Trajectory sharpening under quantum reference frame transformations

Abstract

Quantum reference frame (QRF) transformations relate the description of a physical system to different quantum observers, but their connection to concrete experimental phenomena remains largely unclear. Here we show that the trajectory-sharpening effect obtained by correlating a probed system with an effective negative-mass reference admits a natural interpretation as a QRF transformation to the frame of a negative-mass system. Using a minimal toy model of free particles with signed masses, we identify the Einstein-Podolsky-Rosen-type observables responsible for this effect with a pair of commuting relational observables that, under the instantaneous QRF transformation, are mapped to operators local to distinct subsystems. We then show that the corresponding relational trajectory is reproduced by the dynamical QRF transformation, where the transformed relational position acquires an explicit time dependence. We introduce QRF-invariant sharpness measures for relational trajectories and derive monogamy relations showing that a trajectory cannot be made simultaneously arbitrarily sharp with respect to two distinct negative-mass frames. We further show that trajectory sharpness can instead reside in a collective reference mode encoded across two physical systems, even when neither constituent alone provides an arbitrarily sharp reference, so that sharpness can be distributed rather than localized in a single constituent frame. These results provide an operational, experimentally relevant interpretation of QRF transformations for continuous-variable systems.