Quantum Markov Semigroups with Hamiltonian Terms and Zeno Effect*

Quantum Markov semigroups (QMSs) model time-evolution of open quantum systems in dissipative environments. It is known that any finite-dimensional, continuous QMS with a particular detailed balance condition induces exponential decay of a state's relative entropy to its fixed point projection, following a modified logarithmic-Sobolev inequality (MLSI). In contrast, questions have lingered about semigroups that include a non-trivial Hamiltonian term (precluding detailed balance) and discrete time analogs. We generalize some ideas from semigroups to discrete processes that may include rotations. We then show counter-examples to MLSI and barriers to decay for continuous semigroups with Hamiltonians. When a Hamiltonian shrinks the overall fixed point subspace, strong noise induces a generalized Zeno effect. Counter-intuitively, decay to the long-time fixed point sometimes slows with increasing noise strength, as the Zeno and long-time limits differ. We analytically lower bound rates of convergence to long-time and Zeno fixed points. We numerically analyze simple examples. Finally, we present an experiment run on an IBM Quantum device, in which frequently depolarizing one qubit protects an interacting neighbor from dephasing.