Energy fluctuation relations and repeated quantum measurements

TL;DR

This paper reviews the statistical description of energy fluctuations in quantum systems under repeated measurements, analyzing fluctuation theorems, asymptotic behaviors, and effects of the quantum Zeno regime, with applications to simple quantum systems.

Contribution

It provides a comprehensive analysis of energy fluctuation relations under repeated quantum measurements, including conditions for fluctuation theorem validity and asymptotic properties.

Findings

Fluctuation theorem holds under certain conditions despite measurement randomness.

In the thermodynamic limit, the system tends to a maximally mixed state unless specific invariance conditions are met.

Energy fluctuation relations are modified in the quantum Zeno regime.

Abstract

In this review paper, we discuss the statistical description in non-equilibrium regimes of energy fluctuations originated by the interaction between a quantum system and a measurement apparatus applying a sequence of repeated quantum measurements. To properly quantify the information about energy fluctuations, both the exchanged heat probability density function and the corresponding characteristic function are derived and interpreted. Then, we discuss the conditions allowing for the validity of the fluctuation theorem in Jarzynski form $\langle e^{-\beta Q}\rangle = 1$, thus showing that the fluctuation relation is robust against the presence of randomness in the time intervals between measurements. Moreover, also the late-time, asymptotic properties of the heat characteristic function are analyzed, in the thermodynamic limit of many intermediate quantum measurements. In such a limit, the quantum system tends to the maximally mixed state (thus corresponding to a thermal state with infinite temperature) unless the system's Hamiltonian and the intermediate measurement observable share a common invariant subspace. Then, in this context, we also discuss how energy fluctuation relations change when the system operates in the quantum Zeno regime. Finally, the theoretical results are illustrated for the special cases of two- and three-levels quantum systems, now ubiquitous for quantum applications and technologies.