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Open Systems & Information Dynamics

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

Jinshu Chen, Siqi Fan

Published Jun 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s1230161225500064

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The quantum exclusion semigroup constructed from quantum Bernoulli noise decomposes into distinct actions on diagonal and off-diagonal operator spaces. Its restriction to the diagonal subspace corresponds to a classical Markov process. In this paper, we establish explicit contraction rates for this semigroup under the Wasserstein-1 distance. For arbitrary initial states, the Wasserstein distance between evolved states decays exponentially with a rate combining the Wasserstein curvature of the classical Markov process and the decay rate of quantum off-diagonal operators. The contraction features a prefactor constant containing a quantum correction term that quantifies how coherence affects the convergence dynamics.
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We study a class of generic quantum Markov semigroups on the algebra of all bounded operators on a Hilbert space h arising from the stochastic limit of a discrete system with generic Hamiltonian HS, acting on h, interacting with a Gaussian, gauge invariant, reservoir. The selfadjoint operator HS determines a privileged orthonormal basis of h. These semigroups leave invariant diagonal and off-diagonal bounded operators with respect to this basis. The action on diagonal operators describes a classical Markov jump process. We construct generic semigroups from their formal generators by the minimal semigroup method and discuss their conservativity (uniqueness). When the semigroup is irreducible we prove uniqueness of the equilibrium state and show that, starting from an arbitrary initial state, the semigroup converges towards this state. We also prove that the exponential speed of convergence of the quantum Markov semigroup coincides with the exponential speed of convergence of the classical (diagonal) semigroup towards its unique invariant measure. The exponential speed is computed or estimated in some examples.