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Entropy

Non-Orthogonality Measure for a Collection of Pure Quantum States

Kentaro Kato

Published Apr 21, 2022• 7 citations

Abstract

Modern optical communication technology can realize a large-scale multilevel (or M-ary) optical signal. Investigating the quantum mechanical nature of such a large-scale M-ary optical signal is essential for a unified understanding of quantum information science and optical communication technology. This article focuses on the quantum-mechanical non-orthogonality for a collection of pure quantum states and proposes a non-orthogonality index based on the least squares error criterion in quantum detection theory. First, we define the index for linearly independent signals, and the proposed index is analyzed through numerical simulations. Next, the index is applied to a highly large-scale M-ary phase-shift keying (PSK) coherent state signal. Furthermore, the index is compared with the capacity of the pure state channel with the PSK signal. As a result, it is shown that a highly large-scale M-ary PSK coherent state signal exhibits a quantum nature even when the signal transmission power is very high. Thus, the theoretical characterization of a highly large-scale M-ary coherent state signal based on the proposed index will be the first step toward a better understanding of cutting-edge optical communication technologies such as the quantum stream cipher Y00.

DOI: 10.3390/e24050581

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The limitations of cloning and discriminating quantum states are related to the non-orthogonality of the states. Hence, understanding the collective features of quantum states is essential for the future development of quantum communications technology. This paper investigates the non-orthogonality of different coherent-state signal constellations used in quantum communications, namely phase-shift keying (PSK), quadrature-amplitude modulation (QAM), and a newly defined signal named the sunflower-like (SUN) coherent-state signal. The non-orthogonality index (NOI) and the average probability of correct detection (detection probability) are numerically computed. Results show that PSK NOI increases faster than QAM and SUN as the number of signals increases for a given number of signal photons. QAM and SUN exhibit similar NOI and detection probability, behaving similarly to randomly generated signals for a larger number of signals. Approximation formulas are provided for the detection probability as a function of NOI for each signal type. While similar to QAM, SUN signal offers potential advantages for applications requiring uniform signal-space distribution. The findings provide valuable insights for designing useful quantum signal constellations.

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.

LDPC-Coded

M

-ary PSK Optical Coherent State Quantum Communication

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The fundamental performance limits and channel capacity of optical communications systems operating over the free space channel will be examined using quantum detection theory. The performance of the optimum quantum receiver for on-off keying (OOK) and optical binary phase shift keying (BPSK) is first examined as a pure state (no noise) problem. The classical capacity of the binary symmetric channel for these two modulation schemes is then evaluated for the optimum quantum receiver by making use of the concept of quantum measurement states. The performance of M-ary pulse position modulation, which requires a product state representation, is evaluated along with the performance of certain 'dense signal sets.' Performance comparisons with classical techniques shows over 5 dB improvement in some cases when quantum detection is employed. As a further application of the quantum detection theory, the capacity of the binary channel with on-off keyed modulation and quantum detection is evaluated, and shown to exceed the capacity obtained with classical photon counting.

A unified treatment of typical optical modulation formats is given toward a future development of the quantum enigma cipher. In our analysis, the intensity shift keying (ISK), amplitude shift keying (ASK), phase shift keying (PSK), and quadrature amplitude modulation (QAM) coherent state signals are compared in terms of the masking number of potential signals through the numerical calculations of the square-root measurement. Through this comparison, we see that the use of ISK coherent state signal is the best choice for cryptographic purpose among these formats in achieving an appropriate balance between the number of signal photons and the number of quantum state signals.