On the role of Hermite-like polynomials in the Fock representations of Gaussian states

The expansion of quantum states and operators in terms of Fock states plays a fundamental role in the field of continuous-variable quantum mechanics. In particular, for general single-mode Gaussian operators and Gaussian noisy states, many different approaches have been used in the evaluation of their Fock representation. In this paper a natural approach has been applied using exclusively the operational properties of the Hermite and Hermite-like polynomials and showing their fundamental role in this field. Closed-form results in terms of polynomials, exponentials, and simple algebraic functions are the major contribution of the paper.Comment: 19 pages, 5 figure

Gaussian states and geometrically uniform symmetry

Quantum Gaussian states can be considered as the majority of the practical quantum states used in quantum communications and more generally in quantum information. Here we consider their properties in relation with the geometrically uniform symmetry, a property of quantum states that greatly simplifies the derivation of the optimal decision by means of the square root measurements. In a general framework of the $N$-mode Gaussian states we show the general properties of this symmetry and the application of the optimal quantum measurements. An application example is presented, to quantum communication systems employing pulse position modulation. We prove that the geometrically uniform symmetry can be applied to the general class of multimode Gaussian states

Exact Spectral Analysis of Single-h and Multi-h CPM Signals through PAM decomposition and Matrix Series Evaluation

In this paper we address the problem of closed-form spectral evaluation of CPM. We show that the multi-h CPM signal can be conveniently generated by a PTI SM. The output is governed by a Markov chain with the unusual peculiarity of being cyclostationary and reducible; this holds also in the single-h context. Judicious reinterpretation of the result leads to a formalization through a stationary and irreducible Markov chain, whose spectral evaluation is known in closed-form from the literature. Two are the major outcomes of this paper. First, unlike the literature, we obtain a PSD in true closed-form. Second, we give novel insights into the CPM format.Comment: 31 pages, 10 figure

The list-chromatic index of K 6

We prove that the list-chromatic index and paintability index of K"6 is 5. That indeed @g"@?^'(K"6)=5 was a still open special case of the List Coloring Conjecture. Our proof demonstrates how colorability problems can numerically be approached by the use of computer algebra systems and the Combinatorial Nullstellensatz

Efficient Optimal Minimum Error Discrimination of Symmetric Quantum States

This paper deals with the quantum optimal discrimination among mixed quantum states enjoying geometrical uniform symmetry with respect to a reference density operator $\rho_0$. It is well-known that the minimal error probability is given by the positive operator-valued measure (POVM) obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator $\Pi_0$, and maximizing $\textrm{Tr}(\rho_0 \Pi_0)$. In this paper, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a semidefinite positive operator $X$ commuting with the symmetry operator and such that $X >= \rho_0$. The new formulation gives a deeper insight into the optimization problem and allows to obtain closed-form analytical solutions, as shown by a simple but not trivial explanatory example. Besides the theoretical interest, the result leads to semidefinite programming solutions of reduced complexity, allowing to extend the numerical performance evaluation to quantum communication systems modeled in Hilbert spaces of large dimension.Comment: 5 pages, 1 Table, no figure

Colouring the Petals of a Graph

A petal graph is a connected graph G with maximum degree three, minimum degree two, and such that the set of vertices of degree three induces a 2{regular graph and the set of vertices of degree two induces an empty graph. We prove here that, with the single exception of the graph obtained from the Petersen graph by deleting one vertex, all petal graphs are Class 1. This settles a particular case of a conjecture of Hilton and Zhao

Quantum communications

This book demonstrates that a quantum communication system using the coherent light of a laser can achieve performance orders of magnitude superior to classical optical communications Quantum Communications provides the Masters and PhD signals or communications student with a complete basics-to-applications course in using the principles of quantum mechanics to provide cutting-edge telecommunications. Assuming only knowledge of elementary probability, complex analysis and optics, the book guides its reader through the fundamentals of vector and Hilbert spaces and the necessary quantum-mechanical ideas, simply formulated in four postulates. A turn to practical matters begins with and is then developed by: ·         development of the concept of quantum decision, emphasizing the optimization of measurements to extract useful information from a quantum system; ·         general formulation of a transmitter–receiver system ·         particular treatment of the most popular quantum communications systems—OOK, PPM, PSK and QAM; ·         more realistic performance evaluation introducing thermal noise and system description with density operators; ·         consideration of scarce existing implementations of quantum communications systems and their difficulties with suggestions for future improvement; and ·         separate treatment of quantum information with discrete and continuous states. Quantum Communications develops the engineering student’s exposure to quantum mechanics and shows physics students that its theories can have practically beneficial application in communications systems. The use of example and exercise questions (together with a downloadable solutions manual for instructors) will help to make the material presented really sink in for students and invigorate subsequent research