[Paper Review] Dissipation, quantum coherence, and asymmetry of finite-time cross-correlations

Tan Van Vu, Van Tuan Vo+1 more|arXiv (Cornell University)|May 29, 2023

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper establishes finite-time thermodynamic bounds on the asymmetry of cross-correlations in classical and quantum Markov processes, showing that asymmetry is upper bounded by dissipation (entropy production) in classical systems and by both dissipation and quantum coherence in quantum systems. It proves linear growth in asymmetry over time for classical systems and exponential decay in the long-time limit, with quantum coherence enabling finite asymmetry even when entropy production vanishes.

ABSTRACT

Recent studies have revealed a deep connection between the asymmetry of cross-correlations and thermodynamic quantities in the short-time limit. In this study, we address the finite-time domain of the asymmetry for both open classical and quantum systems. Focusing on Markovian dynamics, we show that the asymmetry observed in finite-time cross-correlations is upper bounded by dissipation. We prove that, for classical systems in a steady state with arbitrary operational durations, the asymmetry exhibits, at most, linear growth over time, with the growth speed determined by the rates of entropy production and dynamical activity. In the long-time regime, the asymmetry exhibits exponential decay, with the decay rate determined by the spectral gap of the transition matrix. Remarkably, for quantum cases, quantum coherence is equally important as dissipation in constraining the asymmetry of correlations. We demonstrate an example where only quantum coherence bounds the asymmetry while the entropy production rate vanishes. Furthermore, we generalize the short-time bounds on correlation asymmetry, as reported by Shiraishi [Phys. Rev. E 108, L042103 (2023)] and Ohga et al. [Phys. Rev. Lett. 131, 077101 (2023)], to encompass finite-time scenarios. These findings offer novel insights into the thermodynamic aspects of correlation asymmetry.

Motivation & Objective

• To understand the thermodynamic constraints on correlation asymmetry in finite-time Markov processes.
• To extend short-time bounds on asymmetry to the full finite-time domain for both classical and quantum systems.
• To investigate the role of quantum coherence in constraining asymmetry when entropy production is zero.
• To generalize existing short-time bounds to broader dynamical regimes, including steady-state and long-time limits.
• To unify the thermodynamic description of asymmetry across discrete, continuous, and quantum dynamics.

Proposed method

• Analytical derivation of asymmetry bounds using master equations for Markov jump processes with time-independent transition matrices.
• Application of matrix exponential identities and integral representations to express correlation asymmetry in terms of transition matrix eigenvalues and eigenvectors.
• Use of operator norm inequalities and complex analysis bounds to upper-bound the asymmetry in terms of entropy production and dynamical activity.
• Derivation of long-time asymptotics via spectral gap analysis of the transition matrix, showing exponential decay of asymmetry.
• Extension to open quantum systems using Lindblad master equations to isolate the role of quantum coherence in asymmetry.
• Demonstration of a nontrivial case where quantum coherence alone bounds asymmetry despite zero entropy production.

Figure 1: Numerical illustration of the thermodynamic bounds on the asymmetry of cross-correlations in terms of (a) entropy production [cf. Eq. ( 7 )] and (b) thermodynamic affinity [cf. Eq. ( 14 )] in a three-state biochemical oscillation [ 34 ] . Observables $a$ and $b$ are randomly sampled in the

Experimental results

Research questions

• RQ1 How does correlation asymmetry evolve over finite time in classical Markov processes, and what thermodynamic quantities constrain it?
• RQ2 What is the role of quantum coherence in limiting asymmetry when entropy production vanishes?
• RQ3 Can short-time bounds on asymmetry be generalized to the entire finite-time domain?
• RQ4 How do spectral properties of the transition matrix, such as the spectral gap, affect long-time asymmetry decay?
• RQ5 In what scenarios does quantum coherence become the dominant constraint on asymmetry, independent of dissipation?

Key findings

• For classical Markov jump processes, correlation asymmetry grows at most linearly in time, with the growth rate determined by entropy production and dynamical activity.
• In the long-time limit, asymmetry decays exponentially, with the decay rate set by the spectral gap of the transition matrix.
• In quantum systems, asymmetry is bounded by both dissipation and quantum coherence, with coherence alone able to constrain asymmetry when entropy production is zero.
• A specific example is constructed where vanishing entropy production coexists with finite asymmetry due solely to quantum coherence.
• The paper generalizes prior short-time bounds on asymmetry to finite-time regimes, unifying classical and quantum descriptions.
• The derived bounds are universal across discrete- and continuous-time dynamics, and apply to both classical and open quantum systems.

Figure 2: Numerical illustration of the inevitable role of quantum coherence in restricting the asymmetry of cross-correlations in the three-level maser. Observables are given by $A=\outerproduct{1}{1}$ and $B=\outerproduct{3}{3}$ . Parameters are $\Delta=0.01$ , $\Omega=1$ , $\alpha_{h}=1$ , $\alph