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

Abstract

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.

15 pages, 5 figures, the new version includes the n-mode case

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• Wigner non-negative states that verify the Wigner entropy conjecture
• Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces
• Quantum information in Riemannian spaces
• Wigner entropy conjecture and the interference formula in quantum phase space
• Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture