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A refinement of the RobertsonSchrodinger uncertainty principle and a Hirschman-Shannon inequality for Wigner distributions
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Abstract(s)
We propose a refinement of the Robertson–Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition, it makes a direct connection with modern harmonic analysis, since it involves the Wigner transform and its symplectic Fourier transform, which is the radar ambiguity function. As a by-product of the refined RSUP, we derive inequalities involving the entropy and the covariance matrix of Wigner distributions. These inequalities refine the Shanon and the Hirschman inequalities for the Wigner distribution of a mixed quantum state
. We prove sharp estimates which critically depend on the purity of
and which are saturated in the Gaussian case.
Description
Keywords
Wigner distribution Uncertainty principles Entropic relations
Citation
Dias, N. C., de Gosson, M. A., & Prata, J. N. (2019). A refinement of the Robertson–Schrödinger uncertainty principle and a Hirschman–Shannon inequality for Wigner distributions. Journal of Fourier Analysis and Applications, 25(1), 210-241