  The minimum probability of error (MPE) measurement discriminates between a
set of candidate quantum states with the minimum average error probability
allowed by quantum mechanics. Conditions for a measurement to be MPE were
derived by Yuen, Kennedy and Lax (YKL). MPE measurements have been found for
states that form a single orbit under a group action, i.e., there is a
transitive group action on the states in the set. For such state sets, termed
geometrically uniform (GU) by Forney, it was shown that the `pretty good
measurement' (PGM) attains the MPE. Even so, evaluating the actual probability
of error (and other performance metrics) attained by the PGM on a GU set
involves inverting large matrices, and is not easy in general. 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). MPE measurements 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 MPE 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 common
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 the YKL conditions numerically for arbitrary state
sets. We show how to evaluate MPE measurements for CGU states for some examples
relevant to quantum-limited classical optical communication.
