A constraint on local definitions of quantum internal energy

TL;DR

This paper investigates the possibility of defining a universal, local internal energy for open quantum systems, revealing that such a definition must involve second-order derivatives of the system's state to be consistent.

Contribution

It establishes a fundamental limit on local internal energy definitions in quantum thermodynamics, requiring second-order derivatives for consistency with global energy.

Findings

Local internal energy must involve second derivatives of the density operator.

A universal local internal energy definition cannot be first-order only.

Implications for quantum thermodynamics and energy accounting are discussed.

Abstract

Recent advances in quantum thermodynamics have been focusing on ever more elementary systems of interest, approaching the limit of a single qubit, with correlations, strong coupling and non-equilibrium environments coming into play. Under such scenarios, it is clear that fundamental physical quantities must be revisited. This article questions whether a universal definition of internal energy for open quantum systems may be devised, setting limits on its possible properties. We argue that, for such a definition to be regarded as local, it should be implemented as a functional of the open system's reduced density operator and its time derivatives. Then we show that it should involve at least up to the second-order derivative, otherwise failing to recover the previously-known internal energy of the "universe". Possible implications of this general result are discussed.