Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system

TL;DR

This paper demonstrates that infinitesimal quantum reference frames are sufficient to determine the asymmetry properties of quantum systems, simplifying the analysis by reducing complex conditions to a single entropic criterion evaluated at the maximally mixed state.

Contribution

It shows that a complete set of entropic asymmetry conditions can be condensed into a single condition using infinitesimal reference frames, eliminating the need for macroscopic frames.

Findings

Asymmetry conditions have extensive redundancy and can be simplified.

Infinitesimal reference frames suffice to determine quantum system asymmetry.

Provides closed-form conditions for minimal depolarization to achieve covariant channels.

Abstract

Symmetry principles are fundamental in physics, and while they are well understood within Lagrangian mechanics, their impact on quantum channels has a range of open questions. The theory of asymmetry grew out of information-theoretic work on entanglement and quantum reference frames, and allows us to quantify the degree to which a quantum system encodes coordinates of a symmetry group. Recently, a complete set of entropic conditions was found for asymmetry in terms of correlations relative to infinitely many quantum reference frames. However, these conditions are difficult to use in practice and their physical implications unclear. In the present theoretical work, we show that this set of conditions has extensive redundancy, and one can restrict to reference frames forming any closed surface in the state space that has the maximally mixed state in its interior. This in turn implies that asymmetry can be reduced to just a single entropic condition evaluated at the maximally mixed state. Contrary to intuition, this shows that we do not need macroscopic, classical reference frames to determine the asymmetry properties of a quantum system, but instead infinitesimally small frames suffice. Building on this analysis, we provide simple, closed conditions to estimate the minimal depolarization needed to make a given quantum state accessible under channels covariant with any given symmetry group.