Universal constraint for relaxation rates of semigroups of qubit Schwarz maps

Unital qubit Schwarz maps interpolate between positive and completelypositive maps. It is shown that relaxation rates of qubit semigroups of unitalmaps enjoying Schwarz property satisfy the universal constraint which providesa modification of the corresponding constraint known for completely positivesemigroups. As an illustration we consider two paradigmatic qubit semigroups:Pauli dynamical maps and phase covariant dynamics. This result has twointeresting implications: it provides a universal constraint for the spectra ofqubit Schwarz maps and gives rise to a necessary condition for a Schwarz qubitmap to be Markovian.