Universal constraint for relaxation rates of semigroups of qubit Schwarz maps
Dariusz Chruściński, Gen Kimura, Farrukh Mukhamedov
TL;DR
The paper establishes a universal relaxation-rate constraint for qubit semigroups of unital Schwarz maps, showing that the standard CPTP bound $\Gamma_k \le \tfrac{1}{2}\Gamma$ generalizes to $\Gamma_k \le \tfrac{2}{3}\Gamma$ when Schwarz-positivity replaces complete positivity. It develops the formalism for Schwarz maps, analyzes Pauli and phase-covariant qubit semigroups to demonstrate tightness, and extends the result to general qubit Schwarz semigroups, using Bloch-operator form to relate relaxation rates to spectral properties. The authors derive spectral constraints for unital $\alpha$-positive maps (with $\alpha\in\{1,3/2,2\}$) and provide necessary conditions for Markovianity in the Schwarz case, including a demonstration that the bounds are tight but not universally sufficient. These results unify positivity, Schwarz, and CP regimes, offer tests for experimental verification, and hint at extensions to higher-dimensional systems.
Abstract
Unital qubit Schwarz maps interpolate between positive and completely positive maps. It is shown that relaxation rates of qubit semigroups of unital maps enjoying Schwarz property satisfy the universal constraint which provides a modification of the corresponding constraint known for completely positive semigroups. As an illustration we consider two paradigmatic qubit semigroups: Pauli dynamical maps and phase covariant dynamics. This result has two interesting implications: it provides a universal constraint for the spectra of qubit Schwarz maps and gives rise to a necessary condition for a Schwarz qubit map to be Markovian.