G-Circulant Quantum Markov Semigroups

Abstract

We broaden the study of circulant Quantum Markov Semigroups (QMS). First, we introduce the notions of [Formula: see text]-circulant GKSL generator and [Formula: see text]-circulant QMS from the circulant case, corresponding to [Formula: see text], to an arbitrary finite group [Formula: see text]. Second, we show that each [Formula: see text]-circulant GKSL generator has a block-diagonal representation [Formula: see text], where [Formula: see text] is a [Formula: see text]-circulant matrix determined by some [Formula: see text]. Denoting by [Formula: see text] the subgroup of [Formula: see text] generated by the support of [Formula: see text], we prove that [Formula: see text] has its own block-diagonal matrix representation [Formula: see text] where [Formula: see text] is an irreducible [Formula: see text]-circulant matrix and [Formula: see text] is the index of [Formula: see text] in [Formula: see text]. Finally, we exploit such block representations to characterize the structure, steady states, and asymptotic evolution of [Formula: see text]-circulant QMSs.