Kinetic Uncertainty Relation (KUR)

• KUR is a set of precision bounds for stochastic observables defined by dynamical activity rather than entropy production.
• It applies to Markov jump processes, first-passage times, and extends to open quantum and collective many-body systems.
• Derivations use fluctuation-response inequalities and Fisher information, offering complementary insights to thermodynamic uncertainty relations.

The kinetic uncertainty relation (KUR) is a family of precision bounds for stochastic observables in which the controlling cost is dynamical activity rather than entropy production. In its classical form, the activity is the mean number of jumps in a Markov jump process, sometimes described as frenesy, and the KUR bounds the attainable signal-to-noise ratio or relative fluctuation of time-integrated observables by that activity (Terlizzi et al., 2018). Subsequent work has extended the notion to first-passage observables, stochastic clocks, open quantum systems, coherent transport, and, more recently, collective dissipative quantum many-body systems (Yunoki et al., 7 Apr 2026).

1. Classical statement and scope

For discrete-state continuous-time Markov jump processes, the KUR can be written in finite-time form as

$$g_X(t)=\frac{\dot X_t^2}{\langle \Delta X_t^2\rangle/t}\le \kappa(t), \qquad \kappa(t)=\frac{\langle \mathcal{A}_t\rangle}{t},$$

or equivalently

$$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$

where $\mathcal{A}_t$ is the total number of jumps up to time $t$ (Terlizzi et al., 2018). In this formulation, precision is controlled by how kinetically active the process is, independently of any thermodynamic interpretation.

A related current formulation uses a generalized anti-symmetric time-integrated current $\mathcal{J}_d$ and the total dynamical activity $A_\tau$: $$A_\tau \ge \frac{(\nabla[\mathcal{J}_d])^2}{\mathrm{Var}[\mathcal{J}_d]},$$ with $\nabla=\tau\partial_\tau-v\partial_v$ (Nishiyama, 2022). This emphasizes that the KUR is naturally expressed as an upper bound on precision $(\text{mean})^2/\text{variance}$ by a purely kinetic object.

The classical KUR differs structurally from the thermodynamic uncertainty relation (TUR). The TUR uses entropy production, whereas the KUR uses activity. The data explicitly state that the KUR applies even when entropy production is zero and even for genuinely irreversible processes for which entropy flow may not be defined, making it applicable to equilibrium systems, nonequilibrium systems, and irreversible population dynamics alike (Terlizzi et al., 2018).

2. Derivation routes and relation to thermodynamic bounds

One derivation of the KUR is based on the fluctuation-response inequality of Dechant and Sasa under a global rescaling of all transition rates $k_{ij}\to (1+\alpha)k_{ij}$; under this perturbation, the Kullback-Leibler cost reduces to the mean activity, yielding a purely kinetic bound (Terlizzi et al., 2018). This route is trajectory-based and does not require local detailed balance.

A more general framework derives KURs directly from stochastic representations. For an observable $$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$0 and an auxiliary zero-mean stochastic observable $$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$1, Cauchy-Schwarz yields a family of uncertainty relations, including

$$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$2

with $$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$3 collecting finite-time and protocol-dependent corrections (Kwon et al., 2024). In this framework, the KUR emerges as an intrinsic consequence of the trajectory-level stochasticity rather than of an auxiliary deterministic evolution equation.

The KUR is complementary to the TUR, and recent work formalizes this complementarity through the unified thermodynamic-kinetic uncertainty relation (TKUR),

$$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$4

where $$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$5 is the total entropy production, $$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$6 the total dynamical activity, and $$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$7 the inverse of $$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$8 (Vo et al., 2022). The limiting regimes recover the TUR near equilibrium and the KUR far from equilibrium. A further information-theoretic analysis shows that the TKUR is the tightest member of a class of bounds built from entropy production, dynamical activity, and observation time, and rewrites it as an inequality between two Kullback-Leibler divergences (Nishiyama, 2022).

3. Sharper classical bounds, first-passage formulations, and stochastic clocks

A central development in the recent literature is that the standard KUR is not generally the tight classical bound in steady state. The clock uncertainty relation (CUR) replaces activity by the inverse mean residual time,

$$\frac{(t\dot X_t)^2}{\mathrm{Var}(X_t)}\le \langle \mathcal{A}_t\rangle,$$9

whereas the standard KUR reads $\mathcal{A}_t$0 with $\mathcal{A}_t$1 (Prech et al., 2024). Since $\mathcal{A}_t$2, the CUR is provably tighter, and the data further state that it can be saturated by explicit counting observables with weights $\mathcal{A}_t$3 (Prech et al., 2024).

An asymptotic refinement, described as an ultimate KUR, replaces the inverse arithmetic mean of decay rates by a harmonic-mean bound: $\mathcal{A}_t$4 This improved bound is always tighter than the activity-based KUR and can always be saturated by the optimal flux

$\mathcal{A}_t$5

(Macieszczak, 2024). The same work extends these results to semi-Markov processes, including quantum reset processes.

The KUR also has a first-passage-time formulation. For a time-homogeneous CTMC and the first passage time $\mathcal{A}_t$6 of a time-integrated current through a threshold, the bound

$\mathcal{A}_t$7

holds, with $\mathcal{A}_t$8 the total number of jumps up to the stopping time (Hiura et al., 2021). The data explicitly note that this result is valid for any threshold and for general time-homogeneous Markov chains, not only in asymptotic large-threshold regimes.

4. Quantum generalizations and mechanisms behind violations of classical bounds

In open quantum systems, the classical activity-based KUR can fail, and the literature identifies several distinct mechanisms. A stochastic-representation framework extends KURs to Markovian open quantum systems by unraveling the dynamics and yields bounds that are described as physically more accessible and tighter in regimes where quantum effects are significant (Kwon et al., 2024).

A metrological route introduces the Fisher information of time-rescaled quantum trajectories. The corresponding trajectory Fisher information is upper bounded by the SLD Fisher information, which can be written as

$\mathcal{A}_t$9

where $t$0 is the dynamical activity and $t$1 is a quantum correction (Nakajima et al., 2023). This gives a computable upper bound even when direct trajectory-level Fisher-information evaluation is intractable.

A further refinement is the unraveling-dependent $t$2-KUR. For jump unravelings,

$t$3

while for diffusive unravelings the prefactor becomes $t$4 (Prech et al., 2024). In the data, $t$5 recovers the classical KUR, and classical-bound violations are possible only when $t$6. This directly shows that coherence does not act in a measurement-independent way: the precise role of coherence depends on the unraveling.

The dependence on decoherence mechanisms is explicit in maser heat engines. Two three-level maser configurations, often treated as classically equivalent, behave differently once spontaneous emission is included: KUR violations occur only in Model II because Model I has faster decoherence, while Model II lacks the spontaneous-emission terms in its decoherence rate and thus supports longer-lived coherence (Singh et al., 26 Aug 2025). The data further state that sub-Poissonian Fano factors are necessary but not sufficient for violation.

Two additional quantum extensions enlarge the kinetic sector itself. The response KUR for Lindblad steady states,

$t$7

adds a perturbation-induced inter-subspace transition term $t$8 to the dynamical-activity contribution (Liu et al., 9 Jan 2025). The susceptibility-kinetic uncertainty relation introduces a partial dynamical activity from the quantum Fisher information associated with coupling rescaling and gives

$t$9

with a susceptibility term that is experimentally accessible by tuning the system-reservoir coupling (Palmqvist et al., 1 Jul 2026).

5. Quantum transport, multiple observables, and finite-time regimes

In coherent multi-terminal transport, the transport literature distinguishes sharply between classical, bosonic, and fermionic regimes. In the classical limit of scattering theory,

$\mathcal{J}_d$0

so the particle-current noise acts as the activity (Palmqvist et al., 2024). In the quantum regime, bosonic bunching produces positive quantum noise, so the classical KUR persists but can be loose; the same work derives a bosonic KUR-like relation with an explicit bandwidth correction. Fermionic antibunching produces negative quantum noise, and the classical KUR can then be violated; the corresponding fermionic KUR-like relation involves the minimum reflection probability $\mathcal{J}_d$1 (Palmqvist et al., 2024).

At strong coupling, the notion of activity must itself be generalized. A steady-state QKUR valid at arbitrary coupling uses a generalized dynamical activity defined from exchange-rate fluctuations and yields

$\mathcal{J}_d$2

with explicit Green-function and Landauer-Büttiker expressions (Blasi et al., 19 May 2025). The same data state that this bound is never violated and becomes tight far from equilibrium, in contrast to classical KURs at strong coupling.

Transport KURs can also be combined with thermodynamic bounds. In multi-terminal quantum transport, a combined thermodynamic-kinetic uncertainty relation remains valid for fermionic and bosonic systems and even in the absence of time-reversal symmetry, with observables such as currents, noise, bandwidth, and reflection coefficients entering directly; this formulation can be inverted to estimate entropy production from transport data (Palmqvist et al., 7 Apr 2025).

For multiple observables, a multidimensional quantum KUR uses a Fisher information matrix. For two currents, the determinant of the covariance matrix is bounded below by a denominator containing $\mathcal{J}_d$3, where the off-diagonal Fisher element $\mathcal{J}_d$4 vanishes for classical stochastic dynamics and acts as a quantum signature of correlations (Moreira et al., 2024). The data state that the multidimensional bound can even be saturated when the observables are perfectly correlated.

Finite-time behavior adds another layer of nonuniqueness. In a two-qubit entanglement engine, there are more than one possible definitions for KURs at finite times; they coincide in steady state but produce different violation regions in transients (Bourgeois et al., 2024). This indicates that, away from the steady-state limit, the precise identification of current, noise, and activity becomes part of the formulation rather than merely of the application.

6. Collective dissipative many-body systems

A major recent extension concerns collective dissipative quantum many-body systems, where the earlier quantum KUR literature had remained confined to single-body settings. For an ensemble of $\mathcal{J}_d$5 interacting spin-$\mathcal{J}_d$6 particles with collective dissipation,

$\mathcal{J}_d$7

and for the time-integrated number of quantum jumps

$\mathcal{J}_d$8

the many-body KUR takes the form

$\mathcal{J}_d$9

Here $A_\tau$0 is a many-body dynamical activity and $A_\tau$1 is a computable upper bound expressed through normalized magnetizations $A_\tau$2, the mean-field evolution matrix $A_\tau$3, and the system parameters $A_\tau$4 (Yunoki et al., 7 Apr 2026).

The physical decomposition is explicit in the data. The term proportional to $A_\tau$5 is the classical dynamical activity due to collective jump events, while the term proportional to $A_\tau$6 is a coherent quantum contribution from Hamiltonian-induced dynamics. Both scale linearly with $A_\tau$7, so the lower bound on the relative fluctuation scales as $A_\tau$8. This is the cooperative enhancement mechanism: precision is collectively amplified and uncertainty is macroscopically suppressed as the particle number grows.

The results were validated numerically in the stationary phase $A_\tau$9, the critical regime $$A_\tau \ge \frac{(\nabla[\mathcal{J}_d])^2}{\mathrm{Var}[\mathcal{J}_d]},$$0, and the boundary time crystal phase $$A_\tau \ge \frac{(\nabla[\mathcal{J}_d])^2}{\mathrm{Var}[\mathcal{J}_d]},$$1. The data state that the analytical bounds hold in all three phases, that the critical regime gives the best achieved precision, and that the boundary time crystal phase offers the potential for the highest ultimate precision. Within the supplied literature, this is the first theoretical description of precision bounds in collective dissipative quantum many-body systems for arbitrary $$A_\tau \ge \frac{(\nabla[\mathcal{J}_d])^2}{\mathrm{Var}[\mathcal{J}_d]},$$2, and it establishes a many-body counterpart of the KUR in a regime where the relevant activity is both collective and coherence-assisted (Yunoki et al., 7 Apr 2026).