Replica thermodynamic trade-off relations: Entropic bounds on network diffusion and trajectory observables

TL;DR

The paper introduces replica Markov processes to derive entropic trade-off relations for nonlinear probability-functionals, notably Rényi entropies. By constructing K-replica dynamics, it proves bounds on relative moments that depend on the dynamical activity, then converts these into entropic bounds for a random walk on a network and for trajectory observables, with the diffusion bound depending only on the initial local escape rate. It extends these results to quantum systems via continuous measurement, defining a quantum dynamical activity B(τ) and obtaining quantum Rényi bounds for trajectory observables. Overall, the work provides computable, entropic constraints on diffusion and trajectory-uncertainty that complement existing variance-based thermodynamic uncertainty relations.

Abstract

We introduce replica Markov processes to derive thermodynamic trade-off relations for nonlinear functions of probability distributions. In conventional thermodynamic trade-off relations, the quantities of interest are linear in the underlying probability distribution. Some important information-theoretic quantities, such as Rényi entropies, are nonlinear; however, such nonlinearities are generally more difficult to handle. Inspired by replica techniques used in quantum information and spin-glass theory, we construct Markovian dynamics of identical replicas and derive a lower bound on relative moments in terms of the dynamical activity. We apply our general result to two scenarios. First, for a random walker on a network, we derive an upper bound on the Rényi entropy of the position distribution of the walker, which quantifies the extent of diffusion on the network. Remarkably, the bound is expressed solely in terms of escape rate from the initial node, and thus depends only on local information. Second, we consider trajectory observables in Markov processes and obtain an upper bound on the Rényi entropy of the distribution of these observables, again in terms of the dynamical activity. This provides an entropic characterization of uncertainty that generalizes existing variance-based thermodynamic uncertainty relations.

Figures (5)

• Figure 1: (a) Single process scenario. This scenario is usually considered in stochastic thermodynamics. Given a Markov process, we consider stochastic trajectories, which are a random realization of the Markov process. We are interested in stochastic quantities associated with trajectories, whose statistics are linear in the trajectory probability. (b) $K=2$ replica process scenario. We consider two identical Markov processes. The state space is a product space of each Markov process and the trajectories are defined on the product space. We are interested in stochastic quantities of the replica process, which become nonlinear quantities in the original single process.
• Figure 2: Illustration of (a) the network diffusion bound [Eq. \ref{['eq:renyi_alpha_bound_state']}] and (b) the trajectory observable bound [Eq. \ref{['eq:renyi_alpha_bound']}]. (a) Diffusion in a random walk on a network. A random walker starts from a node in the network, which is represented by dark red. At time $t=\tau$, the random walker diffuses over the network and the probability $p_\mu(\tau)$ of the walker position is expressed by red, where darker color corresponds to higher probability. The network diffusion bound provides upper bound to the Rényi entropy of $p_\mu(\tau)$. (b) Trajectory observable bound in a Markov process. For the Rényi entropy bound for trajectory observables, we consider a trajectory $\Gamma)$ and its associated observable $N(\Gamma)$. The Rényi entropy is calculated for the distribution $N(\Gamma)$.
• Figure 3: Illustration of the dependence of the Rényi entropic bound. The Rényi entropic bound for the network diffusion depends on $\sum_{\mu(\mu\ne\mu_{0})}W_{\mu\mu_{0}}$, where $B_{\mu_0}$ is the initial state, the local escape rate from $B_{\mu_0}$. The local escape rate from $B_{\mu_0}$ is the sum of transition rates to which the nodes reachable from the initial state with one jump. Suppose the initial state is $B_1$, which is colored with dark red. Then, the reachable states are represented by purple nodes. Therefore, the local escape rate becomes $\sum_{\mu(\mu\ne\mu_{0})}W_{\mu\mu_{0}} = W_{21}+W_{31}+W_{41}$.
• Figure 4: Numerical verification of the Rényi entropy bound in Eq. \ref{['eq:renyi_alpha_bound_state']}. The Rényi entropy $H_\alpha[p_\mu(\tau)]$ versus $\frac{\alpha\tau}{\alpha-1}\sum_{\mu(\mu\neq\mu_{0})}W_{\mu\mu_{0}}$ is plotted for (a) $\alpha=1.5$, (b) $\alpha=2$, and (c) $\alpha=3$. Random realizations are shown with points and the dashed line indicates the equality case of Eq. \ref{['eq:renyi_alpha_bound_state']}. In (a)-(c), the simulations begin by selecting the number of state $D$ between $3$ and $50$. Then, the transition rate $W_{\mu^\prime\mu}$ is randomly produced, and $\tau$ is chosen within the range of $0.1$ to $10$.
• Figure 5: Numerical verification of the Rényi entropy bound in Eq. \ref{['eq:renyi_alpha_bound']}. The Rényi entropy $H_\alpha[P(N_\mathrm{c})]$ versus $\alpha \tau \mathfrak{a}(t=0)/(\alpha-1)$ is plotted for (a) $\alpha=1.5$, (b) $\alpha=2$, and (c) $\alpha=3$. Random realizations are shown with points and the solid line indicates the equality case of Eq. \ref{['eq:renyi_alpha_bound']}. In (a)-(c), classical simulations begin by selecting the number of states $D$ between $2$ and $5$. Then, the transition rate $W_{\mu^\prime\mu}$ is randomly produced, and $\tau$ is chosen within the range of $0.1$ to $10$. The observable $N_\mathrm{c}(\Gamma)$ is the number of total jumps within $[0,\tau]$. The Rényi entropy of the observable is estimated by averaging the results of $10^3$ simulations.