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Entropy-power uncertainty relations: Towards a tight inequality for all Gaussian pure states

par Hertz, Anaëlle ;Jabbour, Michael ;Cerf, Nicolas
Référence Journal of Physics A: Mathematical and Theoretical, 50, 38, 385301
Publication Publié, 2017-08

Article révisé par les pairs

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Résumé :

We show that a proper expression of the uncertainty relation for a pair of canonically-conjugate continuous variables relies on entropy power, a standard notion in Shannon information theory for real-valued signals. The resulting entropy-power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to Bialynicki-Birula and Mycielski, but can be further extended to rotated variables. Hence, based on a reasonable assumption, we give a partial proof of a tighter form of the entropy-power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity. Interestingly, it implies the generalized (rotation-invariant) Schrödinger-Robertson uncertainty relation exactly as the original entropy-power uncertainty relation implies Heisenberg relation. It is saturated for all Gaussian pure states, in contrast with hitherto known entropic formulations of the uncertainty principle.