Fidelity of quantum interferometers

Abstract

For a generic interferometer, the conditional probability density distribution, p(φ|m), for the phase φ 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 ∆φ ∼ 1/ √ N (the standard limit), or ∆φ ∼ 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 φ 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. We find the surprising result that an entangled N00N state input leads to a lower fidelity than a Fock state input, for the same photon number. Introduction.Phase sensitivity of interferometers has been a topic of research for many years because of interest in the fundamental limitations of measurement [1, 2], gravitational-wave detection [3], and optical [4,5], atom [6], and Bose-Einstein condensate(BEC)-based gyroscopes [7,8,9]. Recently, applications to sensors are being explored [10,11].The phase sensitivity of interferometers is believed to be limited by quantum fluctuations [12], and the phase sensitivity of various interferometers has been explored for different types of input states, such as squeezed states [12,13], and number states [14,15,16,17,18,19,20,21,22]. In all the above cases, the phase sensitivity ∆φ has been discussed in terms of two limits, known as the standard limit, ∆φ SL = 1/ √ N , and the Heisenberg limit[23], ∆φ HL = 1/N , where N is the number of particles that enter the interferometer during each measurement cycle. These arguments are based on results of standard estimation theory [24] which connects an ensemble of measurement outcomes, m i , i = 1, 2, · · · , M , with corresponding phases, φ i , through a theoretical relation m = m(φ). An example of the theoretical relation associated with some quantum observable is m(φ) = φ| m|φ , where the state is parameterized by a single parameter φ. Standard estimation theory predicts that the standard deviation, ∆φ, of the probability distribution for the phase φ, is related to the standard deviation in the measurements, ∆m, by [24]

Evaluation of counterfactuality in counterfactual communication protocols

“…is the probability of M i for some specific θ and ψ in , and p(θ) is the a priori probability distribution of the parameter θ [18,35]. The Shannon mutual information provides a measure of how much information about θ that can be obtained through knowledge of the measurement outcomes of a specific experiment.…”

Phase estimation with nonunitary interferometers: Information as a metric

“…In the case of the measurement problem, quantum fluctuations in the initial state and in the channel (interferometer) and the type of measurement determine the amount of information that is obtained about the parameter φ from the measurements. The fidelity has been applied to compare the use of Fock states and N00N states when no prior information is present about the phase [26] and when there is significant prior information about the phase [55]. The complementary measures of fidelity and Fisher information may be contrasted as follows.…”

“…Perhaps the simplest generic measurement problem consists of determining the relative phase shift between two arms of an interferometer from measurements made at the output ports of the interferometer [20][21][22][23][24][25]. This phase shift may be related to a classical external field incident on a phase shifter in one arm of the interferometer, in which case the interferometer can be used as a sensor of the field [26]. The determination of the phase shift is a specific example of the more general problem of parameter estimation, whose goal is to determine one or more parameters from measurements [27][28][29][30][31][32][33][34].…”

Tried-and-true binary strategy for angular displacement estimation based upon fidelity appraisal

“…Therefore, we try to appraise the two strategies by replacing standard deviation with fidelity estimation. In this paper, we report using fidelity [25][26][27] (the Shannon mutual information between the angular displacement θ and the measuring outcomes) to appraise two binary detection strategies whose sensibilities are on a par with each other under the standard deviation metric. A simple understanding for the mutual information is the amount of sender's original information which can be obtained by receiver from the acquired distortion information.…”