Accessible information and optimal strategies for real symmetrical quantum sources

We study the problem of optimizing the Shannon mutual information for sources of real quantum states, i.e., sources for which there is a basis in which all of the states have only real components. We consider in detail the sources EM of M equiprobable quantum bit (qubit) states lying symmetrically around the great circle of real states on the Bloch sphere and give a variety of explicit optimal strategies. We also consider general real group-covariant sources for which the group acts irreducibly on the subset of all real states and prove the existence of a real group-covariant optimal strategy, extending a theorem of Davies [E. B. Davies, IEEE. Inf. Theory IT-24, 596 (1978)]. Finally, we propose an optical scheme to implement our optimal strategies, simple enough to be realized with present technology.