Impact of quadrature measurement on quantum coherence

Abstract

We examine the behavior of quadrature coherence under the measurement of the same field quadrature. This is carried out with the help of a beam splitter, which implies the contribution of an auxiliary field state impinging at the other input port. To this end we consider the linear input-output transformation of a lossless beam splitter to relate input and output coherences, measured in terms of the $l_1$-norm. After obtaining a general input-output relation between coherences we apply the result to Gaussian and number states. For Gaussian states we obtain that coherence does not depend on the measurement outcome, and that the average coherence always equals the coherence of the reduced state, showing no average effect on coherence of the measurement. On the other hand, for number states the output coherence depends on the measurement, decreasing the relative coherence with increasing photon number. Finally, we consider relative-entropy as a measure of coherence to show that for number states and coherence measures other than the $l_1$-norm the average coherence no longer equals the coherence of the output reduced state.