Phase-noise limitations on single-photon cross-phase modulation with differing group velocities

A framework is established for evaluating {\sc cphase} gates that use single-photon cross-phase modulation (XPM) originating from the Kerr nonlinearity. Prior work Phys. Rev. A {\bf 73,} 062305 (2006)], which assumed that the control and target pulses propagated at the same group velocity, showed that the causality-induced phase noise required by a non-instantaneous XPM response function precluded the possibility of high-fidelity $\pi$-radian conditional phase shifts. The framework presented herein incorporates the more realistic case of group-velocity disparity between the control and target pulses, as employed in existing XPM-based fiber-optical switches. Nevertheless, the causality-induced phase noise identified in [Phys. Rev. A {\bf 73,} 062305 (2006)] still rules out high-fidelity $\pi$-radian conditional phase shifts. This is shown to be so for both a reasonable theoretical model for the XPM response function and for the experimentally-measured XPM response function of silica-core fiber.