G-Circulant Quantum Markov Semigroups

We broaden the study of circulant Quantum Markov Semigroups (QMS). First, we introduce the notions of G-circulant GKSL generator and G-circulant QMS from the circulant case, corresponding to Z(n), to an arbitrary finite group G. Second, we show that each G-circulant GKSL generator has a block-diagonal representation Q circle times 1(G), where Q is a G-circulant matrix determined by some a is an element of l(2)(G). Denoting by H the subgroup of G generated by the support of alpha, we prove that Q has its own block-diagonal matrix representation (Q) over tilde circle times 1(r) where (Q) over tilde is an irreducible H-circulant matrix and r is the index of H in G. Finally, we exploit such block representations to characterize the structure, steady states, and asymptotic evolution of G-circulant QMSs.