Measuring the quality of a quantum reference frame: the relative entropy of frameness

TL;DR

This paper introduces the relative entropy of frameness as a new, easily computable measure of quantum reference frame quality, linking it to G-asymmetry and exploring its implications for quantum information processing.

Contribution

It defines a novel measure of quantum reference frame quality, the relative entropy of frameness, and establishes its relation to G-asymmetry and information bounds.

Findings

The relative entropy of frameness equals the G-asymmetry.

It provides an upper bound on mutual information about group elements.

The regularized measure is zero, indicating no interconversion rate.

Abstract

In the absence of a reference frame for transformations associated with a group G, any quantum state that is non-invariant under the action of G may serve as a token of the missing reference frame. We here introduce a novel measure of the quality of such a token: the relative entropy of frameness. This is defined as the relative entropy distance between the state of interest and the nearest G-invariant state. Unlike the relative entropy of entanglement, this quantity is straightforward to calculate and we find it to be precisely equal to the G-asymmetry, a measure of frameness introduced by Vaccaro et al. It is shown to provide an upper bound on the mutual information between the group element encoded into the token and the group element that may be extracted from it by measurement. In this sense, it quantifies the extent to which the token successfully simulates a full reference frame. We also show, that despite a suggestive analogy from entanglement theory, the regularized relative entropy of frameness is zero and therefore does not quantify the rate of interconversion between the token and some standard form of quantum reference frame. Finally, we show how these investigations yield a novel approach to bounding the relative entropy of entanglement.