Optimal measurements for symmetric quantum states with applications to optical quantum communication

Minimum probability of error (MPE) measurements are quantum mechanical measurements that discriminate between a set of candidate states and achieve the minimum error allowed by quantum mechanics. Conditions for a measurement to be an MPE measurement have been derived by Yuen, Kennedy and Lax. But finding explicit measurements that satisfy these conditions is a hard problem in general and even in cases where an explicit measurement is known, calculating the MPE of the measurement is not easy. In some cases, MPE measurements have been found such as when the states form a single orbit under a group action i.e., there is a transitive group action on the states. For such state sets, termed Geometrically Uniform (GU) in [1], it was shown that the 'pretty good measurement' (PGM) is optimal. However, calculating the MPE and other performance metrics for the PGM involves inverting large matrices, and it is therefore not easy in general to evaluate. Our first contribution is a formula for the MPE and conditional probabilities of GU sets using group representation theory. Next, we consider sets of pure states that have multiple orbits under the group action. Such states are termed compound geometrically uniform (CGU). CGU sets appear in many practical problems in quantum communication and imaging. For example, they include all linear codes formed using pure-state modulation constellations, which are known to achieve the ultimate (Holevo) capacity of optical communication. Optimal (MPE) measurement for general CGU sets are not known. In this paper, we show how our representation-theoretic description of optimal measurements for GU sets naturally generalizes to the CGU case. We show how to compute the optimal measurement for CGU sets by reducing the problem to solving a few simultaneous equations. The number of equations depends on the sizes of the multiplicity space of irreducible representations. For many group representations (such as those of several practical good linear codes), this is much more tractable than solving large semi-definite programs - which is what is needed to solve for MPE measurements for an arbitrary set of pure states using the Yuen-Kennedy-Lax conditions [2]. We show examples of the evaluation of optimal measurements for CGU states.