Postselection and Energy Conservation

Measurements, of observables that do not commute with energy, can shift the expected energy of a measured qubit. Owing to energy conservation, such a shift should be accompanied by an equal-and-opposite energy shift of the apparatus, on average. However, it is less obvious how the energy of an apparatus shifts on a case-by-case basis, i.e., for particular qubit measurement outcomes (postselections). We compute this shift using two measurement models that respect energy conservation explicitly: a toy quantum clock model, and a more realistic model utilizing the Jaynes-Cummings interaction. Our main findings are that the postselected energy shift of the apparatus does not, in general, balance that of the measured qubit (and may even be anomalous, exceeding the level spacing of the qubit), and that the results depend on the particular measurement implementation (as opposed to the targeted measurement). The clock energy shift also explicitly contains the qubit's energy weak value, despite a lack of deliberate weak measurements. We compare our Jaynes-Cummings model results with the experimental findings of [J. Stevens, et al, Phys. Rev. Lett. 129, 110601 (2022)], which analyzed the energetics of single qubit gates, and find agreement in the appropriate regime.