Anomalous energy exchanges and Wigner function negativities in a single qubit gate

Anomalous weak values and Wigner function's negativity are well known witnesses of quantum contextuality. We show that these effects occur when analyzing the energetics of a single qubit gate generated by a resonant coherent field traveling in a waveguide. The buildup of correlations between the qubit and the field is responsible for bounds on the gate fidelity, but also for a nontrivial energy balance recently observed in a superconducting setup. In the experimental scheme, the field is continuously monitored through heterodyne detection and then post-selected over the outcomes of a final qubit's measurement. The post-selected data can be interpreted as field's weak values and can show anomalous values in the variation of the field's energy. We model the joint system dynamics with a collision model, gaining access to the qubit-field entangled state at any time. We find an analytical expression of the quasi-probability distribution of the post-selected heterodyne signal, i.e. the conditional Husimi-Q function. The latter grants access to all the field's weak values: we use it to obtain that of the field's energy change and display its anomalous behaviour. Finally, we derive the field's conditional Wigner function and show that anomalous weak values and Wigner function's negativities arise for the same values of the gate's angle.