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

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

2009-01-07

arxiv.org/abs/0901.0943v2

Phys. Rev. A 80, 012307 (2009)

Physics

Quantum Physics

12 pages; many improvements in v2 including a weakening of the assumptions of the main theorem and better upper bounds for bot

Scientific paper

10.1103/PhysRevA.80.012307

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.

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