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Quantum Optics Journal of the European Optical Society Part B

Quantum phase detection and digital communication

Michael J. W. Hall, Ian Fuss

Published Jun 1, 1991• 34 citations
Publisher

Abstract

A canonical description of electromagnetic phase, related to both the Susskind-Glogower and Pegg-Barnett phase formalisms is applied to determine the quantum limits for digital communication based on phase detection. It is proved that canonical phase detection is superior to all other shift-invariant phase detection methods, in its ability to accurately resolve signals into phase bins under any given energy constraint. Under a bounded energy constraint, the optimal signal states are characterized in terms of discrete prolate spheroidal sequences and an asymptotic formula is given for the minimum error rate. Numerical results are given to provide comparison between canonical and heterodyne phase detection. A new method of deriving the statistics of (ideal) heterodyne detection for a given image-band field is presented.

DOI
10.1088/0954-8998/3/3/002

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IEEE Journal of Quantum Electronics•Jul 1, 1984•199 citations
A canonical description of electromagnetic phase, related to both the Susskind-Glogower and Pegg-Barnett phase formalisms is applied to determine the quantum limits for digital communication based on phase detection. It is proved that canonical phase detection is superior to all other shift-invariant phase detection methods, in its ability to accurately resolve signals into phase bins under any given energy constraint. Under a bounded energy constraint, the optimal signal states are characterized in terms of discrete prolate spheroidal sequences and an asymptotic formula is given for the minimum error rate. Numerical results are given to provide comparison between canonical and heterodyne phase detection. A new method of deriving the statistics of (ideal) heterodyne detection for a given image-band field is presented.
Phase measurements on a single-mode radiation field are examined from a system-theoretic viewpoint. Quantum estimation theory is used to establish the primacy of the Susskind-Glogower (SG) phase operator; its phase eigenkets generate the probability operator measure (POM) for maximum likelihood phase estimation. A commuting observables description for the SG-POM on a signal x apparatus state space is derived. It is analogous to the signal-band x image-band formulation for optical heterodyne detection. Because heterodyning realizes the annihilation operator POM, this analogy may help realize the SG-POM. The wave function representation associated with the SG POM is then used to prove the duality between the phase measurement and the number operator measurement, from which a number-phase uncertainty principle is obtained, via Fourier theory, without recourse to linearization. Fourier theory is also employed to establish the principle of number-ket causality, leading to a Paley-Wiener condition that must be satisfied by the phase-measurement probability density function (PDF) for a single-mode field in an arbitrary quantum state. Finally, a two-mode phase measurement is shown to afford phase-conjugate quantum communication at zero error probability with finite average photon number. Application of this construct to interferometric precision measurements is briefly discussed.
The double-homodyne and the heterodyne detection schemes for phase shifts between two synchronous modes of the electromagnetic field are analyzed in the framework of quantum estimation theory. The probability operator-valued measures (POM's) of the detectors are evaluated and compared with the ideal one in the limit of strong local reference oscillator. The present operational approach leads to a reasonable definition of phase measurement, whose sensitivity is actually related to the output r.m.s. noise of the photodetector. We emphasize that the simple-homodyne scheme does not correspond to a proper phase-shift measurements as it is just a zero-point detector. The sensitivity of all detection schemes are optimized at fixed energy with respect to the input state of radiation. It is shown that the optimal sensitivity can be actually achieved using suited squeezed states.
The Shapiro-Wagner phase measurement is addressed here. It is clarified that this concept includes an infinite number of operator realizations of quantum phase measurements, which may be characterized using polar and spectral decomposition. For accurate phase measurement, the regimes of ideal and optimized phase resolution are specified. The former is equivalent to the Susskind-Glogower or Hermitian phase concepts and needs infinite energy on the auxiliary input port. The ultimate phase resolution cannot surpass the limit 1/n, n being the total average number of photons entering both the signal and image input ports.
Phase measurements on a single-mode radiation field of annihilation operator a^ are examined from a system-theoretic viewpoint. Quantum estimation theory is used to establish the primacy of the Susskind-Glogower (SG) phase operator, exp^(i\ensuremath{\varphi})==(a^a${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${)}^{\mathrm{\ensuremath{-}}1/2}$a^; its phase eigenkets generate the probability operator measure (POM) for maximum-likelihood phase estimation. When used with an optimum input state, the SG-POM produces a local phase accuracy \ensuremath{\delta}\ensuremath{\varphi}\ensuremath{\sim}1/N2{\mathit{N}}^{2}N2, where N is the average photon number of the state. This performance is far superior to the \ensuremath{\delta}\ensuremath{\varphi}\ensuremath{\sim}1/ \ensuremath{\surd}N and \ensuremath{\delta}\ensuremath{\varphi}\ensuremath{\sim}1/N behaviors of coherent-state and optimized squeezed-state interferometers, respectively. A commuting-observables description for the SG-POM on an extended, signal\ifmmode\times\else\texttimes\fi{}apparatus, state space is derived. It is analogous to the signal-band\ifmmode\times\else\texttimes\fi{}image-band formulation for optical heterodyne detection. Because heterodyne detection realizes the a^-POM, this analogy may help realize the SG-POM. Two new classes of nonclassical states are identified through study of the SG operator---the coherent phase states and the squeezed phase states. These are direct analogs of the familiar coherent and squeezed states associated with the annihilation operator. Linear system theory is used to elicit a broader class of states---the rational phase states---which includes the coherent and squeezed phase states as special cases, and affords a filter-theory derivation of the minimum average-energy quantum state with prescribed phase-measurement statistics. Fourier theory is also used to prove a number-phase uncertainty relation without recourse to the usual linearization procedures, and to show that the Pegg-Barnett phase operator has the same measurement statistics as the SG-POM for arbitrary quantum states.
The quantum limits on simultaneous phase and squared-amplitude measurements made via optical heterodyne detection on a single-mode radiation field are established. The analysis proceeds from a fully quantum mechanical treatment of heterodyning with ideal photon detectors. A high mean field uncertainty principle is proven for simultaneous phase and squared-amplitude observations under the condition that the signal and image band states are independent, and the image band has zero mean. Operator representations are developed which show that no such principle applies when arbitrary signal/image band dependence is permitted, although the mean observations are no longer functions of the signal field alone. A multimode two-photon coherent state illustrating this behavior at finite energy is exhibited. Potential applications for the resulting improved accuracy measurements are briefly described.