Exponential Convergence Rate of the Quantum Exclusion Semigroup Constructed by Quantum Bernoulli Noise

The quantum exclusion semigroup constructed from quantum Bernoulli noise can be decomposed into actions on the diagonal and off-diagonal operator spaces. Its action on the diagonal subspace describes a classical Markov process. This paper analyzes the convergence properties of this semigroup. A key result shows that the exponential convergence rate of the quantum semigroup (quantified via the spectral gap) matches precisely the convergence rate of its classical diagonal counterpart towards the unique invariant measure. Furthermore, we provide explicit computations or estimates of the exponential convergence rate in specific examples.