Theory of quantum pulse position modulation and related numerical problems

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.Index Terms-Quantum detection, linear programming, square root measurement (SRM), least square measurement (LSM), geometrically uniform symmetry (GUS), thermal noise, pulse position modulation (PPM).

Quantum Communications Systems

“…The considerations we just made "in words" can be translated to "formulas," but this is not so simple, because the symmetry operator S is not separable, but it operates between the K factors of the composite Hilbert space H = H ⊗K 0 . For the symmetry operator, the following result applies, recently demonstrated in [13]. For example, for n = 2 and K = 3 (3-PPM) we have and, developing the products, we obtain the 16×16 matrix (remember that the tensor product for matrices becomes Kronecker's product, see Sect.…”

“…Here we shall propose an original method based on the SRM and on the property of quantum PPM of verifying the GUS [13]. It seems odd that such property has not been remarked by other authors; because, on one hand, it is very intuitive, and, on the other hand, it makes it possible to directly achieve the same optimal result.…”

“…The explicit evaluation of such eigenvectors is long and cumbersome, and is developed in [13], using the fact that for the PPM the operator S is a permutation matrix. The important thing is that this evaluation can be done "analytically" for every n and K, without resorting to numeric evaluation, which could be prohibitive for high values of N = n K .…”

Toward optimum efficiency in a quantum receiver for coded ppm

“…each the tensor product of M pure, coherent states ( [5]). Some examples from an M=8 signal set are shown here: We assume the highly simplified model with no extraneous noises.…”

Quantum Communications Systems with Thermal Noise

“…Among them, the SRM method applied to mixed states and the analysis in the presence of noise of QAM, PSK, and PPM quantum systems [1,2]. In some numerical computations we will find it convenient to use the compression technique introduced at the end of Chap.…”