The kinetic uncertainty relation (KUR) constrains fluctuations of individual currents in classical nonequilibrium systems in terms of the dynamical activity, which quantifies the average number of transitions per unit of time. Here, we derive a multi-current KUR (MKUR) establishing the precision limit for multiple currents generated in classical stochastic processes in the nonequilibrium steady state. We then apply our theory to identify quantum devices whose correlated current fluctuations surpass this classical limit by comparing the quantum system to a classical emulator with the same average currents in the steady state. In this way, violations of the emulator's MKUR have a clear physical meaning, i.e., that the quantum-coherent evolution reduces joint current fluctuations below what is possible for a classical Markovian system with the same energy-level structure and incoherent resources. Since the MKUR incorporates correlations between currents, its violation pinpoints nonclassical fluctuations in parameter regimes where the single-current KUR is satisfied. We illustrate the essential and complementary role of these bounds for characterizing joint current precision with two examples: a coherently driven two-level system and an experimentally motivated model of a three-level heat engine subject to parasitic environmental couplings.
The kinetic uncertainty relation (KUR) constrains fluctuations of individual currents in classical nonequilibrium systems in terms of the dynamical activity, which quantifies the average number of transitions per unit of time. Here, we derive a multi-current KUR (MKUR) establishing the precision limit for multiple currents generated in classical stochastic processes in the nonequilibrium steady state. We then apply our theory to identify quantum devices whose correlated current fluctuations surpass this classical limit by comparing the quantum system to a classical emulator with the same average currents in the steady state. In this way, violations of the emulator's MKUR have a clear physical meaning, i.e., that the quantum-coherent evolution reduces joint current fluctuations below what is possible for a classical Markovian system with the same energy-level structure and incoherent resources. Since the MKUR incorporates correlations between currents, its violation pinpoints nonclassical fluctuations in parameter regimes where the single-current KUR is satisfied. We illustrate the essential and complementary role of these bounds for characterizing joint current precision with two examples: a coherently driven two-level system and an experimentally motivated model of a three-level heat engine subject to parasitic environmental couplings.
Thermodynamic (TUR) and kinetic (KUR) uncertainty relations are fundamental bounds constraining the fluctuations of current observables in classical, nonequilibrium systems. Several works have verified, however, violations of these classical bounds in open quantum systems, motivating the derivation of new quantum TURs and KURs that account for the role of quantum coherence. Here, we go one step further by deriving multidimensional KUR and TUR for multiple observables in open quantum systems undergoing Markovian dynamics. Our derivation exploits a multiparameter metrology approach, in which the Fisher information matrix plays a central role. Crucially, our bounds are tighter than previously derived quantum TURs and KURs for single observables, precisely because they incorporate correlations between multiple observables. We also find an intriguing quantum signature of correlations that is captured by the off-diagonal element of the Fisher information matrix, which vanishes for classical stochastic dynamics. By considering two examples, namely a coherently driven qubit system and the three-level maser, we demonstrate that the multidimensional quantum KUR bound can even be saturated when the observables are perfectly correlated.
The kinetic uncertainty relation (KUR) bounds the signal-to-noise ratio of stochastic currents in terms of the number of transitions per unit time, known as the dynamical activity. This bound was derived in a classical context and can be violated in the quantum regime due to coherent effects. However, the precise connection between KUR violations and quantum coherence has so far remained elusive, despite significant investigation. In this Letter, we solve this problem by deriving a modified bound that exactly pinpoints how, and when, coherence might lead to KUR violations. Our bound is sensitive to the specific kind of unraveling of the quantum master equation. It, therefore, allows one to compare quantum jumps and quantum diffusion, and understand, in each case, how quantum coherence affects fluctuations. We illustrate our result on a double quantum dot, where the electron current is monitored either by electron jump detection or with continuous diffusive charge measurement.
Kinetic uncertainty relations bound current precision of stochastic processes by dynamical activity. The extension of these bounds to quantum systems has been impeded by coherence, strong system-reservoir coupling, and the subtlety of defining dynamical activity in the quantum regime. Here, we introduce a partial dynamical activity through the quantum Fisher information associated with the rescaling of the system-reservoir coupling and show that it bounds current precision via a universal susceptibility-kinetic uncertainty relation. The general validity of this relation for any open quantum system is guaranteed by the natural contribution of a susceptibility term, which is experimentally accessible by tuning the system-reservoir coupling strength. We show how the partial dynamical activity encompasses previous definitions of activity in the weak-coupling Markovian limit and that it provides an information-geometric interpretation of correlator-based activities. We illustrate the tight constraint on precision that our bound provides with the example of steady-state transport through a double quantum dot, where quantum effects invalidate previously developed kinetic uncertainty relations. We expect our bound to provide a powerful tool for optimizing precision in arbitrary quantum systems.
Kinetic uncertainty relations (KURs) set fundamental limits on the precision of nonequilibrium transport by bounding the signal-to-noise ratio of currents in terms of the dynamical activity, a quantity that counts exchange events between a system and its reservoirs. This framework is well established in the weak-coupling regime, where transport occurs via well-defined, particlelike tunneling processes. At strong coupling, however, quantum coherence challenges both the validity of standard KURs and the notion of activity itself. In this Letter, we introduce a generalized definition of dynamical activity valid at arbitrary system-reservoir coupling, and show that it leads to a breakdown of standard KURs at strong coupling. Building on this result, we derive and prove a novel uncertainty relation, denoted quantum KUR (QKUR), which provides a genuine quantum extension of KUR, accounting for intrinsic quantum coherent contributions of the generalized activity. We demonstrate that the generalized activity reduces to the standard master-equation definition in the weak-coupling regime for generic systems of N coupled quantum dots described by a quadratic Hamiltonian, and analyze the resulting QKUR bound in paradigmatic quantum-coherent mesoscopic devices, including single- and double-quantum dot systems and a quantum point contact.
Entanglement engines are autonomous quantum thermal machines designed to generate entanglement from the presence of a particle current flowing through the device. In this work, we investigate the functioning of a two-qubit entanglement engine beyond the steady-state regime. Within a master equation approach, we derive the time-dependent state, the particle current, as well as the associated current correlation functions. Our findings establish a direct connection between coherence and internal current, elucidating the existence of a critical current that serves as an indicator for entanglement in the steady state. We then apply our results to investigate kinetic uncertainty relations (KURs) at finite times. We demonstrate that there is more than one possible definition for KURs at finite times. Although the two definitions agree in the steady-state regime, they lead to different parameter ranges for violating KUR at finite times.
Precision in nonequilibrium processes comes at a thermodynamic cost: suppressing fluctuations generally requires increased dissipation. Thermodynamic uncertainty relations (TURs) make this trade-off quantitative by linking current fluctuations to entropy production in classical stochastic dynamics. In the decade since its discovery, the canonical steady-state TUR and its finite-time generalizations have become a cornerstone of non-equilibrium thermodynamics, constraining the performance of molecular machines and allowing heat dissipation to be inferred from observable fluctuations. Whether the canonical TUR can be violated in a controlled quantum device remains an outstanding experimental question, in part because doing so requires resolving extremely small steady-state currents as well as their fluctuations. Here we experimentally show that steady-state quantum transport can surpass the precision permitted by the canonical TUR. We observe this violation in a superconducting quantum thermal machine coupled to a microwave waveguide acting as a cold bath and to a classical noise source providing an effective infinite-temperature bath. We observe a TUR ratio Q=1.71±0.17, in violation of the classical bound Q≥2. Our results demonstrate a fundamental distinction between classical and quantum thermodynamics, paving the way for quantum thermal devices that achieve enhanced precision at reduced energy cost.