Quantum Phase Measurements Research Articles

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

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.

Maximum-likelihood analysis of multiple quantum phase measurements.

Shapiro, Shepard, and Wong [Phys. Rev. Lett. 62, 2377 (1989)] suggested that a scheme of multiple phase measurements, using quantum states with minimum ``reciprocal peak likelihood,'' could achieve a phase sensitivity scaling as 1/${\mathit{N}}_{\mathrm{tot}}^{2}$, where ${\mathit{N}}_{\mathrm{tot}}$ is the mean number of photons availavle for all measurements. We have simulated their scheme for as many as 240 measurements and have found optimum phase sensitivities for 3\ensuremath{\le}${\mathit{N}}_{\mathrm{tot}}$\ensuremath{\le}120. A power-law fit to the simulated data yields a phase sensitivity that scales as 1/${\mathit{N}}_{\mathrm{tot}}^{0.85\ifmmode\pm\else\textpm\fi{}0.01}$. We conclude that reciprocal peak likelihood is not a good measure of sensitivity.

Phase measurement in quantum optics

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.

Quantum phase measurements with infinite peak-likelihood and zero phase information

Maximum likelihood estimation of an unknown c-number phaseshift is considered, along the lines established by Shapiro, Shepard and Wong, Phys. Rev. Lett., vol.62, p.2377, 1989. An input state is exhibited whose reciprocal peak likelihood vanishes, delta phi to 0, while its phase-measurement probability density converges, in distribution, to uniform, p( phi mod Phi ) to d1/2 pi .

Quantum phase measurement: A system-theory perspective.

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.