Invariant theoretic approach to uncertainty relations for quantum systems

TL;DR

This paper develops a covariant framework for deriving quantum uncertainty relations based on positive semidefiniteness, emphasizing symmetry actions and applying it to multi-mode and single-mode systems with detailed analysis of higher moments.

Contribution

It introduces a general covariant method for quantum uncertainty relations, unifying various cases and emphasizing symmetry invariance, including multi-mode and polynomial observables.

Findings

Derived $Sp(2n, R)$-covariant multi-mode uncertainty relations.

Analyzed fourth order moments with Lorentz covariance.

Established covariance of uncertainty relations under symmetry operations.

Abstract

We present a general framework and procedure to derive uncertainty relations for observables of quantum systems in a covariant manner. All such relations are consequences of the positive semidefiniteness of the density matrix of a general quantum state. Particular emphasis is given to the action of unitary symmetry operations of the system on the chosen observables, and the covariance of the uncertainty relations under these operations. The general method is applied to the case of an $n$-mode system to recover the $Sp(2n,\,R)$-covariant multi mode generalization of the single mode Schr\"{o}dinger-Robertson Uncertainty Principle; and to the set of all polynomials in canonical variables for a single mode system. In the latter situation, the case of the fourth order moments is analyzed in detail, exploiting covariance under the homogeneous Lorentz group $SO(2,\,1)$ of which the symplectic group $Sp(2,\,R)$ is the double cover.