Maximum likelihood estimation of an unknown c-number phaseshift is considered, along the lines established by Shapiro, Shepard and Wong, Phys. Rev. Lett., vol.62, p.2377, 1989. An input state is exhibited whose reciprocal peak likelihood vanishes, delta phi to 0, while its phase-measurement probability density converges, in distribution, to uniform, p( phi mod Phi ) to d 1/2 pi .

Quantum Optics Journal of the European Optical Society Part BFeb 1, 199222 citations

Exponential decrease in phase uncertainty

Wolfgang P. Schleich, Jonathan P. Dowling, R. J. Horowicz

Physical Review ASep 1, 199131 citations

Physical Review Letters

Maximum-likelihood analysis of multiple quantum phase measurements

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

Published Oct 12, 1992• 72 citations

Publisher

Abstract

{\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.

DOI: 10.1103/physrevlett.69.2153

Related papers

Maximum-likelihood statistics of multiple quantum phase measurements

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

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.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.

Physical Review AMar 1, 199392 citations

Maximum-likelihood statistics of multiple quantum phase measurements

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

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.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.

Annual Meeting Optical Society of AmericaJan 1, 19901 citation

Phase measurements at the theoretical limit

Dominic William Berry, Howard M. Wiseman

It is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode. The best previously published technique [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] is an adaptive technique that introduces a phase variance that scales as ${n}^{\ensuremath{-}1.5},$ where $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ is the mean photon number of the state. This is far above the minimum intrinsic quantum phase variance of the state, which scales as ${n}^{\ensuremath{-}2}.$ It has been shown that a lower limit to the phase variance that is introduced scales as $\mathrm{ln}(\mathrm{n\ifmmode \bar{}\else \={}\fi{}})/{n}^{2}.$ Here we introduce an adaptive technique that attains this theoretical lower limit.It is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode. The best previously published technique [H. M. Wiseman and R. B. Killip, Phys. Rev. A 57, 2169 (1998)] is an adaptive technique that introduces a phase variance that scales as ${n}^{\ensuremath{-}1.5},$ where $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ is the mean photon number of the state. This is far above the minimum intrinsic quantum phase variance of the state, which scales as ${n}^{\ensuremath{-}2}.$ It has been shown that a lower limit to the phase variance that is introduced scales as $\mathrm{ln}(\mathrm{n\ifmmode \bar{}\else \={}\fi{}})/{n}^{2}.$ Here we introduce an adaptive technique that attains this theoretical lower limit.

arXiv:quant-ph/0002059Physical Review ADec 12, 200017 citations

Quantum signal processing for quantum phase estimation: Fourier transform versus maximum likelihood approaches

François Chapeau‐Blondeau, Étienne Belin

Annals of TelecommunicationsSep 5, 20207 citations

Quantum phase measurements with infinite peak-likelihood and zero phase information

Z. Hradil, Jeffrey H. Shapiro

Quantum Optics Journal of the European Optical Society Part BFeb 1, 199222 citations

Exponential decrease in phase uncertainty

Wolfgang P. Schleich, Jonathan P. Dowling, R. J. Horowicz

The phase probability curve of a recently proposed photon state consists of a broad background with a sharp central peak [J. H. Shapiro, S. R. Shepard, and N. Wong, Phys. Rev. Lett. 62, 2377 (1989)]. These authors argue that the inverse peak-height phase uncertainty \ensuremath{\delta}cphi of this distribution decreases inversely as the square of the mean photon number 〈m〉---an improvement over either coherent or highly squeezed states. We show that the width \ensuremath{\Delta}cphi of the best-fitting Gaussian to the central peak--a measure of phase uncertainty tailored to this narrow feature--decreases exponentially with increasing 〈m〉. The importance of this result may be offset by the observation that the area under this peak also vanishes very rapidly.The phase probability curve of a recently proposed photon state consists of a broad background with a sharp central peak [J. H. Shapiro, S. R. Shepard, and N. Wong, Phys. Rev. Lett. 62, 2377 (1989)]. These authors argue that the inverse peak-height phase uncertainty \ensuremath{\delta}cphi of this distribution decreases inversely as the square of the mean photon number 〈m〉---an improvement over either coherent or highly squeezed states. We show that the width \ensuremath{\Delta}cphi of the best-fitting Gaussian to the central peak--a measure of phase uncertainty tailored to this narrow feature--decreases exponentially with increasing 〈m〉. The importance of this result may be offset by the observation that the area under this peak also vanishes very rapidly.

Physical Review ASep 1, 199131 citations

{\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.

{\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.