Self-restricting Noise and Exponential Relative Entropy Decay Under Unital Quantum Markov Semigroups

TL;DR

This paper investigates the decay behavior of open quantum systems under combined dissipative and Hamiltonian dynamics, revealing conditions for exponential decay and introducing the concept of self-restricting noise.

Contribution

It demonstrates that exponential decay can re-emerge in unital quantum Markov semigroups despite initial failures, and introduces the concept of self-restricting noise where strong dissipation limits noise spread.

Findings

Exponential decay reappears at finite times in certain quantum semigroups.

Counterexamples show failure of CMLSI-like decay in combined dissipative and Hamiltonian processes.

Strong dissipation can bound the rate of decay, preventing noise from spreading beyond initial subspaces.

Abstract

States of open quantum systems often decay continuously under environmental interactions. Quantum Markov semigroups model such processes in dissipative environments. It is known that finite-dimensional quantum Markov semigroups with GNS detailed balance universally obey complete modified logarithmic Sobolev inequalities (CMLSIs), yielding exponential decay of relative entropy to a subspace of fixed point states. We analyze continuous processes that combine dissipative with Hamiltonian time-evolution, precluding this notion of detailed balance. First, we find counterexamples to CMLSI-like decay for these processes and determine conditions under which it fails. In contrast, we prove that despite its absence at early times, exponential decay re-appears for unital, finite-dimensional quantum Markov semigroups at finite timescales. Finally, we show that when dissipation is much stronger than Hamiltonian time-evolution, the rate of eventual, exponential decay toward the semigroup's decoherence-free subspace is bounded inversely in the decay rate of the dissipative part alone. Dubbed self-restricting noise, this inverse relationship arises when strong damping suppresses effects that would otherwise spread noise beyond its initial subspace.