Uncertainty relations in stochastic processes: An information inequality approach

Abstract

The thermodynamic uncertainty relation is an inequality stating that it is impossible to attain higher precision than the bound defined by entropy production. In statistical inference theory, information inequalities assert that it is infeasible for any estimator to achieve an error smaller than the prescribed bound. Inspired by the similarity between the thermodynamic uncertainty relation and the information inequalities, we apply the latter to systems described by Langevin equations and derive the bound for the fluctuation of thermodynamic quantities. When applying the Cramér-Rao inequality, the obtained inequality reduces to the fluctuation-response inequality. We find that the thermodynamic uncertainty relation is a particular case of the Cramér-Rao inequality, in which the Fisher information is the total entropy production. Using the equality condition of the Cramér-Rao inequality, we find that the stochastic total entropy production is the only quantity which can attain equality in the thermodynamic uncertainty relation. Furthermore, we apply the Chapman-Robbins inequality and obtain a relation for the lower bound of the ratio between the variance and the sensitivity of systems in response to arbitrary perturbations.

Thermodynamic uncertainty relations in a linear system

“…A connection between discrete and continuous time uncertainty relations is shown in [34]. One can also see similar uncertainty relations in the context of discrete processes [35], multidimensional systems [36], Brownian motion in the tilted periodic potential [37], general Langevin systems [38], molecular motors [39], run and tumble processes [40], biochemical oscillations [41], interacting oscillators [42], effect of magnetic field [43], linear response [44], measurement and feedback control [45], information [46], underdamped Langevin dynamics [47], timedelayed Langevin systems [48], various systems [49], etc.. Recently, Hasegawa et al [50] found an uncertainty relation for the time-asymmetric observable for the system driven by a time-symmetric driving protocol using the steady state fluctuation theorem.…”

Thermodynamic uncertainty relations under arbitrary control protocols

“…Our results serve as a useful tool for estimation tasks in general Langevin systems. Information inequalities have successfully been applied to derive many important thermodynamic bounds, such as the sensitivity-precision trade-off [19], a quantum TUR [41], and the speed limit [42]. Extending our approach to other classical and quantum systems or finding a hyperaccurate observable [43] would be interesting.…”

“…With respect to attaining the derived bounds, one can show that the exact equality condition cannot be attained (from the equality condition of the Cauchy-Swartz inequality) [12]. Unlike original TUR, which is saturated near equilibrium [19], attaining the derived bounds is not ensured in such regimes. As shown later, the bounds become tight for long observation times.…”

“…For a given trajectory , let P θ ( ) denote the path probability of observing in the auxiliary dynamics. According to the Cramér-Rao inequality [19], the precision of the observable φ is bounded by the Fisher information as…”

“…Subsequently, the violation of the original bound has been found for other dynamics, e.g., for discrete-time Markov chains [9], transport systems [10], and underdamped dynamics [11,12]. TURs have been intensively refined in other contexts that include both classical and quantum systems [12][13][14][15][16][17][18][19][20][21][22][23]. A remarkable application of TURs is the estimation of entropy production [24].…”