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Some entropic extensions of the uncertainty principle

Abstract : In connection with the uncertainty principle in quantum mechanics (Heisenberg) or in time-frequency analysis (Heisenberg-Gabor), we study its formulation in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula and Zozor et al. These results can be considered as generalizations of the Heisenberg inequalities in the sense that they measure the mutual uncertainty of a random variable (or wave function) and its conjugated random variable (or Fourier transformed wave function) through their associated Rényi entropies with conjugated indexes. We consider here the more general case where the entropic indexes are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indexes alpha and beta in the plane (alpha,beta). Our results explain and extend a recent study by Luis, where states with quantum fluctuations below the Gaussian case are discussed at the single point (2,2).
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https://hal.archives-ouvertes.fr/hal-00282363
Contributor : Steeve Zozor <>
Submitted on : Tuesday, May 27, 2008 - 2:02:52 PM
Last modification on : Wednesday, May 13, 2020 - 4:30:04 PM

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Steeve Zozor, Mariela Portesi, Christophe Vignat. Some entropic extensions of the uncertainty principle. ISIT 2008, Jul 2008, Toronto, Canada. pp.1676-1680, ⟨10.1109/ISIT.2008.4595273⟩. ⟨hal-00282363⟩

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