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Physical Review Letters

Optimal Quantum Measurements for Phase Estimation

Barry C. Sanders, Gerard J Milburn

Published Oct 16, 1995• 242 citations
Publisher

Abstract

Quantum information theory is applied to practical interferometer-based phase measurements to deduce the optimal phase measurement scheme with two optical modes. Optimal phase measurements, given ideal input states, reveal an asymptotic $1/n$ decrease in phase uncertainty $\ensuremath{\Delta}\ensuremath{\theta}$ for $n$ the mean photon number of the input state. In contradistinction to previous schemes for realizing the number-phase uncertainty limit, the $1/n$ limit achieved here is independent of the interferometer phase shift; prior information about the expected phase shift is not necessary to attain this limit. These results apply more generally to su(2) and so(3) phase parameter estimation.

DOI
10.1103/physrevlett.75.2944

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Quantum information theory is applied to practical interferometer-based phase measurements to deduce the optimal phase measurement scheme with two optical modes. Optimal phase measurements, given ideal input states, reveal an asymptotic 1/n1/n1/n decrease in phase uncertainty \ensuremathΔ\ensuremathθ\ensuremath{\Delta}\ensuremath{\theta}\ensuremathΔ\ensuremathθ for nnn the mean photon number of the input state. In contradistinction to previous schemes for realizing the number-phase uncertainty limit, the 1/n1/n1/n limit achieved here is independent of the interferometer phase shift; prior information about the expected phase shift is not necessary to attain this limit. These results apply more generally to su(2) and so(3) phase parameter estimation.
Quantum information theory, applied to optical interferometry, yields a 1/n scaling of phase uncertainty Δφ independent of the applied phase shift φ, where n is the number of photons in the interferometer. This 1/n scaling is achieved provided that the output state is subjected to an optimal phase measurement. We establish this scaling law for both passive (linear) and active (nonlinear) interferometers and identify the coefficient of proportionality. Whereas a highly nonclassical state is required to achieve optimal scaling for passive interferometry, a classical input state yields a 1/n scaling of phase uncertainty for active interferometry.
We derive optimal N-photon two-mode input states for interferometric phase measurements. Under canonical measurements the phase variance scales as ${N}^{\ensuremath{-}2}$ for these states, as compared to ${N}^{\ensuremath{-}1}$ or ${N}^{\ensuremath{-}1/2}$ for states considered by previous authors. We prove that it is not possible to realize the canonical measurement by counting photons in the outputs of the interferometer, even if an adjustable auxiliary phase shift is allowed in the interferometer. However, we introduce a feedback algorithm based on Bayesian inference to control this auxiliary phase shift. This makes the measurement close to a canonical one, with a phase variance scaling slightly above ${N}^{\ensuremath{-}2}.$ With no feedback, the best result (given that the phase to be measured is completely unknown) is a scaling of ${N}^{\ensuremath{-}1}.$ For optimal input states having up to four photons, our feedback scheme is the best possible one, but for higher photon numbers more complicated schemes perform marginally better.
Summary form only given. Interferometry is the basis of many high-precision measurements. The ultimate limit to the precision is due to quantum effects. This limit is most easily explored for a Mach-Zehnder interferometer The outputs of this device can be measured to yield an estimate /spl phi/ of the phase difference /spl phi/ between the two arms of the interferometer. It is well known that this can achieve the standard quantum limit for phase variance when an N-photon number state enters one input port. Several authors have proposed ways of reducing the phase variance to the Heisenberg limit. In this paper we show that there is a physical measurement scheme using photodetectors and feedback that is almost as good as the optimal measurement. The fundamental idea is that, in addition to the unknown phase we wish to measure, /spl phi/, in one arm of the interferometer, we introduce a known phase shift, /spl Phi/, into the other arm of the interferometer. After each photodetection we adjust this introduced phase shift in order to minimize the expected uncertainty of our best phase estimate /spl phi/ after the next photodetection. This feedback process is uniquely defined using Bayesian statistics, and can be solved exactly for moderate photon numbers.
Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.
We give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.
Quantum stochastic phase estimation has many applications in the precise measurement of various physical parameters. Similar to the estimation of a constant phase, there is a standard quantum limit for stochastic phase estimation, which can be obtained with the Mach–Zehnder interferometer and coherent input state. Recently, it has been shown that the stochastic standard quantum limit can be surpassed with nonclassical resources such as squeezed light. However, practical methods to achieve quantum enhancement in the stochastic phase estimation remain largely unexplored. Here we propose a method utilizing the SU(1,1) interferometer and coherent input states to estimate a stochastic optical phase. As an example, we investigate the Ornstein–Uhlenback stochastic phase. We analyze the performance of this method for three key estimation problems: prediction, tracking, and smoothing. The results show significant reduction of the mean square error compared with the Mach–Zehnder interferometer under the same photon number flux inside the interferometers. In particular, we show that the method with the SU(1,1) interferometer can achieve fundamental quantum scaling, achieve stochastic Heisenberg scaling, and surpass the precision of the canonical measurement.