Phase estimation at the quantum Cramér-Rao bound via parity detection

Abstract

In this letter, we show that for all the so-called path-symmetric states, the measurement of parity of photon number at the output of an optical interferometer achieves maximal phase sensitivity at the quantum Cramer-Rao bound. Such optimal phase sensitivity with parity is attained at a suitable bias phase, which can be determined a priori. Our scheme is applicable for local phase estimation.PACS numbers: 42.50. St, 42.50.Dv, 42.50.Ex, 42.50.Lc Interferometry is a vital component of various precision measurement, sensing, and imaging techniques. It works based on mapping the quantity of interest on to the unknown phase of a system and estimating the latter; for example, the relative phase between the two modes or "arms" of an optical interferometer. Optical interferometry, often described in the Mach-Zehnder configuration, in general differs in the strategies of state preparation and detection. The conventional choice is to use a coherent light source and intensity difference detection. Assuming the unitary phase acquisition operator to be:(wheren a ,n b are the number operators associated with the modes,) the phase sensitivity of the conventional Mach-Zehnder interferometer (MZI) is bounded by the shot noise limit (SNL) δφ = 1/ √n (forn photons in the coherent state on average); whereas, protocols of quantum interferometry promise enhanced phase sensitivities by prescribing the use of states with nonclassical photon correlations and detection strategies that probe the particle nature of the output light, for example, via number counting [1]. As for such nonclassical states, the proposal to squeeze the vacuum state entering the unused port of the conventional MZI, by Caves in 1981, resulted in the first instance of a quantum-enhanced MZI capable of operating below the SNL [2]. Others, such as the twin-Fock state [3], the maximally path-entangled N00N state (|N, 0 + |0, N )/ √ 2, which reach the Heisenberg limit (HL) δφ = 1/N , were later proposed [4].Much of the latest experimental efforts in quantum interferometry have been focussed on attaining the HL [5]. The theory of quantum phase estimation aids in identifying potential schemes for such phase sensitivities [6]. It is based on the information-theoretical concept of the Cramer-Rao bound [7]. For the form of unitary phase acquisition considered in Eq. (1), the quantum CramerRao bound (QCRB)-an attribute of the quantum state input to the MZI alone (independent of detection)-at its best, can reach Heisenberg scaling in the absence of photon losses. On the other hand, the classical CramerRao bound (CCRB)-an attribute of the combination of a quantum state and detection strategy-can optimally reach the QCRB of the state. Uys and Meystre derived optimal quantum states, whose CCRB with number counting-based detection attains the HL in the absence of photon losses [8]. Lee et al. included photon losses to the problem and worked out the optimal inputs [9]. Meanwhile, Dorner et al. found optimal inputs in the presence of photon losses for the generic optimal detecti...

Adaptive phase estimation with two-mode squeezed vacuum and parity measurement

“…Since the photon number in a TMSV state is always even, the parity signals are the same in both the output ports, and therefore can be detected by performing photon-counting at only one port. It turns out that parity detection is sufficient to achieve the Cramér-Rao bound in the case where the state is path symmetric [27], which is the case here. Propagation of the light through a MZI imprints phase information on the state that is retrieved by measuring parity at the output of the MZI.…”

“…For these states, it was previously shown that parity detection yields a phase sensitivity, estimated from the Cramér-Rao bound, beyond the Heisenberg limit. Since then, it was found that parity detection attains the Cramér-Rao bound for a wide range of states including TMSV states [38].…”

Quantum interferometry with binary-outcome measurements in the presence of phase diffusion

“…(6) remains valid for arbitrary input state and is independent from the presence of noises. Previously, Seshadreesan et al [13] found that for the parity measurement, the inversion estimator can reach the CR bound. This measurement is a special case of photon counting at one of two output ports with the outcomes µ ± = ±1 (see below).…”

“…The parity measurement [10][11][12][13] at the output port c, described by the parity operatorΠ = (−1)ˆc †ĉ , groups the photon counting {n, m} into binary outcomes ±1, according to even or odd number of photons n at that port c. Such a kind of data processing provides an optimal phase estimator for the input path-symmetric states [13]. The conditional probabilities P(±1|φ) are obtained by a sum of P(n, m|φ) over the even or the odd n's, namely…”

“…It originates from atomic spectroscopy with an ensemble of trapped ions [10] and was discussed in the context of optical interferometry by Gerry [11] and subsequently by others [12,13]. Recently, Gao et al [14] proposed that the parity measurement can lead to the super-resolution in the coherentlight MZI.…”