Quantum Frames of Reference and the Noncommutative Values of Observables

TL;DR

This paper explores how quantum reference frame transformations affect the values of observables, introducing noncommutative values to account for quantum fluctuations and entanglement, with applications to qubit systems.

Contribution

It develops a relational formulation of quantum reference frame transformations incorporating noncommutative observable values, including discrete spectrum cases.

Findings

Noncommutative values provide a definite description of observable changes.

Quantum fluctuations and entanglement are integral to reference frame transformations.

Analysis includes a case example with qubit systems.

Abstract

Based on a recent relational formulation of quantum reference frame transformations, especially with a case of quantum spatial translations in particular, we analyzed how the `value' of an observable for a fixed state change. That is the exact analog of the classical description, for example, of the value of the $x$-coordinate for a particle decrease by 2 units when we perform a translation of the reference frame putting the new origin at $x=2$. The essence of the quantum reference frame transformations is to have the quantum fluctuations, and even entanglement, of the physical object which serves as the (new) reference frame, taken into account. We illustrate how the recently introduced notion of the noncommutative values of quantum observables gives such a definite description successfully. Formulations, and an analysis of a case example in qubit systems, of analog transformations for observables with a discrete or finite spectrum is also presented. Issues about the evolving picture of the symmetry system of all quantum reference frame transformations discussed.