Operational thermodynamic uncertainty relations for time-dependent driving: a priori criterion for the tightest observable and extension to driven open quantum systems

For classical overdamped Langevin or Markov-jump dynamics driven by an arbitrary time-dependent protocol from an arbitrary initial state, Koyuk and Seifert [Phys. Rev. Lett. 125, 260604 (2020)] established an operational thermodynamic uncertainty relation (TUR): the mean, fluctuations, and response to a slight rescaling of the driving speed of any current or state observable yield a model-free lower bound on the mean total entropy-production rate. The task addresses two questions that this work posed as unresolved. First, there is no known a priori criterion for selecting which observable yields the tightest bound for a given driven process; later literature has charted bound quality only in the fast- and slow-driving limits, proved that correlation-based combinations tighten bounds, established model-based optimality for steady-state Markov jump processes via the pseudo entropy production (a cost computable only from the model's transition rates), and built data-driven machine-learning estimators, but an operational a priori selection criterion remains open. Second, whether and in what form the response-based relation extends to time-dependently driven open quantum systems is unresolved: the naive classical inequality is known to be violated by quantum coherence, and existing quantum precision bounds are either restricted to steady states or to slowly driven near-quasistatic regimes, or replace entropy production by activity-, asymmetry-, or quasiprobability-type costs, so the existence, form, and mechanism of a universal operational relation in the driven quantum regime remain open. Both components jointly determine how tight and how universal the operational TUR inference framework for time-dependent driving is.

Background

Stochastic thermodynamics describes small systems far from equilibrium in terms of stochastic trajectories. Its central quantity, the entropy production, measures dissipation and time-irreversibility, but is hard to determine without a microscopic model. Thermodynamic uncertainty relations (TURs) provide a way around this: for dynamics that are, on some possibly unobserved level, an overdamped Langevin process or a continuous-time Markov jump process, the relative fluctuations of any time-integrated current are lower-bounded by the entropy production. In the original form for non-equilibrium steady states (NESSs, time-independent driving with stationary currents), $D_J \sigma / J^2 \ge 1$, where $J$ is the mean of a time-integrated current, $D_J$ its scaled variance, and $\sigma$ the mean entropy-production rate. Because the bound requires only the mean and fluctuations of any single observable, it became a tool for thermodynamic inference: estimating dissipation in molecular motors, living systems, and other systems whose internal dynamics are unknown.

Koyuk and Seifert extended this to the broad remaining class of processes: systems driven by an arbitrary time-dependent control protocol $\lambda(vt)$, with overall speed parameter $v$, from an arbitrary initial state over a finite observation time $T$. They showed that for any current $J(T,v)$ with fluctuations $D_J(T,v)$, the mean total entropy-production rate satisfies $\sigma(T,v) \ge [J(T,v) + \Delta J(T,v)]^2 / D_J(T,v)$, where the differential operator $\Delta \equiv T\partial_T - v\partial_v$ encodes the response of the observable to a small change of the observation time and of the driving speed. Crucially, they also derived an analogous bound for state observables, $\sigma(T,v) \ge [\Delta A(T,v)]^2 / D_A(T,v)$, where $A$ is either the value of a state function at the final time or its time average along the trajectory, so that even observing only the final state of the system yields a bound. The operational character is the key feature: the inputs are the observable's mean, fluctuations, and response to a slight rescaling of the driving speed, all measurable without any knowledge of the system's dynamics or its coupling to the control. This distinguishes these genuine TURs from 'generalized' TURs that follow from the fluctuation theorem alone but are weaker and become trivial at long times.

Two limitations of this framework were posed as open in the original work and define the problem. First, tightness and observable selection: the bound holds for every observable but is generically not saturated, and the tightest bound comes from different observable classes in different regimes (in the paper's protein-unfolding example, a current observable is best at intermediate driving speeds, while state observables dominate at fast driving). The quality factor, i.e. the fraction of the true entropy production captured by the bound, must currently be explored by scanning many candidate observables, since no criterion is known that predicts a priori which observable will be strongest. Second, the quantum regime: the proof rests on classical trajectory ensembles and their behavior under time reversal, and the derivation of an analogous operational relation for time-dependently driven open quantum systems (systems coupled to reservoirs and controlled by time-dependent Hamiltonians or jump operators) was posed as an open challenge to theory. This is known to be subtle: for open quantum systems, coherence can reduce the thermodynamic cost of precision below the classical bound, so the naive transfer of the classical inequality fails and any valid quantum relation must incorporate genuinely quantum corrections. Existing quantum precision bounds typically hold for steady states, hold only in slowly driven near-quasistatic regimes, or replace entropy production by other costs such as the dynamical activity (a measure of the typical frequency of stochastic or quantum jumps), an asymmetry term between forward and backward processes, or quasiprobabilities (joint distributions of non-commuting observables that can take negative values).

Problem Statement

The problem concerns the operational thermodynamic uncertainty relations (TURs) for time-dependent driving derived by Koyuk and Seifert, and consists of two linked components whose common objective is to establish how tight and how universal this model-free inference framework is. Both components are mandatory: each may be settled either constructively or by a rigorous impossibility result, and the task is resolved in full only when both components are settled.

Component 1 (a priori tightness criterion). For classical overdamped Langevin dynamics or continuous-time Markov jump processes driven by an arbitrary time-dependent protocol $\lambda(vt)$ with speed parameter $v$ from an arbitrary initial state over observation time $T$, the Koyuk-Seifert relation gives, for any time-integrated current, $\sigma(T,v) \ge [J(T,v) + \Delta J(T,v)]^2 / D_J(T,v)$ with $\Delta \equiv T\partial_T - v\partial_v$, and an analogous bound $\sigma(T,v) \ge [\Delta A(T,v)]^2 / D_A(T,v)$ for state observables $A$ (final-time or time-averaged state functions), where $D_J$ and $D_A$ are the scaled variances of the corresponding trajectory observables, $\sigma$ is the mean total entropy-production rate, and the inputs are the observable's mean, fluctuations, and response to a slight rescaling of the driving speed. Establish, or rigorously disprove the existence of, a criterion that identifies a priori — from operationally accessible data, without solving the system's dynamics — which observable (current, state function, or combination) yields the tightest bound for a given driven process, going beyond the known limiting analyses of fast and slow driving. Consistent with the Background, such a criterion must predict the strongest observable without the exhaustive scan over candidate observables that constitutes current practice: merely measuring each candidate's mean, fluctuations, and speed-response and outputting whichever candidate maximizes the computed bound value reproduces the status quo and does not qualify as an a priori criterion.

Component 2 (extension to driven open quantum systems). Determine whether a universal, operationally accessible precision-dissipation relation of this response-based form — a lower bound on entropy production obtained from the mean, fluctuations, and speed-response of trajectory observables — exists for time-dependently driven open quantum systems under arbitrary protocols and from arbitrary initial states, and if so establish its explicit form and the mechanism by which quantum corrections (coherence, measurement backaction, quasiprobability negativity) enter. Any answer must be consistent with the established violations of the naive classical inequality in coherently driven quantum systems and must reduce to the classical relation in the appropriate limit.

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