Fidelity of quantum interferometers

Abstract

For a generic interferometer, the conditional probability density distribution $p(\ensuremath{\phi}\ensuremath{\mid}m)$, for the phase $\ensuremath{\phi}$ given measurement outcome $m$ will generally have multiple peaks. Therefore, the phase sensitivity of an interferometer cannot be adequately characterized by the standard deviation, such as $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕\sqrt{N}$ (the standard limit), or $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}\ensuremath{\sim}1∕N$ (the Heisenberg limit). We propose an alternative measure of phase sensitivity---the fidelity of an interferometer---defined as the Shannon mutual information between the phase shift $\ensuremath{\phi}$ and the measurement outcomes $m$. As an example application of interferometer fidelity, we consider a generic optical Mach-Zehnder interferometer, used as a sensor of a classical field. For the case where there exists no a priori information on the phase shift, we find the surprising result that maximally entangled state input leads to a lower fidelity than Fock state input, for the same photon number.