Self-consistency of the two-point energy measurement protocol

The Two-Point energy Measurement (TPM) protocol defines the work done on a system undergoing unitary evolution as the difference in energy measurement outcomes performed before and after such evolution. By noting that energy measurements on the system can be modelled as a unitary premeasurement interaction between the system and a measurement apparatus, followed by measurement of the apparatus by a pointer observable, we show that it is possible to design a measurement scheme for the TPM protocol on the system that simultaneously acts as a TPM protocol for the compound of system-plus-apparatus so as to reveal the total work distribution. We further demonstrate that: (i) the average total work will be the change in average energy, given the total unitary evolution, for all initial system states and system unitary processes; and (ii) the total work distribution will be identical to the system-only work distribution, for all system states, if and only if the unitary premeasurements conserve the total energy of system-plus-apparatus for all system states.