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arXiv:2607.12264

Heisenberg Scaling in Many-Body Kinetic Uncertainty Relation via Quantum Feedback

Hayato Yunoki, Yoshihiko Hasegawa

quant-phSubmitted Jul 14, 2026• Announced Jul 15, 2026• 0 citations
v1 · Jul 14, 2026v2 · Sep 28, 2026
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Abstract

Precision is a central figure of merit for quantum devices, including quantum clocks whose performance is determined by the stability of counting events. Kinetic uncertainty relations set fundamental limits on the precision of such counting observables, showing that their fluctuations cannot be suppressed without increasing the activity of the system. While many-body effects offer a natural route to enhanced performance, it remains unclear how far they can enhance counting precision. In quantum metrology, Heisenberg scaling refers to the suppression of estimation variance as $1/N^2$ with the particle number $N$. This raises the question of whether the fluctuation of counting observables can exhibit an analogous Heisenberg-like $1/N^2$ scaling, but no protocol for achieving it has been established. We establish a protocol that achieves this scaling by applying quantum feedback to a superradiant spin ensemble. Because the superradiant enhancement of activity is transient, the scaling of counting precision becomes achievable only when it is controlled by feedback. We establish this result analytically through a many-body kinetic uncertainty relation and feedback-modified mean-field equations, and show it by numerical simulations. Our results demonstrate that feedback can turn collective dissipation into a resource for Heisenberg scaling of counting precision.

Comments
8 pages, 2 figures; 9 pages of supplementary material

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Precision is a central figure of merit for quantum devices, including quantum clocks whose performance is determined by the stability of counting events. Kinetic uncertainty relations set fundamental limits on the precision of such counting observables, showing that their fluctuations cannot be suppressed without increasing the activity of the system. While many-body effects offer a natural route to enhanced performance, it remains unclear how far they can enhance counting precision. In quantum metrology, Heisenberg scaling refers to the suppression of estimation variance as 1/N21/N^21/N2 with the particle number NNN. This raises the question of whether the fluctuation of counting observables can exhibit an analogous Heisenberg-like 1/N21/N^21/N2 scaling, but no protocol for achieving it has been established. We establish a protocol that achieves this scaling by applying quantum feedback to a superradiant spin ensemble. Because the superradiant enhancement of activity is transient, the scaling of counting precision becomes achievable only when it is controlled by feedback. We establish this result analytically through a many-body kinetic uncertainty relation and feedback-modified mean-field equations, and show it by numerical simulations. Our results demonstrate that feedback can turn collective dissipation into a resource for Heisenberg scaling of counting precision.
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Feedback can convert continuously monitored quantum jumps into a thermodynamic resource. We formulate full counting statistics for open quantum systems under unital jump feedback by incorporating the feedback maps into the tilted generator. The resulting trajectory ensemble determines both current fluctuations and the Fisher information of the measurement record. We show that feedback can enhance reservoir-parameter estimation and clock precision without necessarily changing average thermodynamic currents. This enhanced precision is not bounded by reservoir entropy production alone. By embedding the reduced dynamics in an enlarged measurement-feedback process, we derive a feedback-modified thermodynamic uncertainty relation in which the information entropy production of the feedback apparatus supplies the missing cost. A charge-monitored double quantum dot illustrates the framework: jump-conditioned feedback improves thermometry and chemical-potential sensing, and stabilizes a quantum clock defined by output-current ticks.
The thermodynamic uncertainty relation posits that higher thermodynamic costs are essential for a system to function with greater precision. Recent discussions have expanded thermodynamic uncertainty relations beyond classical non-equilibrium systems, investigating how quantum characteristics can be utilized to improve precision. In this Letter, we explore how quantum feedback, a control technique used to manipulate quantum systems, can enhance the precision. Specifically, we derive a quantum thermodynamic uncertainty relation for feedback control under jump measurement, which provides the lower bound to the scaled variance of the number of jumps. We find that the presence of feedback control can increase the accuracy of continuous measured systems, which is verified with numerical simulations. Moreover, we derive a quantum thermodynamic uncertainty relation for feedback control under homodyne detection.
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.