Non-classical states of light find applications in enhancing the performance of optical interferometric experiments, with notable example of gravitational wave-detectors. Still, the presence of decoherence hinders significantly the performance of quantum-enhanced protocols. In this review, we summarize the developments of quantum metrology with particular focus on optical interferometry and derive fundamental bounds on achievable quantum-enhanced precision in optical interferometry taking into account the most relevant decoherence processes including: phase diffusion, losses and imperfect interferometric visibility. We introduce all the necessary tools of quantum optics as well as quantum estimation theory required to derive the bounds. We also discuss the practical attainability of the bounds derived and stress in particular that the techniques of quantum-enhanced interferometry which are being implemented in modern gravitational wave detectors are close to the optimal ones.

Simultaneous estimation of multiple parameters in quantum metrological models is complicated by factors relating to the (i) existence of a single probe state allowing for optimal sensitivity for all parameters of interest, (ii) existence of a single measurement optimally extracting information from the probe state on all the parameters, and (iii) statistical independence of the estimated parameters. We consider the situation when these concerns present no obstacle, and for every estimated parameter the variance obtained in the multiparameter scheme is equal to that of an optimal scheme for that parameter alone, assuming all other parameters are perfectly known. We call such models compatible. In establishing a rigorous theoretical framework for investigating compatibility, we clarify some ambiguities and inconsistencies present in the literature and discuss several examples to highlight interesting features of unitary and nonunitary parameter estimation, as well as deriving new bounds for physical problems of interest, such as the simultaneous estimation of phase and local dephasing.

We discuss the role of an external phase reference in quantum interferometry. We point out inconsistencies in the literature with regard to the use of the quantum Fisher information (QFI) in phase estimation interferometric schemes. We discuss the interferometric schemes with and without an external phase reference and show a proper way to use QFI in both situations.PACS numbers: 03.65. Ta, 06.20.Dk Laws of quantum mechanics impose fundamental bounds on measurement precisions of basic physical quantities such as position, momentum, energy, time, phase etc. Theses bounds follow from the structure of the theory itself which contrasts the situation encountered in classical physics where measurement uncertainties are due to factors which in principle may be eliminated by improving the quality of measurement procedures. One of the most important measurement techniques where such bounds have been analyzed is optical interferometry [1].In a generic interferometric measurement using a Mach-Zehnder setup and classical light sources the precision of estimating the relative phase delay ϕ inside the interferometer is bounded by the so called standard quantum limit (SQL) δϕ ≥ 1/ √ N , where N is average number of photon-counts. At the fundamental quantum level, the bound is a result of an independent probabilistic behavior of individual photon propagating through the interferometer.Breaching the SQL requires the use of special nonclassical states of light were photons can no longer be regarded as independent. One of the first proposals in this direction was the idea to mix coherent light with the squeezed vacuum at the input beam splitter of the Mach-Zehnder interferometer [2]. Thanks to the reduced vacuum fluctuations in one of the quadratures of the squeezed state it is possible to improve the precision beyond the SQL. This observations prompted the search for more fundamental bounds on achievable precision, which would be obeyed by all quantum states [3].In general, looking for the optimal phase estimation protocols is difficult since one needs to optimize over the input state |ψ in that is fed into the interferometer, the measurement {Π n } that is performed at the output and the estimator ϕ(n) -a function that assigns a phase value to a given measurement outcome. One of the popular ways to obtain useful bounds in quantum metrology, without the need for Figure 1. An interferometric scheme with coherent and squeezed vacuum states interfered at a beam-splitter, with arbitrary quantum measurement potentially involving an additional reference beam. In general, the QFI bounds on the phase estimation precision depend on the way the interferometer phase delay is modeled: (i) phase shift only in the upper arm, (ii) phase shift distributed symmetrically, (iii) phase shifts defined with respect to an additional reference beam.cumbersome optimization, is to use the concept of the quantum Fisher information (QFI) [4].The purpose of this paper is to give a proper interpretation to the bounds obtained via the QFI and point out conf...

We show that quantification of the performance of quantum-enhanced measurement schemes based on the concept of quantum Fisher information (QFI) yields results that are asymptotically equivalent to those from the rigorous Bayesian approach, provided generic uncorrelated noise is present in the setup. At the same time, we show that for the problem of decoherence-free phase estimation this equivalence breaks down, and the achievable estimation uncertainty calculated within the Bayesian approach is larger by a factor of π than that predicted from the QFI even in the large prior knowledge (small parameter fluctuation) regime, where the QFI is conventionally regarded as a reliable figure of merit. We conjecture that an analogous discrepancy is present in the arbitrary decoherence-free unitary parameter estimation scheme, and propose a general formula for the asymptotically achievable precision limit. We also discuss protocols utilizing states with an indefinite number of particles, and show that within the Bayesian approach it is legitimate to replace the number of particles with the mean number of particles in the formulas for the asymptotic precision, which as a consequence provides another argument for proposals based on the properties of the QFI of indefinite particle number states leading to sub-Heisenberg precisions not being practically feasible.

We consider the problem of characterising the spatial extent of a composite light source using the superresolution imaging technique when the centroid of the source is not known precisely. We show that the essential features of this problem can be mapped onto a simple qubit model for joint estimation of a phase shift and a dephasing strength.

Erratum: Compatibility in multiparameter quantum metrology [Phys. Rev. A 94, 052108 (2016)]

We demonstrate that the optimal states in lossy quantum interferometry may be efficiently simulated using low rank matrix product states. We argue that this should be expected in all realistic quantum metrological protocols with uncorrelated noise and is related to the elusive nature of the Heisenberg precision scaling in the asymptotic limit of large number of probes.

Abstract:We analyze the performance of on-off keying (OOK) and its restricted version pulse position modulation (PPM) over a lossy narrowband optical channel under the constraint of a low average photon number, when direct detection is used at the output. An analytical approximation for the maximum PPM transmission rate is derived, quantifying the effects of photon statistics on the communication efficiency in terms of the g (2) second-order intensity correlation function of the light source. Enhancement attainable through the use of sub-Poissonian light is discussed. Comput. Math. 5(4), 329-359 (1996). 14. K. Tsujino, D. Fukuda, G. Fujii, S. Inoue, M. Fujiwara, M. Takeoka, and M. Sasaki, "Sub-shot-noise-limit discrimination of on-off keyed coherent signals via a quantum receiver with a superconducting transition edge sensor," Opt. Express 18(8), 8107-8114 (2010). 15. J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, "Optical codeword demodulation with error rates below the standard quantum limit using a conditional nulling receiver," Nature Photon. 6(6), 374-379 (2012). 16. S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris, "Homodyne detection as a near-optimum receiver for phase-shift-keyed binary communication in the presence of phase diffusion," Phys. Rev. A 87(5), 050303(R) (2013).