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Cambridge University Press eBooks

Minimal uncertainty and Wigner transforms

Jean-Pierre Antoine, Romain Murenzi, Pierre Vandergheynst, S. Twareque Ali

Published Sep 23, 2004• 0 citations
Publisher
DOI
10.1017/cbo9780511543395.009

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It was conjectured by Wehrl and proven by Lieb that the Shannon differential entropy of the Husimi function of a continuous-variable quantum state is lower bounded by the value it takes on coherent states. This corroborates the intuition, familiar from Heisenberg's uncertainty relations, that coherent states are minimal uncertainty states. The analogous statement for the Wigner function would be that the Wigner differential entropy of states with a positive Wigner function is bounded below by the entropy of pure Gaussian states. This was conjectured a decade ago, but recently disproven. Here, we prove the analogous statement for Kirkwood-Dirac distributions with mutually unbiased bases and for the Gross-Wigner distributions. For these discrete systems, the entropy-minimizing role is played by the pure states with positive quasiprobability distributions. Further, we prove that Kirkwood-Dirac distributions with large nonpositivity have large uncertainty.