Browsing by Author "Dias, Nuno C."
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- On a Recent Conjecture by Z. Van Herstraeten and N. J. Cerf for the Quantum Wigner EntropyPublication . Dias, Nuno C.; Prata, João N.We address a recent conjecture stated by Z. Van Herstraeten and N. J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positiveWigner functions. Moreover, we are able to prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant, which is not very distant from the constant suggested by Van Herstraeten and Cerf. We also prove an analogous result for another conjecture stated by the same authors for the R´enyi entropy of positive Wigner functions. As a by-product we prove a new inequality for the radar-ambiguity function (and for the Wigner distribution) which is reminiscent of Lieb’s inequalities.
- What is the Wigner function closest to a given square integrable function?Publication . Ben-Benjamin, J.S.; Cohen, J.; Dias, Nuno C.; Loughlin, P.; Prata, João N.We consider an arbitrary square integrable function F on the phase-space and look for the Wigner function closest to it with respect to the L2 norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert--Schmidt operator withWeyl symbol F. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a bona fide Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results, we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics.