measure (POM) of the Mark-II scheme in the Fock basis. Then, we develop two SEP methods based on the statistics of the received PSK symbol and the error introduced by the Mark-II measurement. The first method derives the phase probability density induced by the squeezed state noise and incorporates the additional Mark-II phase uncertainty through an angular convolution. Since this convolution does not admit a simple closed form, we also introduce an effective tangential-variance model, which yields a closed form SEP expression in terms of the Owen's T-function. Numerical results show that phase squeezing substantially reduces the SEP of M-PSK compared to coherent state transmission, with greater gains for higher constellation orders. Notably, for the investigated scenario, squeezing can almost double the photon efficiency of M-PSK as the mean number of transmitted photons increases. Finally, the proposed approximations closely follow the Mark-II POM analysis, typically within an accuracy of 2-4 photons, and therefore provide accurate and computationally efficient tools for analyzing phase squeezed quantum M-PSK communication.
For efficient use of energy in quantum communication, it is best to use discrete signals that approach the classical capacity achieved by a continuous input of coherent states. Analytical treatment of the classical communication channel requires discretization, and among such discrete signals, using signals as close as possible to the quantum channel capacity is ideal. The class of binary phase shift keying (BPSK) coherent-state signals has been proven to be optimal in the case of binary discretization; however, for multiary discretization with three or more signals, the optimal signals are unknown. In this study, we consider the case of ternary discretization. We numerically show that 3PSK is the optimal signal constellation in two dimensions.
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.