Uncertainty Relations Relative to Phase-Space Quantum Reference Frames

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.