Theory of Quantum Pulse Position Modulation and Related Numerical Problems

TL;DR

This paper investigates quantum pulse position modulation (PPM) in noisy and noise-free environments, comparing detection methods and highlighting the advantages of quantum detection over classical approaches.

Contribution

It formulates the quantum PPM states for pure and mixed cases, compares detection strategies, and emphasizes the role of symmetry in optimizing error probability calculations.

Findings

Quantum detection significantly outperforms classical methods.

Convex linear programming and square root measurement are computationally efficient.

Symmetry properties simplify the analysis of quantum PPM error probabilities.

Abstract

The paper deals with quantum pulse position modulation (PPM), both in the absence (pure states) and in the presence (mixed states) of thermal noise, using the Glauber representation of coherent laser radiation. The objective is to find optimal (or suboptimal) measurement operators and to evaluate the corresponding error probability. For PPM, the correct formulation of quantum states is given by the tensorial product of m identical Hilbert spaces, where m is the PPM order. The presence of mixed states, due to thermal noise, generates an optimization problem involving matrices of huge dimensions, which already for 4-PPM, are of the order of ten thousand. To overcome this computational complexity, the currently available methods of quantum detection, which are based on explicit results, convex linear programming and square root measurement, are compared to find the computationally less expensive one. In this paper a fundamental role is played by the geometrically uniform symmetry of the quantum PPM format. The evaluation of error probability confirms the vast superiority of the quantum detection over its classical counterpart.