Quantum Uncertainty Relations for Thermodynamic Energy Flows

The Heisenberg uncertainty relation, which links the uncertainties of the position and momentum of a particle, has an important footprint on the quantum behavior of a physical system. Analogous to this principle, we propose that thermodynamic currents associated with work, heat, and internal energy satisfy their own uncertainty relations. To formalize this idea, we represent these currents by well-defined Hermitian operators, constructed so that their expectation values match the corresponding average currents. Because these operators generally do not commute, the resulting quantum currents differ fundamentally from their classical counterparts. Using the Robertson-Schrödinger uncertainty relation, we derive various uncertainty relations that link different thermodynamic flows. We further illustrate this approach by applying it to quantum batteries, where we derive an energy-power uncertainty relationship and show how measurements affect the fluctuations.

💡 Research Summary

The manuscript introduces a systematic operator‑based framework for describing thermodynamic currents—work, heat, and internal‑energy rates—in quantum systems, and derives uncertainty relations that bind the fluctuations of these currents. The authors begin by considering a generic open quantum system described by a total Hamiltonian (H_{\text{tot}}(t)=H_S(t)\otimes\mathbb I_E+\mathbb I_S\otimes H_E+V_{SE}). By demanding that the expectation values of newly defined Hermitian operators (\hat W(t)) (work rate) and (\hat Q(t)) (heat flow) coincide with the standard average rates (\dot W(t)=\text{Tr}_S{\dot H_S(t)\rho_S(t)}) and (\dot Q(t)=\text{Tr}_S{H_S(t)\dot\rho_S(t)}), they obtain the explicit forms
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