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

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 , the cyclic group of M elements. Our result provides an operational interpretation of 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 and M ⩾ 4, it is equal to a quantity that is not additive in general, but instead can be super-additive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analogue.