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Stochastics

Contraction rates for quantum exclusion semigroups in Wasserstein-1 distance

Jinshu Chen, Jie Hao

Published Feb 11, 2026• 0 citations
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Abstract

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.

DOI
10.1080/17442508.2026.2627603

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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.
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.
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.
Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy the canonical anti-commutation relation (CAR) in equal time. This paper aimed to discuss the classical reduction and ergodicity of quantum exclusion semigroups constructed by QBN. We first study the classical reduction of the quantum semigroups to an Abelian algebra of diagonal elements and the space of off-diagonal elements. We then provide an explicit representation formula by separating the action on off-diagonal and diagonal operators, on which they are reduced to the semigroups of classical Markov chains. Finally, we prove that the asymptotic behavior of the quantum semigroups is equivalent to one of its associated Markov chains, and that the semigroups restricted to the off diagonal space of operators have a zero limit.
Quantum exclusion semigroup constructed form quantum Bernoulli noises can be written on diagonal and off-diagonal operators space, respectively. The diagonal parts describe a classical Markov process. In this paper, we first produce a decomposition into the sum of irreducible components determined by the class states of the associated classical Markov process and prove that each irreducible component is recurrent or transient if and only if its diagonal restriction is recurrent or transient. We then give conditions for existence of invariant states and study properties of invariant states such as detailed balance, uniqueness and convergence towards as time approaches infinity.
In this study, we first deal with the canonical anti-commutation relation algebra in which the Bernoulli creation and annihilation operators generate the structure of ultra hyperfinite C[Formula: see text]-algebra. This observation simplifies the construction of the minimal quantum Markov semigroup construct from quantum Bernoulli noises by taking a filter through the family gamma of finite subsets of the natural numbers. As a result, we obtain a Feller quantum Markov semigroup defined on the C[Formula: see text]-algebra, which in turn is extended to the entire algebra of bounded operators. Then, one condition for us to obtain a faithful normal invariant state of this semigroup is an equilibrium state (it satisfies the detailed balance conditions). Finally, we use the quantum Wasserstein distance to discuss the W-entropy rate of this invariant state.