Self-consistency of the two-point energy measurement protocol

TL;DR

This paper investigates the self-consistency of the two-point energy measurement protocol in quantum thermodynamics, showing conditions under which the total work distribution aligns with the system-only work distribution.

Contribution

It demonstrates that the TPM protocol is self-consistent when the measurement apparatus starts in an energy eigenstate and explores conditions for the total work distribution to match the system-only distribution.

Findings

Self-consistency is achieved if the apparatus is initially in an energy eigenstate.

Total work distribution satisfies the unmeasured first law when apparatus Hamiltonian matches the pointer observable.

Total work distribution equals the system-only distribution if unmeasured work due to system-apparatus coupling is zero.

Abstract

A thermally isolated quantum system undergoes unitary evolution by interacting with an external work source. The two-point energy measurement (TPM) protocol defines the work exchanged between the system and the work source by performing ideal energy measurements on the system before, and after, the unitary evolution. However, the ideal energy measurements used in the TPM protocol ultimately result from a unitary coupling with a measurement apparatus, which requires an interaction with an external work source. For the TPM protocol to be self-consistent, we must be able to perform the TPM protocol on the compound of system plus apparatus, thus revealing the total work distribution, such that when ignoring the apparatus degrees of freedom, we recover the original TPM work distribution for the system of interest. In the present manuscript, we show that such self-consistency is satisfied so long as the apparatus is initially prepared in an energy eigenstate. Moreover, we demonstrate that if the apparatus Hamiltonian is equivalent to the "pointer observable", then: (i) the total work distribution will satisfy the "unmeasured" first law of thermodynamics for all system states and system-only unitary processes; and (ii) the total work distribution will be identical to the system-only work distribution, for all system states and system-only unitary processes, if and only if the unmeasured work due to the unitary coupling between system and apparatus is zero for all system states.