Phase-space noncommutative extension of the Robertson-Schroedinger formulation of Ozawa's uncertainty principle

TL;DR

This paper explores how noncommutative quantum mechanics modifies Ozawa's uncertainty principle, revealing potential violations and incompatibilities that could be experimentally tested or challenge existing models.

Contribution

It derives a matrix form of Ozawa's uncertainty principle within noncommutative phase-space and analyzes the NC corrections for specific measurement interactions.

Findings

NC corrections can cause violations of OUP.

Some measurement models become incompatible with NC quantum mechanics.

Potential experimental tests for OUP violation are identified.

Abstract

We revisit Ozawa's uncertainty principle (OUP) in the framework of noncommutative (NC) quantum mechanics. We derive a matrix version of OUP accommodating any NC structure in the phase-space, and compute NC corrections to lowest order for two measurement interactions, namely the Backaction Evading Quadrature Amplifier and Noiseless Quadrature Transducers. These NC corrections alter the nature of the measurement interaction, as a noiseless interaction may acquire noise, and an interaction of independent intervention may become dependent of the object system. However the most striking result is that noncommutativity may lead to a violation of the OUP itself. The NC corrections for the Backaction Evading Quadrature Amplifier reveal a new term which may potentially be amplified in such a way that the violation of the OUP becomes experimentally testable. On the other hand, the NC corrections to the Noiseless Quadrature Transducer shows an incompatibility of this model with NC quantum mechanics. We discuss the implications of this incompatibility for NC quantum mechanics and for Ozawa's uncertainty principle.