Alignment of reference frames and an operational interpretation for the G-asymmetry

TL;DR

This paper characterizes optimal quantum states and measurements for aligning reference frames associated with specific groups, providing an operational interpretation for the G-asymmetry and revealing quantum superadditivity phenomena.

Contribution

It offers an information-theoretic interpretation of G-asymmetry and analyzes the asymptotic behavior of reference frame alignment for groups U(1) and Z_M.

Findings

Accessible information approaches the Holevo bound with many copies.

Alignment rate equals the regularized G-asymmetry.

Quantum superadditivity occurs for Z_M with M ≥ 4.

Abstract

We determine the quantum states and measurements that optimize the accessible information in a reference frame alignment protocol associated with the groups U(1), corresponding to a phase reference, and $\mathbb{Z}_M$, the cyclic group of $M$ elements. Our result provides an operational interpretation for the $G$-asymmetry which is information-theoretic and which was thus far lacking. In particular, we show that in the limit of many copies of the bounded-size quantum reference frame, the accessible information approaches the Holevo bound. This implies that the rate of alignment of reference frames, measured by the (linearized) accessible information per system, is equal to the regularized, linearized $G$-asymmetry. The latter quantity is equal to the number variance in the case where $G=U(1)$. Quite surprisingly, for the case where $G=\mathbb{Z}_{M}$ and $M\geq 4$, it is equal to a quantity that is not additive in general, but instead can be superadditive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analog.