1,338 research outputs found
Bell inequality with an arbitrary number of settings and its applications
Based on a geometrical argument introduced by Zukowski, a new multisetting
Bell inequality is derived, for the scenario in which many parties make
measurements on two-level systems. This generalizes and unifies some previous
results. Moreover, a necessary and sufficient condition for the violation of
this inequality is presented. It turns out that the class of non-separable
states which do not admit local realistic description is extended when compared
to the two-setting inequalities. However, supporting the conjecture of Peres,
quantum states with positive partial transposes with respect to all subsystems
do not violate the inequality. Additionally, we follow a general link between
Bell inequalities and communication complexity problems, and present a quantum
protocol linked with the inequality, which outperforms the best classical
protocol.Comment: 8 pages, To appear in Phys. Rev.
Multivariable q-Racah polynomials
The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson
polynomials is studied for parameters satisfying a truncation condition such
that the orthogonality measure becomes discrete with support on a finite grid.
For this parameter regime the polynomials may be seen as a multivariable
counterpart of the (one-variable) -Racah polynomials. We present the
discrete orthogonality measure, expressions for the normalization constants
converting the polynomials into an orthonormal system (in terms of the
normalization constant for the unit polynomial), and we discuss the limit leading to multivariable Racah type polynomials. Of special interest is the
situation that lies on the unit circle; in that case it is found that there
exists a natural parameter domain for which the discrete orthogonality measure
(which is complex in general) becomes real-valued and positive. We investigate
the properties of a finite-dimensional discrete integral transform for
functions over the grid, whose kernel is determined by the multivariable
-Racah polynomials with parameters in this positivity domain.Comment: AMS-LaTeX v1.2, 38 page
From "Dirac combs" to Fourier-positivity
Motivated by various problems in physics and applied mathematics, we look for
constraints and properties of real Fourier-positive functions, i.e. with
positive Fourier transforms. Properties of the "Dirac comb" distribution and of
its tensor products in higher dimensions lead to Poisson resummation, allowing
for a useful approximation formula of a Fourier transform in terms of a limited
number of terms. A connection with the Bochner theorem on positive definiteness
of Fourier-positive functions is discussed. As a practical application, we find
simple and rapid analytic algorithms for checking Fourier-positivity in 1- and
(radial) 2-dimensions among a large variety of real positive functions. This
may provide a step towards a classification of positive positive-definite
functions.Comment: 17 pages, 14 eps figures (overall 8 figures in the text
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