Moments of the Wigner Distribution and a Generalized Uncertainty Principle

TL;DR

This paper derives a generalized uncertainty principle based on the moments of the Wigner distribution, extending the conventional uncertainty principle to all orders and highlighting implications for quantum state reconstruction.

Contribution

It introduces a canonically invariant, explicit set of constraints on Wigner distribution moments, generalizing the uncertainty principle beyond second order.

Findings

Provides a new set of moment constraints for Wigner distributions

Generalizes the uncertainty principle to all moments

Notes potential application in quantum state tomography

Abstract

The nonnegativity of the density operator of a state is faithfully coded in its Wigner distribution, and this places constraints on the moments of the Wigner distribution. These constraints are presented in a canonically invariant form which is both concise and explicit. Since the conventional uncertainty principle is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. Possible application in quantum state reconstruction using optical homodyne tomography is noted.