In their seminal paper, Caves and Schumaker presented a new formalism for quantum optics, intended to serve as building block describing two-photon processes, in terms of new, generalized qudratures. The important, revolutionary concept was that it fundamentally two-mode, i.e. the related observables could not be attributed any single one comprising modes, but rather complex quadrature only bot...

1. Introduction. The estimation of quadrature errors for analytic functions has been considered by Davis and Rabinowitz [1]. An estimate for the error of the Gaussian quadrature formula for analytic functions was obtained by Davis [2]. McNamee [3] has also discussed the estimation of error of the Gauss-Legendre quadrature for analytic functions. Convergence of the Gaussian quadratures was discu...

Szegö quadrature formulas represent the analogue on the unit circle of the well-known Gauss–Christoffel quadratures for intervals of the real line. In the present work, Szegö formulas are revised and an application to the computation of the Fourier transform given. 2006 Elsevier Inc. All rights reserved.

We provide a simple description of the most general collective Gaussian attack in continuous-variable quantum cryptography. In the scenario of such general attacks, we analyze the asymptotic secret-key rates which are achievable with coherent states, joint measurements of the quadratures and one-way classical communication.

In this paper we introduce a system of nonlinear ordinary differential equations which in a particular case reduces to Volterra’s system. We found in two simplest cases the complete sets of the integrals of motion using Nambu–Poisson reformulation of the Hamiltonian dynamics. In these cases we have solved the systems by quadratures.