Qubit degradation due to cross-phase-modulation photon-number measurement

Kerr effect cross-phase-modulation between signal and probe beams has been proposed and demonstrated as a means for quantum nondemolition (QND) measurement of the signal beam's photon number by homodyne detection of the probe beam. Recently, such QND measurements have been suggested as providing a new route to photonic quantum computation. For a signal-beam qubit, whose computational basis is the vacuum and one-photon states, the QND detector's error probability is shown to be determined by the qubit fidelity after the cross-phase-modulation interaction, with the former increasing from 0 to 1/2 as the latter increases from 2/3 to 1. This relationship is shown to hold regardless of the probe beam's input state. It also applies, without modification, when the cross-phase-modulation interaction is replaced with a more general unitary transformation.