Uncertainty Relations Relative to Phase-Space Quantum Reference Frames

TL;DR

This paper investigates how quantum reference frames affect Heisenberg's uncertainty relations, revealing that relative to a quantum frame, position and momentum can appear compatible, leading to new frame-dependent uncertainty bounds.

Contribution

It introduces a covariant phase-space observable for quantum reference frames and derives novel, frame-relative uncertainty relations, bridging quantum reference frames and incompatibility.

Findings

Position and momentum appear compatible relative to a QRF.

Standard uncertainty relations are recovered in the classical limit.

New frame-relative uncertainty relations are derived.

Abstract

We study Heisenberg's uncertainty relation relative to a quantum reference frame (QRF). We introduce the QRF as a covariant phase-space observable, show that when described relative to it, position and momentum appear compatible, and derive novel, frame-relative uncertainty relations. This is achieved by constructing a joint observable for position and momentum, and calculating the variances of its margins. We then verify that in the classical limit of the QRF, the standard uncertainty relations are recovered, fortifying claims that standard quantum theory must be understood relative to an external, classical frame. These results may open up new research directions at the interface between QRFs and incompatibility.