Abstract
Autonomous engines operating at the nano-scale can be prone to deleterious fluctuations in the heat and particle currents which increase, for fixed power output, the more reversible the operation regime is. This fundamental trade-off between current fluctuations and entropy production forms the basis of the recently formulated thermodynamic uncertainty relations (TURs). However, these relations have so far only been derived for classical Markovian systems and can be violated in the quantum regime. In this paper we show that the geometry of quantum non-equilibrium steady-states alone, already directly implies the existence of a TUR, but with a looser bound. The geometrical nature of this result makes it extremely general, establishing a fundamental limit for the thermodynamics of precision. Our proof is based on the McLennan-Zubarev ensemble, which provides an exact description of non-equilibrium steady-states. We first prove that the entropy production of this ensemble can be expressed as a quantum relative entropy. The TURs are then shown to be a direct consequence of the quantum Cramer-Rao bound, a fundamental result from parameter estimation theory. By combining techniques from many-body physics and information sciences, our approach also helps to shed light on the delicate relationship between quantum effects and current fluctuations in autonomous machines.
Highlights
Autonomous machines, whether classical or quantum, generically operate in nonequilibrium conditions
We show that the geometry of quantum nonequilibrium steady states alone directly implies the existence of Thermodynamic uncertainty relations (TURs), but with a looser bound, which is not violated by the above recent findings
Pietzonka and Seifert (PS) showed that, for steady-state engines described by classical Markovian stochastic processes, the application of TUR to the work current, i.e., the power P ≡ JW = JL − JR leads straightforwardly to the upper bound Eq (2), with η = P / JL being the efficiency of the engine and ηC being the Carnot efficiency corresponding to TL = TH and TR = TC
Summary
Autonomous machines, whether classical or quantum, generically operate in nonequilibrium conditions. Is the TUR expected to have implications for the functioning of biological clocks [33] and control techniques [34], it was demonstrated recently that it has significant consequences for the operation of autonomous machines It follows from Eq (1) that the fluctuations in the output power of an engine operating between two reservoirs at temperatures TC and TH > TC is bounded by [35]. While the precise mechanisms responsible for these violations are still not fully understood, this opens up an interesting perspective, as it would in principle allow one to use quantum effects to reduce the deleterious current fluctuations without compromising the engine’s efficiency and output power [40,41,42] These violations naturally lead one to ask, to what extent, are TURs really a universal feature of nonequilibrium steady states. A more systematic study of these two bounds in other physical platforms and models will be pursued in a forthcoming work
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