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 ${\mathcal{E}}_M$ 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.

• Received 22 December 1998

DOI:https://doi.org/10.1103/PhysRevA.59.3325