Remarks on the Fact that the Uncertainty Principle Does Not Determine the Quantum State

TL;DR

This paper explores the limitations of the uncertainty principle in determining quantum states, introduces new results for Gaussian Wigner distributions, and relates them to phase-space coarse-graining using quantum blobs.

Contribution

It provides a novel proof connecting Hardy’s uncertainty principle with Gaussian Wigner distributions and clarifies the role of the uncertainty principle in quantum state characterization.

Findings

Established a new relation between density matrices and the uncertainty principle.

Proved a new result for Gaussian-dominated Wigner distributions using Hardy's uncertainty principle.

Linked phase-space coarse-graining to quantum blobs.

Abstract

We discuss the relation between density matrices and the uncertainty principle; this allows us to justify and explain a recent statement by Man'ko et al. We thereafter use Hardy's uncertainty principle to prove a new result for Wigner distributions dominated by a Gaussian and we relate this result to the coarse-graining of phase-space by "quantum blobs".