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

TL;DR

This work extends the theory of exponential entropy decay in quantum Markov dynamics beyond detailed balance by incorporating Hamiltonian drift into dissipative Markov semigroups. It establishes that adding a Hamiltonian to a dissipator with CMLSI can preserve or destroy CMLSI depending on commutation with the fixed-point projection, and in general there exists a decoherence-free subspace toward which decay occurs, even when CMLSI fails at early times. For unital, finite-dimensional QMS, a universal finite-time exponential-like decay bound is proved with a timescale-dependent constant ετ, revealing a robust decay mechanism beyond strict CMLSI. Moreover, a self-restricting-noise phenomenon is shown: in regimes where dissipation dominates, the eventual decay rate is bounded inversely by the dissipative strength, highlighting how strong damping can inhibit noise spread, a behavior with potential implications for quantum control and error suppression.

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.

Figures (2)

• Figure 1: Relative entropy of a 4-qubit spin chain to (fully decayed) complete mixture: (1) Spin chain illustration. The noised qubit is on top and shaded yellow. The 3 qubits below are shaded green with nearest-neighbor interactions. (2) Relative entropy vs. time with input $(\hat{1}/2)^{\otimes 3} \otimes |{0}\rangle\langle{0}|$, where the legend notes $\gamma$ in Equation \ref{['eq:chainl']}. (3) Relative entropy vs. time averaged over 50 randomly selected input densities.
• Figure 2: Relative entropy of qubit $B$'s state to the fixed point, which for this input is completely mixed, computed numerically and compared to theoretical lower bounds. The decay rate is $\lambda_0$ as in Equation \ref{['eq:exprl']}. The plot is scaled logarithmically on the horizontal axis and linearly on the vertical.

Theorems & Definitions (56)

• Proposition 1.1: Introduction Version of Proposition \ref{['prop:decayhaml']}
• Theorem 1.2: Introduction Version of Theorem \ref{['thm:maindecay']}
• Theorem 1.3: Introduction Version of Theorem \ref{['thm:zvscmlsi']}
• Lemma 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof