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Physica Scripta

Information theory and optimal phase measurement

K. R. W. Jones

Published Jan 1, 1993• 9 citations
Publisher

Abstract

Using a single mode analysis, we assess the utility of Shannon information as a robust performance criterion for phase detection. To provide a sharp test, we apply it to the Shapiro, Shepard and Wong optimal phase state. This has well-known pathologies, and is thus a good hurdle. Numerical studies show that the SSW-state information gain converges rapidly to the finite value ⟨Δ I ⟩ ∼ 0.966 bits at infinite photon number. For large photon numbers N the truncated phase state scheme yields ⟨Δ I ⟩ ∼ log 2 (2 N + 1) − 1.220 bits whereas the coherent state scheme returns ⟨Δ I ⟩ ∼ log 2 N 1/2 − 1.604 bits. Numerical studies confirm these asymptotic formulae and reveal an interesting crossover regime where the coherent state scheme proves superior to the truncated phase state scheme at low photon numbers N ⩽ 10. The anomalous behaviour of the SSW-state is studied with the aid of a simple scaling argument.

DOI
10.1088/0031-8949/1993/t48/015

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Using a single mode analysis, we assess the utility of Shannon information as a robust performance criterion for phase detection. To provide a sharp test, we apply it to the Shapiro, Shepard and Wong optimal phase state. This has well-known pathologies, and is thus a good hurdle. Numerical studies show that the SSW-state information gain converges rapidly to the finite value ⟨Δ I ⟩ ∼ 0.966 bits at infinite photon number. For large photon numbers N the truncated phase state scheme yields ⟨Δ I ⟩ ∼ log 2 (2 N + 1) − 1.220 bits whereas the coherent state scheme returns ⟨Δ I ⟩ ∼ log 2 N 1/2 − 1.604 bits. Numerical studies confirm these asymptotic formulae and reveal an interesting crossover regime where the coherent state scheme proves superior to the truncated phase state scheme at low photon numbers N ⩽ 10. The anomalous behaviour of the SSW-state is studied with the aid of a simple scaling argument.
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.
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.
Optical phase measurement is a simple example of a quantum--limited measurement problem with important applications in metrology such as gravitational wave detection. The formulation of optimal strategies for such measurements is an important test-bed for the development of robust statistical methods for instrument evaluation. However, the class of possible distributions exhibits extreme pathologies not commonly encountered in conventional statistical analysis. To overcome these difficulties we reformulate the basic variational problem of optimal phase measurement within a Bayesian paradigm and employ the Shannon information as a robust figure of merit. Single-mode performance bounds are discussed, and we invoke a general theorem that reduces the problem of finding the multi-mode performance bounds to the bounding of a single integral, without need of the central limit theorem.
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.
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 $\ensuremath{\Delta}\ensuremath{\theta}$ 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.