Optimal probabilistic measurement of phase

Abstract

When measuring phase of quantum states of light, the optimal single-shot measurement implements projection on the un-physical phase states. If we want to improve the precision further we need to accept a reduced probability of success, either by implementing a probabilistic measurement or by probabilistically manipulating the measured quantum state by means of noiseless amplification. We analyze the limits of this approach by finding the optimal probabilistic measurement which, for a given rate of success, maximizes the precision with which the phase can be measured.Phase is a central concept in both classical and quantum optics. It was, however, a matter of lengthy dialogue, before the quantum description of phase was established. The initial attempts of Dirac to treat phase as a canonical conjugate to photon number failed, because it is impossible to represent phase by a quantum mechanical observable [1]. As a consequence, phase can not be projectively measured, it can only be estimated (or guessed) by analyzing the results of other measurements. Despite this, phase states do exist [2] (even if they are not orthogonal) and they were eventually used to construct a well behaved phase operator [3]. Other attempts to describe phase properties of quantum states relied on the measurement-related phase distribution [4]. Both approaches were later reconciled with the fundamental canonical phase distribution [5].The canonical phase distribution characterizes phase properties of a quantum state and it is completely independent of its photon number distribution. It can be used to obtain a wide range of quantities related to phase estimation, but it also determines how much information about the phase of the state can be obtained by performing a measurement only on a single copy of it. True, the ideal canonical phase measurement does not and cannot exist, but several approximative approaches have been suggested [7,8].Aside from improving the actual detector scheme, overall performance of phase measurement can be also enhanced by specific alteration of the measured quantum state. Highly nonclassical quantum state can, in principle, lead to an unparalleled precision [9], while weakly nonclassical states are both beneficial and experimentally feasible [10]. However, if the state is unaccessible prior to phase encoding, we need to rely on operations which can enhance the amount of phase information already carried by the scrutinized state. Such operations are commonly referred to as noiseless amplifiers and a great deal of attention was recently devoted both to the concept [11] and to the experimental realizations [12]. The cost of this improvement comes in the reduced success rate of the operation. The amplification is therefore not very practical when the measurements can be repeated, but ut may be useful when the event to be detected is rare and we need to be certain that the single obtained measurement outcome corresponds to the theoretical value as closely as possible.However, even in the scenarios in which the probabili...

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