Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces

TL;DR

This paper generalizes entropic uncertainty relations for multiple variables, including non-canonical and incompatible measurements, using a matrix of commutators, and extends results to Rényi entropies and covariance-based relations.

Contribution

It introduces a multidimensional entropic uncertainty relation based on a commutator matrix applicable to non-canonical variables and linear canonical transforms, extending previous relations.

Findings

Derived a bound using a commutator matrix determinant.

Extended the relation to Rényi entropies.

Proved a covariance-based uncertainty relation generalizing Robertson's.

Abstract

The uncertainty relation for continuous variables due to Byalinicki-Birula and Mycielski expresses the complementarity between two $n$-uples of canonically conjugate variables $(x_1,x_2,\cdots x_n)$ and $(p_1,p_2,\cdots p_n)$ in terms of Shannon differential entropy. Here, we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two $n$-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured variables. This uncertainty relation also captures the complementarity between any two incompatible linear canonical transforms, the bound being written in terms of the corresponding symplectic matrices in phase space. Finally, we extend this uncertainty relation to R\'enyi entropies and also prove a covariance-based uncertainty relation which generalizes Robertson relation.