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Physical Review A

Maximum-likelihood statistics of multiple quantum phase measurements

Alistair S. Lane, Samuel L. Braunstein, Carlton M. Caves

Published Mar 1, 1993• 92 citations
Publisher

Abstract

Shapiro, Shepard, and Wong [Phys. Rev. Lett. 62, 2377 (1989)] suggested that a scheme of multiple phase measurements, using quantum states with minimum ``reciprocal peak likelihood,'' could achieve a phase sensitivity scaling as 1/${\mathit{N}}_{\mathrm{tot}}^{2}$, where ${\mathit{N}}_{\mathrm{tot}}$ is the mean number of photons available for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3\ensuremath{\le}${\mathit{N}}_{\mathrm{tot}}$\ensuremath{\le}120; a power-law fit to the simulated data yields a phase sensitivity that scales as 1/${\mathit{N}}_{\mathrm{tot}}^{0.82\ifmmode\pm\else\textpm\fi{}0.01}$. By using a combination of numerical and analytical techniques, we extend our results to higher values of ${\mathit{N}}_{\mathrm{tot}}$ than are accessible to our simulations; we find no evidence for phase sensitivities better than the benchmark 1/${\mathit{N}}_{\mathrm{tot}}$ sensitivity of squeezed-state interferometry. We conclude that reciprocal peak likelihood is not a good measure of phase sensitivity. We discuss other factors that are important to phase sensitivity.

DOI
10.1103/physreva.47.1667

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Maximum-likelihood analysis of multiple quantum phase measurements

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Shapiro, Shepard, and Wong [Phys. Rev. Lett. 62, 2377 (1989)] suggested that a scheme of multiple phase measurements, using quantum states with minimum ``reciprocal peak likelihood,'' could achieve a phase sensitivity scaling as 1/${\mathit{N}}_{\mathrm{tot}}^{2}$, where ${\mathit{N}}_{\mathrm{tot}}$ is the mean number of photons availavle for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3\ensuremath{\le}${\mathit{N}}_{\mathrm{tot}}$\ensuremath{\le}120. A power-law fit to the simulated data yields a phase sensitivity that scales as 1/${\mathit{N}}_{\mathrm{tot}}^{0.85\ifmmode\pm\else\textpm\fi{}0.01}$. We conclude that reciprocal peak likelihood is not a good measure of sensitivity.

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Physical Review Letters•Dec 21, 1992•148 citations
Shapiro, Shepard, and Wong [Phys. Rev. Lett. 62, 2377 (1989)] suggested that a scheme of multiple phase measurements, using quantum states with minimum ``reciprocal peak likelihood,'' could achieve a phase sensitivity scaling as 1/Ntot2{\mathit{N}}_{\mathrm{tot}}^{2}Ntot2​, where Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​ is the mean number of photons available for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3\ensuremath{\le}Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​\ensuremath{\le}120; a power-law fit to the simulated data yields a phase sensitivity that scales as 1/Ntot0.82\ifmmode±\else\textpm\fi0.01{\mathit{N}}_{\mathrm{tot}}^{0.82\ifmmode\pm\else\textpm\fi{}0.01}Ntot0.82\ifmmode±\else\textpm\fi0.01​. By using a combination of numerical and analytical techniques, we extend our results to higher values of Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​ than are accessible to our simulations; we find no evidence for phase sensitivities better than the benchmark 1/Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​ sensitivity of squeezed-state interferometry. We conclude that reciprocal peak likelihood is not a good measure of phase sensitivity. We discuss other factors that are important to phase sensitivity.
Shapiro, Shepard, and Wong [Phys. Rev. Lett. 62, 2377 (1989)] suggested that a scheme of multiple phase measurements, using quantum states with minimum ``reciprocal peak likelihood,'' could achieve a phase sensitivity scaling as 1/Ntot2{\mathit{N}}_{\mathrm{tot}}^{2}Ntot2​, where Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​ is the mean number of photons availavle for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3\ensuremath{\le}Ntot{\mathit{N}}_{\mathrm{tot}}Ntot​\ensuremath{\le}120. A power-law fit to the simulated data yields a phase sensitivity that scales as 1/${\mathit{N}}_{\mathrm{tot}}^{0.85\ifmmode\pm\else\textpm\fi{}0.01}$. We conclude that reciprocal peak likelihood is not a good measure of sensitivity.
Recent theoretical studies have considered various possibilities for experimental measurements of phase using quantum states of a finite mean number of photons. Currently, the most experimentally accessible of these possibilities is an interferometric phase measurement using a squeezed state that has minimum variance for the phase quadrature; this possibility can yield, in principle, a phase measurement with uncertainty of order l/(mean number of photons). Shapiro, Shepard, and Wong1 (SSW) have proposed another possibility. They maximize the peak likelihood in an abstract measurement of the Susskind-Glogower phase operator, and they claim a phase sensitivity that goes as l/(mean number of photons).2 The SSW phase probability distribution consists of an extremely narrow peak sitting on a broad base that covers the entire 2 radians of phase; as a result, a single SSW phase measurement has poor sensitivity. Shapiro, Shepard, and Wong argue that their claimed sensitivity can be achieved in a sequence of measurements. We study the statistics of the maximum-likelihood estimator in a sequence of SSW phase measurements, and we report how the uncertainty in the estimator scales with the mean number of photons used in the entire sequence.
The Susskind-Glogower (SG) phase operator is shown to be the maximum-likelihood (ML) quantum measurement of optical phase for all input states. The ML performance of the SG measurement is optimized, at average photon number N, by an input state with number-ket representation of the form ${\mathrm{\ensuremath{\psi}}}_{\mathrm{n}}$=A/(1+n), for 0\ensuremath{\le}n\ensuremath{\le}M<\ensuremath{\infty}. The optimum phase-error behavior, measured by the reciprocal peak-likelihood, is proportional to 1/N2{\mathrm{N}}^{2}N2, as opposed to the 1/ \ensuremath{\surd}N and 1/N dependences of coherent-state and optimum squeezed-state interferometers.
Dispersion and peak likelihood---two performance measures recently used for the description of quantum phase---are compared in this Rapid Communication. The Susskind-Glogower phase operator represents the extremal quantum estimator with respect to both measures. The states with infinite peak likelihood and zero phase information explicitly demonstrate that peak likelihood is not relevant to the phase measurement, whereas dispersion represents a well-behaved performance measure of the phase variable. Peak likelihood restricts the lower limit of the variance of photon number, and the above-mentioned states represent the extremal states of the Shapiro-Shepard [Phys. Rev. A 43, 3795 (1991)] uncertainty relation.
We investigate the quantum mechanical bound to how precisely we can determine a phase shift given only a constraint on the mean total number of photons available. By considering how quickly one can gain information from data analysis, we derive the sensitivity achievable (in principle) for measurements involving even highly non-Gaussian noise. Using these results we calculate the sensitivity of several recent proposals for precision phase measurement, and show that no proposal to date beats the sensitivity believed achievable by squeezed state interferometry.