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Physical Review A

Relation between ideal and feasible phase concepts

Z. Hradil

Published Mar 1, 1993• 24 citations
Publisher

Abstract

The isometry U${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$,U${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$U^\ensuremath{\ne}U^U${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$=1^, which relates the ideal phase measurement to the feasible phase measurement based on heterodyne detection, is specified here. Consequently, the Shapiro-Wagner phase measurement [IEEE J. Quantum Electron. QE-20, 803 (1984)] incorporates the ideal phase concepts of Susskind-Glogower [Physics 1, 49 (1964)] or Pegg-Barnett [Phys. Rev. A 39, 1665 (1989)].

DOI
10.1103/physreva.47.2376

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The isometry U{\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}},U{\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}U^\ensuremath{\ne}U^U{\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}=1^, which relates the ideal phase measurement to the feasible phase measurement based on heterodyne detection, is specified here. Consequently, the Shapiro-Wagner phase measurement [IEEE J. Quantum Electron. QE-20, 803 (1984)] incorporates the ideal phase concepts of Susskind-Glogower [Physics 1, 49 (1964)] or Pegg-Barnett [Phys. Rev. A 39, 1665 (1989)].
Unitary equivalence between the ideal and feasible phases is proven. The ideal phase can be described by the phase operator D${\mathrm{^}}_{\mathrm{RNS}}$, constructed from the relative-number states, and the feasible phase considered here is characterized by the Shapiro-Wagner phase operator D${\mathrm{^}}_{\mathrm{SW}}$. The unitary operator W^ satisfying D${\mathrm{^}}_{\mathit{S}\mathit{W}}$=W^D${\mathrm{^}}_{\mathrm{RNS}}$W^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ and D${\mathrm{^}}_{\mathrm{RNS}}$=W^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$D${\mathrm{^}}_{\mathrm{SW}}$W^ is derived. :
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.
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.