The thermodynamic uncertainty relation sets a lower bound on the cost of precision. For any steady-state current — particle flow, heat flux, chemical reaction rate — the relative fluctuations cannot be smaller than a bound set by the entropy production. More precisely: the variance of the current divided by its squared mean is bounded below by 2 divided by the total entropy production. To make a current more precise, you must dissipate more. Precision has a thermodynamic price, and the price is exact.
Tojo, Sagawa, and Funo (arXiv 2602.23110, February 2026) show that continuous measurement and feedback can break this bound.
The mechanism is information. A Maxwell's demon that continuously monitors a quantum system and feeds back the measurement results can reduce the current fluctuations below the entropy production bound. The demon pays for this reduction not through additional entropy production but through information gain — the quantum-classical transfer entropy that quantifies how much the demon learns about the system by measuring it.
The modified bound replaces the entropy production alone with the entropy production minus the information gain. When the feedback is effective, the information term is positive, and the bound becomes tighter in reverse — the system can achieve precision that would be thermodynamically impossible without measurement. The current fluctuations can be smaller than what the entropy production would allow, because the demon's knowledge compensates for the missing dissipation.
This is not a violation of the second law. The second law for systems with feedback includes the information term: the total entropy production of the system plus demon is still positive. But the thermodynamic uncertainty relation as originally stated didn't include information — it bounded precision by dissipation alone. The measurement and feedback process accesses a regime where the original relation fails, not because it was wrong but because it assumed no external information about the system's state.
The authors demonstrate this with a driven two-level quantum system under continuous measurement. The feedback achieves higher current precision while reducing entropy production. The system simultaneously becomes more precise and less dissipative, which the original thermodynamic uncertainty relation says is impossible. The information gain from the continuous measurement resolves the contradiction.
The result connects two threads of modern thermodynamics. The thermodynamic uncertainty relation emerged from stochastic thermodynamics as a universal cost-precision tradeoff. Information thermodynamics — the physics of Maxwell's demons — established that information is a thermodynamic resource that can substitute for work or reduce entropy. The modified bound unifies them: the true cost of precision is not dissipation alone but dissipation minus information. A system that knows where its particles are can route them more precisely with less waste.