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Annals of Physics

Discrete Coherent States and Probability Distributions in Finite-Dimensional Spaces

D. Galetti, Marcelo A. Marchiolli

Published Aug 1, 1996• 70 citations
Publisher
DOI
10.1006/aphy.1996.0079

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For a qudit, a quantum system on a finite dimensional Hilbert space, we construct minimum uncertainty states and a non-negative quasi probability distribution. They are the discrete version of the coherent states and the Husimi function in the continuous quantum mechanics. The two bases related by the discrete Fourier transformation are identified with the position and momentum bases. We prove an inequality concerning uncertainties of the position and momentum of a qudit. The minimum uncertainty states are the ones which saturate the inequality. Like the coherent states, they are not mutually orthogonal but comprise an overcomplete set in the state space. As the dimension of the Hilbert space goes to infinity, these states approach the coherent states. Using the minimum uncertain states, we construct a non-negative quasi probability distribution for a qudit. Its marginal distributions are smeared out, but we show that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance is assumed.
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We demonstrate how large classes of discrete and continuous statistical distributions can be incorporated into coherent states, using the concept of a reproducing kernel Hilbert space. Each family of coherent states is shown to contain, in a sort of duality, which resembles an analogous duality in Bayesian statistics, a discrete probability distribution and a discretely parametrized family of continuous distributions. It turns out that nonlinear coherent states, of the type widely studied in quantum optics, are a particularly useful class of coherent states from this point of view, in that they contain many of the standard statistical distributions. We also look at vector coherent states and multidimensional coherent states as carriers of mixtures of probability distributions and joint probability distributions.