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Physical Review A

Unitary equivalence between ideal and feasible phases

Masashi Ban

Published Sep 1, 1994• 23 citations
Publisher

Abstract

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. :

DOI
10.1103/physreva.50.2785

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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^ \ifmmode∘\else\textdegree\fi^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}\ifmmode∘\else\textdegree\fi and D{\mathrm{^}}_{\mathrm{RNS}}=W^ \ifmmode∘\else\textdegree\fi^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}\ifmmode∘\else\textdegree\fiD{\mathrm{^}}_{\mathrm{SW}}W^ is derived. :
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)].
A comparative study is performed of two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator, called Shapiro-Wagner and Caves' frame, respectively. This approach is based on the introduction of a convenient operator hatpsi\\hat \\psihatpsi which allows a unified formulation of both cases. For the Shapiro-Wagner scheme, where [hatpsi,hatpsidag]=0[\\hat \\psi, \\hat \\psi^{\\dag}] =0[hatpsi,hatpsidag]=0, quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves' framework, characterized by [hatpsi,hatpsidag]neq0[\\hat \\psi, \\hat \\psi^{\\dag}] \\neq 0[hatpsi,hatpsidag]neq0, within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.