Thermodynamic Uncertainty Relation with Quantum Feedback

arXiv:2602.22651 · quant-ph, cond-mat.mes-hall, cond-mat.stat-mech · Submitted 2026-02-26 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Hi, I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Thermodynamic Uncertainty Relation with Quantum Feedback".

Mira: Here's the gist of the paper. The gist: The authors establish a finite-time thermodynamic uncertainty relation for arbitrary time-integrated currents in open quantum systems under Markovian feedback control, revealing how feedback suppresses fluctuations alongside thermodynamic cost and establishing a fundamental precision bound imposed by information-based control.

Kai: How it works The framework considers an open quantum system weakly coupled to a thermal environment where quantum jumps are continuously monitored, and Markovian feedback is applied instantaneously based on the detected outcomes <ref:2602.22651#pg2>.

Mira: The authors use quantum mutual information to quantify the information exploited by feedback control relevant to current fluctuations <ref:2602.22651#pg3>.

Kai: They derive a second law of thermodynamics for measurement and feedback processes, establishing a finite-time TUR for arbitrary time-integrated currents in terms of entropy production and mutual information [Eq.

Title and authors: Kai: So we're looking at this paper called Thermodynamic Uncertainty Relation with Quantum Feedback. It’s by Ryotaro Honma and Tan Van Vu from Kyoto University. It sounds pretty technical, but the core idea is about how fluctuations in a quantum system are fundamentally limited because of feedback control.

Mira: Yeah, it’s about linking current fluctuations to entropy production. It suggests that when you have continuous monitoring and Markovian feedback in an open quantum system, you get this new way to see the thermodynamic uncertainty relation.

Lev: From an engineering standpoint, I'm wondering how much overhead this feedback part really adds when we try to run it on actual hardware with real qubits and cryogenics.

Kai: That’s a fair question Lev, because the paper sets up this entire framework using continuous measurement and feedback applied instantaneously based on detected outcomes. It’s about how that information exploitation plays into the uncertainty relation.

Mira: Exactly, and what’s interesting is how they use quantum mutual information to measure exactly how much information the feedback control is exploiting to suppress those fluctuations.

The paper's summary: Kai: So, in this Thermodynamic Uncertainty Relation with Quantum Feedback paper, they set up a system weakly coupled to a thermal environment where quantum jumps are continuously monitored and Markovian feedback is applied. They use quantum mutual information to quantify the information exploited by this feedback control that is relevant to current fluctuations.

Mira: And then they derive a second law of thermodynamics specifically for these measurement and feedback processes, which leads them to establish a finite-time TUR for arbitrary time-integrated currents in terms of entropy production and mutual information

Eq. (fifteen): <ref:2602.22651#pg2>.

Lev: So the main result is that this relation recovers the original TUR when you remove the feedback control or if you look at the classical limit, which is good because it shows consistency with what we already know.

Kai: It also shows how feedback control actually suppresses those current fluctuations while incurring a thermodynamic cost that's bounded by that mutual information they quantified.

Mira: Which means even though you’re using feedback to get better precision, there’s still this fundamental limitation imposed by the information-based control itself.

The paper's improvements: Kai: The paper suggests an improvement because they derive a second law of thermodynamics for measurement and feedback processes, which they show means that the total entropy production before feedback is always nonnegative, S(ϱ − t+dt) − S(ϱt) + dSenv ≥ zero fifty-six <ref:2602.22651#pg3,that the total entropy production before feedback is always nonnegative>.

Mira: That’s significant because it implies that even though you can have a decrease in system entropy after the measurement and feedback step, it's only by an amount bounded by the mutual information exploited during that feedback step.

Lev: From my side, this is interesting because it connects the physics of quantum jumps with measurable quantities like mutual information; if we could measure that mutual information directly on hardware, it gives us a concrete way to assess the control cost.

Kai: And then they take this and combine it with other equations to get their improved TUR

Eq. (fifteen): , which is VarJ⟨J⟩two ≥ (one + δJ)two 4AΣ2Φ(Σ2A)two six.

Mira: And that correction term delta J captures the quantum coherence and transient relaxation on current precision, and its sign actually tells you whether the precision gets enhanced or suppressed.

Conclusion: Kai: So to wrap up, these authors established a second law of thermodynamics for continuous measurement and Markovian feedback protocols using entropy production and mutual information. They got the improved TUR

Eq. (fifteen): which shows that accumulated mutual information plays a crucial role in suppressing relative current fluctuations <ref:2602.22651#pg2>.

Mira: It also provides a quantitative framework for understanding the fundamental limits of current precision in open quantum systems under continuous measurement and feedback control, linking it to entropy production and the mutual information between the system and the memory.

Lev: For me, it means that when we think about building real hardware, we need to be very careful because this relation sets a concrete limit on how much precision we can hope to get just from applying feedback.

Kai: We're looking forward to seeing how this framework helps us figure out the practical limits of current precision in systems like the quantum clock models they applied.

Ryotaro Honma, Tan Van Vu

Center for Gravitational Physics and Quantum Information · Yukawa Institute for Theoretical Physics, Kyoto University

quant-ph, cond-mat.mes-hall, cond-mat.stat-mech

Submitted: 2026-02-26

Updated: 2026-10-22

Comments: 8+17 pages, 2+2 figures

Journal ref: Phys. Rev. Lett. 137, 160406 (2026)

DOI: 10.1103/6d8l-94pv

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist: The authors establish a finite-time thermodynamic uncertainty relation for arbitrary time-integrated currents in open quantum systems under Markovian feedback control, revealing how

Key concepts

Thermodynamic Uncertainty Relation (TUR): A fundamental limit on the precision of measuring a physical quantity. This paper derives an improved version that shows how fluctuations are bounded not just by standard thermodynamic costs but also by the information used during feedback control.
Markovian Feedback Control: An instantaneous control strategy applied to an open quantum system based on continuously monitored outcomes (quantum jumps). This feedback uses the measured data to instantaneously adjust the system's evolution, exploiting quantum mutual information.
Quantum Mutual Information: A measure quantifying how much information about a system is gained by observing its environment. In this context, it quantifies the specific information exploited by the feedback control mechanism to suppress current fluctuations.

Terminology

Mutual Information — 2× in this paper · explained in 2 episodes

Summary

The gist: The authors establish a finite-time thermodynamic uncertainty relation for arbitrary time-integrated currents in open quantum systems under Markovian feedback control, revealing how feedback suppresses fluctuations alongside thermodynamic cost and establishing a fundamental precision bound imposed by information-based control.

How it works

The framework considers an open quantum system weakly coupled to a thermal environment where quantum jumps are continuously monitored, and Markovian feedback is applied instantaneously based on the detected outcomes <ref:2602.22651#pg2>. The authors use quantum mutual information to quantify the information exploited by feedback control relevant to current fluctuations <ref:2602.22651#pg3>. They derive a second law of thermodynamics for measurement and feedback processes, establishing a finite-time TUR for arbitrary time-integrated currents in terms of entropy production and mutual information [Eq. (15)] <ref:2602.22651#pg3>. This relation recovers the original TUR in the absence of feedback and in the classical limit <ref:2602.22651#pg3>.

Setup

The setup involves a continuous measurement and feedback process for a finite-dimensional system S weakly coupled to a thermal environment E, where quantum jumps occur and are recorded in a memory M <ref:2602.22651#pg3>. The system dynamics is governed by an effective Hamiltonian Heff defined as H −(i/2) ∑k≥1 L†kLk, where jump operators Lk are involved <ref:2602.22651#pg3>. The conditioned system state after a jump is given by Fk[MkϱtM†k]/ tr Fk[MkϱtM†k], and the unconditional state is updated as ϱt+dt = M0ϱtM†0 + ∑k≥1 Fk[MkϱtM†k] <ref:2602.22651#pg3>. The total entropy production before feedback is always nonnegative, S(ϱ−t+dt) − S(ϱt) + dSenv ≥ 0 <ref:2602.22651#pg4>.

Main Results

The central result is the second law of thermodynamics for measurement and feedback control, derived by combining Eqs. (5) and (11), yielding dSsys + dSenv − dI ≥ 0 <ref:2602.22651#pg3>. This implies that the total entropy may decrease after measurement and feedback control, but only by an amount bounded by the mutual information exploited during the feedback step <ref:2602.22651#pg5>. The final result is a finite-time TUR for arbitrary time-integrated currents in terms of entropy production and mutual information [Eq. (15)] <ref:2602.22651#pg3>.

Application to Quantum Clock Model

The framework is applied to a quantum clock model, demonstrating that the clock precision can be enhanced by feedback control even with a single thermal reservoir <ref:2602.22651#pg3>. The authors show that a persistent tick current emerges due to unitary feedback applied on jump events, and the relative fluctuation Var[J]/ ⟨J⟩2 is consistently lower bounded by 2(1 + δJ)2/Σ <ref:2602.22651#pg6>. This numerically validates the main result (15) <ref:2602.22651#pg6>.

Short-Time Limit and Precision Enhancement

In the short-time limit, it is shown that the correction term δJ = O(τ), meaning it vanishes as τ → 0 <ref:2602.22651#pg23>. The current average scales as ⟨J⟩ ∝ τ in this limit, leading to δJ = O(τ) <ref:2602.22651#pg22>. This demonstrates that the precision of a clock is fundamentally limited by the information-thermodynamic cost per tick and quantum coherence <ref:2602.22651#pg6>.

Comparison with Previous Formulations

The formulation (12) of the second law for measurement and feedback is tighter than existing results, such as [26], because it incorporates the mutual information dI exploited during feedback <ref:2602.22651#pg5>. The authors compare their formulation with Ref. [1], showing that their result provides a tighter bound on the irreversibility of the full measurement and feedback process <ref:2602.22651#pg5>.

Conclusion

In this Letter, they established thermodynamic bounds for open quantum systems undergoing continuous measurement and Markovian feedback control <ref:2602.22651#pg3>. They obtained the improved TUR (17) by combining Eqs. (29), (33), and (32) <ref:2602.22651#pg6>. Their findings provide a quantitative framework for understanding the fundamental limits of current precision in open quantum systems under continuous measurement and feedback control <ref:2602.22651#pg6>.

How it works

The improved TUR is given by Var[J]⟨J⟩2 ≥ (1 + δJ)2 4AΣ2Φ(Σ2A)2 <ref:2602.22651#pg6>. This result immediately implies the original TUR (15) by applying the inequality Φ(x) ≥ √x <ref:2602.22651#pg6>. The correction term δJ captures quantum coherence and transient relaxation on current precision, and its sign can determine whether precision is enhanced or suppressed <ref:2602.22651#pg22>.

Special Cases

In the absence of feedback, the result reduces to the previously established thermokinetic uncertainty relation (TKUR) [3] <ref:2602.22651#pg22>. The short-time limit shows that δJ = O(τ), vanishing as τ → 0 <ref:2602.22651#pg22>. This confirms that for classical systems in this limit, both correction term δJ and the information term I vanish, recovering the original TUR for classical systems [2] <ref:2602.22651#pg22>. The unitality of the CPTP map F[○] is equivalent to S(F[ϱ])−S(ϱ) ≥ 0. This confirms that feedback operations do not induce heat dissipation in this setting <ref:2602.22651#pg5>.

Final Summary

The authors successfully derived a second law of thermodynamics for continuous measurement and Markovian feedback protocols, expressed in terms of entropy production and the mutual information between the system and the memory <ref:2602.22651#pg3>. Building on this result, they obtained the quantum TUR (15) for arbitrary time-integrated currents, showing that in addition to entropy production, the accumulated mutual information between the system and the memory plays a crucial role in suppressing relative current fluctuations <ref:2602.22651#pg3>. They numerically demonstrated these results using a minimal quantum clock model <ref:2602.22651#pg6>. Their findings provide a quantitative framework for understanding the fundamental limits of current precision in open quantum systems under continuous measurement and feedback control <ref:2602.22651#pg6>.

Improvements for AI systems

Precision enhancement in quantum clock models: The improved AI system can operate a quantum clock model with a three-level system even when coupled to a single thermal reservoir by applying unitary feedback conditioned on jump events, leading to persistent currents and enhanced precision as demonstrated in the example.
Fundamental precision bounds for current measurements: The AI can establish new thermodynamic uncertainty relations that link relative current fluctuations to entropy production and information exploited during feedback control, providing a fundamental limit on measurement precision even when negative entropy production occurs.
Information-driven fluctuation suppression: The system can utilize quantum mutual information to quantify the information contribution of Markovian feedback, allowing it to suppress current fluctuations while incurring a thermodynamic cost that is fundamentally limited by this mutual information.
Quantitative performance prediction for thermal machines: The AI can provide a quantitative framework for analyzing the performance trade-offs of thermal machines under feedback control, specifically showing how clock precision is fundamentally limited by the information-thermodynamic cost per tick and quantum coherence.

Abstract

Fluctuations are intrinsic to microscopic systems and impose fundamental limits on nonequilibrium precision, as captured by the thermodynamic uncertainty relation (TUR), which links current fluctuations to entropy production. While feedback control is expected to further suppress fluctuations, its role within the TUR framework has remained unclear, particularly in quantum systems where control is inherently information-driven. In this Letter, we consider open quantum systems weakly coupled to a thermal environment, in which quantum jumps are continuously monitored, and Markovian feedback is applied. Using quantum mutual information to quantify the information contribution induced by feedback, we derive a finite-time TUR for arbitrary time-integrated currents in terms of entropy production and mutual information. Our results uncover how feedback control suppresses fluctuations together with thermodynamic cost and establish a fundamental precision bound imposed by information-based control. As an application, we analyze a quantum clock model and demonstrate that the clock precision can be enhanced by feedback control in the presence of a single thermal reservoir.