Thermodynamic entropic uncertainty relation

TL;DR

The paper addresses how to quantify the trade-off between entropy production and the uncertainty of trajectory-based observables in stochastic thermodynamics by introducing symmetry entropy $\\Lambda[P(\\Phi)] = H[P(\\Phi)] - H[P(|\\Phi|)]$ and proving the thermodynamic entropic uncertainty relation $\\Sigma \\ge \\ln 2 - \\Lambda[P(\\Phi)]$ for time-reversal antisymmetric observables $\\Phi(\\Gamma) = - \\Phi(\\Gamma^\\dagger)$. A corollary, $\\Sigma + H[P(\\Phi)] \\ge \\ln 2$, follows, with refinements when $\\Phi=0$ is allowed, and equivalence to a Jensen–Shannon divergence form $\\mathrm{JS}[P(\\Phi) \\| P(-\\Phi)] = \\ln 2 - \\Lambda[P(\\Phi)]$. The authors derive the bounds using KL monotonicity and Jensen’s inequality, extend to continuous observables, and discuss equality conditions (equilibrium). They apply the framework to the diffusion decision model, showing a fundamental energy–accuracy trade-off in stochastic decision-making for both discrete and continuous outputs, highlighting how increased asymmetry in the observable distribution necessitates higher entropy production. This entropy-centered TUR broadens nonequilibrium thermodynamics’ reach to categorical and continuous observables, with implications for information processing and potential quantum extensions.

Abstract

Thermodynamic uncertainty relations reveal a fundamental trade-off between the precision of a trajectory observable and entropy production, where the uncertainty of the observable is quantified by its variance. In information theory, Shannon entropy is a common measure of uncertainty. However, a clear quantitative relationship between the Shannon entropy of an observable and the entropy production in stochastic thermodynamics remains to be established. In this Letter, we show that an uncertainty relation can be formulated in terms of the Shannon entropy of an observable and the entropy production. We introduce symmetry entropy, an entropy measure that quantifies the symmetry of the observable distribution, and demonstrate that a greater asymmetry in the observable distribution requires higher entropy production. Specifically, we establish that the sum of the entropy production and the symmetry entropy cannot be less than $\ln 2$. As a corollary, we also prove that the sum of the entropy production and the Shannon entropy of the observable is no less than $\ln 2$. As an application, we demonstrate our relation in the diffusion decision model, revealing a fundamental trade-off between decision accuracy and entropy production in stochastic decision-making processes.

Figures (5)

• Figure 1: Conceptual representation of the thermodynamic entropic uncertainty relation. (a) Stochastic thermodynamic process. The thermodynamic entropic uncertainty relation considers a stochastic process, where each state transition is random. (b) Trajectory of the stochastic process shown in (a). $\Gamma$ denotes the time evolution of a realization of the process. $\Gamma^\dagger$ is the time-reversal of $\Gamma$. (c) Probability distribution of observable $\Phi(\Gamma)$. $\Phi(\Gamma)$ is arbitrary, provided it satisfies the time-reversal antisymmetric property [Eq. \ref{['eq:time_reversal']}].
• Figure 2: Examples of the symmetry entropy $\Lambda[P(\Phi)]$. Horizontal axes denote values of $\Phi$ (left column) and $|\Phi|$ (right column). The values of $\Phi$ always form pairs; $-1$ and $1$, and $-2$ and $2$ are two pairs in the examples. Vertical axes denote the probability distribution $P(\Phi)$ (left column) and $P(|\Phi|)$ (right column). (a) Distribution symmetric with respect to $\Phi=0$, where the probabilities of all of the pairs are identical. $P(|\Phi|)$ is different from the original distribution $P(\Phi)$ and thus $\Lambda[P(\Phi)]$ is $\ln 2$. (b) Asymmetric distribution where the probabilities of all of the pairs are fully biased. $P(|\Phi|)$ and $P(\Phi)$ are identical and thus $\Lambda[P(\Phi)]$ is $0$. (c) Asymmetric distribution where probabilities of all of the pairs are fully biased. The distributions $P(|\Phi|)$ and $P(\Phi)$ are different, but they effectively become the same if the labels for the pair $-1$ and $1$ are swapped. Therefore, they are essentially the same distribution. This results in $\Lambda[P(\Phi)]=0$.
• Figure 3: Description of the diffusion decision model. The green trajectory illustrates a realization that exceeds the threshold $\theta$, resulting in a decision of $+1$ ("yes" class). Conversely, the purple trajectory represents a realization that falls below $-\theta$, leading to a decision of $-1$ ("no" class). Trajectories that do not fall into either category correspond to a decision of $0$ ("neutral" class).
• Figure 4: Results for the diffusion decision model with the discrete-output observable [Eq. \ref{['eq:Phi_def_main']}]. Panels (a) and (b) show entropic quantities as a function of parameter $f$ for $\theta = 0$$\theta = 0.5$, respectively, with $A=1$ and $\tau=10$. Panel (c) shows entropic quantities as a function of $\tau$ with $\theta = 0$, $A = 1$, and $f=0.5$. In these figures, the blue dashed line shows the entropy production $\Sigma$, and the purple dot-dashed line represents $H[P(\Phi)]$. Their sum, $\Sigma + H[P(\Phi)]$, is also depicted by the red solid line. The orange dotted line represents $\ln 2$ in (a) and (c) and $(1-P(0))\ln 2$ in (b), which serves as the lower bound for $\Sigma + H[P(\Phi)]$ according to Eq. \ref{['eq:TEUR_binary_P0']}.
• Figure 5: Results for the diffusion decision model with the continuous-output observable [Eq. \ref{['eq:Phi_continuous']}]. Panels (a) and (b) show the entropic quantities versus $f$ for $A = 1$ and $A = 0.1$, respectively, with $\tau$ fixed at $10$. The blue dashed line shows the entropy production $\Sigma$, and the purple dot-dashed line represents $\Lambda[P(\Phi)]$, the symmetry entropy. Their sum, $\Sigma + \Lambda[P(\Phi)]$, is also depicted with the red solid line. The orange dotted line displays $\ln 2$, which serves as the lower bound for $\Sigma + \Lambda[P(\Phi)]$ according to Eq. \ref{['eq:main_result2']}.