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Annual Meeting Optical Society of America

Phase-conjugate quantum communication at zero error probability with finite average photon number

Jeffrey H. Shapiro

Published Jan 1, 1991• 0 citations
Publisher

Abstract

The Susskind-Glogower (SG) phase operator underlies quantum phase for a single-mode field—its phase eigenkets generate the probability operator measure (POM) for maximum-likelihood quantum phase estimation.1 For an arbitrary state, the associated phase representation is the Fourier transform of the more familiar number representation. Thus, because the number representation is a one-sided sequence, the Paley-Wiener theorem constrains the possible phase measurement probability densities for finite-energy single-mode fields,2 precluding error-free phase-modulated communication.3 The SG-POM is not an observable on the signal state space. However, by adjoining an unexcited apparatus mode, the SG-POM can be made as a pair of commuting observables on the extended, signal × apparatus, state space.2 We show that a correlated signal × apparatus state—called a number-product vacuum state—leads to a two-sided Fourier relationship that thwarts the Paley-Wiener theorem. Using this arrangement, with a phase-conjugate modulator, we obtain error-free transmission of K phase shifts at a root-mean-squares photon number of K/2. Use of this construct in interferometric precision measurements is discussed.

DOI
10.1364/oam.1991.mo4

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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.

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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}$a^; its phase eigenkets generate the probability operator measure (POM) for maximum-likelihood phase estimation. When used with an optimum input state, the SG-POM produces a local phase accuracy \ensuremath{\delta}\ensuremath{\varphi}\ensuremath{\sim}1/${\mathit{N}}^{2}$, 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.

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Journal of Modern Optics•May 1, 1993•59 citations
The Susskind-Glogower (SG) phase operator underlies quantum phase for a single-mode field—its phase eigenkets generate the probability operator measure (POM) for maximum-likelihood quantum phase estimation.1 For an arbitrary state, the associated phase representation is the Fourier transform of the more familiar number representation. Thus, because the number representation is a one-sided sequence, the Paley-Wiener theorem constrains the possible phase measurement probability densities for finite-energy single-mode fields,2 precluding error-free phase-modulated communication.3 The SG-POM is not an observable on the signal state space. However, by adjoining an unexcited apparatus mode, the SG-POM can be made as a pair of commuting observables on the extended, signal × apparatus, state space.2 We show that a correlated signal × apparatus state—called a number-product vacuum state—leads to a two-sided Fourier relationship that thwarts the Paley-Wiener theorem. Using this arrangement, with a phase-conjugate modulator, we obtain error-free transmission of K phase shifts at a root-mean-squares photon number of K/2. Use of this construct in interferometric precision measurements is discussed.
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 Paley–Wiener restriction on the statistics of single-mode quantum phase measurements implies that single-mode, phase-modulated quantum communication always has nonzero error probability at finite average photon number. A two-mode formulation is demonstrated which circumvents the Paley–Wiener constraint, leading to a scheme for zero-error probability phase-conjugate quantum communication at finite average photon number. The minimum root-mean-square (RMS) total photon number for error-free K-ary phase-conjugate communication turns out to be K/2, and the state achieving this optimum performance is exhibited. Application of the construct to precision measurements is briefly discussed. Here, the optimum state with RMS photon number K/2 can be used to guarantee that the phase estimate is within ±π/K radians of the true value.
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 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.
In a shot-noise-limited (coherent state) phasesensing interferometer, the root-mean-square (rms) phase error is ~ 1 / N av 1 / 2 where Nav is the average number of detected photons. In an optimized squeezed-state interferometer, the rms phase error goes as 1/Nav. We have considered the phase measurement problem from the viewpoint of quantum estimation theory.1 In particular, for the problem of optimum phase-shift estimation from observation of a single mode of the radiation field under an average photon number constraint, we have found both the optimum quantum state and the optimum quantum measurement. The former turns out to have a truncated number-state expansion whose coefficients, out to the truncation point, fall off as 1/n. The latter turns out to be the Susskind-Glogower phase operator. More important, the resulting phase estimation performance improves as 1 / N av 2 .
Various measures of phase resolution (including variance, entropy, confidence half-width, and reciprocal peak likelihood) are reviewed, and applied to the class of coherent phase states. These states arise naturally as eigenstates of the Susskind-Glogower phase operator, and are found to have near-optimal phase resolution properties for fixed mean photon number [nbar]. The mutual information for a communication channel based on coherent phase states is calculated as log ([nbar] + 1), within 1·5 bits of the theoretical maximum for single-mode channels.