A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions

Abstract

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 ρ. We prove sharp estimates which critically depend on the purity of ρ and which are saturated in the Gaussian case.

Hydrodynamic interpretation of generic squeezed coherent states: A kinetic theory

“…Indeed, the fact that S (q,p) corresponds to a minimum entropy state can be argued within the Schrödinger-Robinson uncertainty principle [59,60]. Namely, the determinant of the covari- 3 Information entropy matches the thermodynamic entropy up to the factor k B for systems in equilibrium as in our case.…”

The non-commutative Robertson–Schrödinger uncertainty principle

“…To prove this property we combine Sylvester's criterion for Hermitian, positive semi-definite matrices [14] with the ordering of eigenvalues. The no negativity of the principal minors 4 of size three leads to…”

“…In the literature [4], [5] some of the above issues have been addressed and partially studied following different paths. A brief outlook of the present work which exhibits our contribution is as follows.…”

Coarse-graining and symplectic non-squeezing

“…An exception to this is the string theory community where the ideas contained in [19], which gave rise to the various versions of Floer homology and quantum cohomology, are used in a variety of contexts, not the least of which are the various statements associated with mirror symmetry [28]. Another exception has been the work of M. de Gosson who has strongly advocated for the use of the symplectic non-squeezing theorem in Physics in general, but especially in the context of the semi-classical limits of quantum systems [29,30,31,32,33,34,35,36,37,38] during the last two decades. Being influenced by his work, we also presented some views on the possible implications of the non-squeezing theorem for the macroscopic time irreversibility of Hamiltonian systems in [39,40].…”