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30/04/2018, 14:00 — 15:00 — Seminar room (2.8.3), Physics Building

Nicolas J. Cerf, *Université Libre de Bruxelles*

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On Shannon entropy in quantum phase space : from entropic uncertainty relations to continuous majorization relations

Shannon differential entropy is a na tural uncertainty measure when considering a continuous-spectrum observable, such as a quadrature component of a mode of the electromagnetic field. I will show that it makes a very useful — though not yet fully explored — tool for the phase-space description of light. After a short survey on the entropic uncertainty relations generalizing the Heisenberg principle with differential entropies instead of variances, I will emphasize the notion of entropy power in this context and formulate new entropic uncertainty relations expressing the balance between multimode Gaussian projective measurements. This will lead me to discuss the differential entropy of a Wigner function, which, for states with non-negative such functions, yields a perfect uncertainty measure that is invariant under all Gaussian unitaries (symplectic transformations and displacements) and is saturated by all Gaussian pure states. I will then consider entropy-power inequalities for a beam splitter and finally discuss some conjectured continuous-variable majorization relations in phase space.