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arXiv:quant-ph/9805028 · Journal of Modern Optics

Retrodictively optimal localizations in phase space

D. M. Appleby

Published Apr 1, 1999• 4 citations
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Abstract

In a previous paper it was shown that the distribution of measured values for a retrodictively optimal simultaneous measurement of position and momentum is always given by the initial state Husimi function. This result is now generalized to retrodictively optimal simultaneous measurements of an arbitrary pair of rotated quadratures ◯ θ 1 and [pcirc] θ 2. It is shown, that given any such measurement, it is possible to find another such measurement, informationally equivalent to the first, for which the axes defined by the two quadratures are perpendicular. It is further shown that the distribution of measured values for such a measurement belongs to the class of generalized Husimi functions most recently discussed by Wünsche and Büzek. The class consists of the subset of Wódkiewicz's operational probability distributions for which the filter reference state is a squeezed vacuum state.

DOI
10.1080/09500349908231306

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In a previous paper it was shown that the distribution of measured values for a retrodictively optimal simultaneous measurement of position and momentum is always given by the initial state Husimi function. This result is now generalized to retrodictively optimal simultaneous measurements of an arbitrary pair of rotated quadratures ◯ θ 1 and [pcirc] θ 2. It is shown, that given any such measurement, it is possible to find another such measurement, informationally equivalent to the first, for which the axes defined by the two quadratures are perpendicular. It is further shown that the distribution of measured values for such a measurement belongs to the class of generalized Husimi functions most recently discussed by Wünsche and Büzek. The class consists of the subset of Wódkiewicz's operational probability distributions for which the filter reference state is a squeezed vacuum state.
The accuracy of the Arthurs-Kelly model of a simultaneous measurement of position and momentum is analysed using concepts developed by Braginsky and Khalili in the context of measurements of a single quantum observable. A distinction is made between the errors of retrodiction and prediction. It is shown that the distribution of measured values coincides with the initial-state Husimi function when the retrodictive accuracy is maximized (this being the result first obtained by Busch, using somewhat different methods). It is further shown that the distribution of measured values is related to the final-state anti-Husimi function (the P representation of quantum optics) when the predictive accuracy is maximized. The disturbance of the system by the measurement is also discussed. A class of minimally disturbing measurements is characterized. It is shown that the distribution of measured values then coincides with one of the smoothed Wigner functions described by Cartwright.
The analytic properties of a class of generalized Husimi functions are discussed, with particular reference to the problem of state reconstruction. The class consists of the subset of Wódkiewicz's operational probability distributions for which the filter reference state is a squeezed vacuum state. The fact that the function is analytic means that perfectly precise knowledge of its values over any small region of phase space provides enough information to reconstruct the density matrix. If, however, one only has imprecise knowledge of its values, then the amplification of statistical errors which occurs when one attempts to carry out the continuation seriously limits the amount of information which can be extracted. To take account of this fact a distinction is made between explicate or experimentally accessible information, and information which is only present in implicate or experimentally inaccessible form. It is shown that an explicate description of various aspects of the system can be found localized on various two-real-dimensional surfaces in complexified phase space. In particular, the continuation of the function to the purely imaginary part of complexified phase space provides an explicate description of the Wigner function.
The analytic properties of a class of generalized Husimi functions are discussed, with particular reference to the problem of state reconstruction. The class consists of the subset of Wódkiewicz's operational probability distributions for which the filter reference state is a squeezed vacuum state. The fact that the function is analytic means that perfectly precise knowledge of its values over any small region of phase space provides enough information to reconstruct the density matrix. If, however, one only has imprecise knowledge of its values, then the amplification of statistical errors which occurs when one attempts to carry out the continuation seriously limits the amount of information which can be extracted. To take account of this fact a distinction is made between explicate or experimentally accessible information, and information which is only present in implicate or experimentally inaccessible form. It is shown that an explicate description of various aspects of the system can be found localized on various two-real-dimensional surfaces in complexified phase space. In particular, the continuation of the function to the purely imaginary part of complexified phase space provides an explicate description of the Wigner function.
What knowledge can be obtained from the record of a continuous measurement about the quantum state of the measured system at the beginning of the measurement? The task of quantum state retrodiction, the inverse of the more common state prediction, is rigorously addressed in quantum measurement theory through retrodictive positive operator-valued measures (POVMs). This introduction to this general framework presents its practical formulation for retrodicting Gaussian quantum states using continuous-time homodyne measurements and applies it to optomechanical systems. We identify and characterize achievable retrodictive POVMs in common optomechanical operating modes with resonant or off-resonant driving fields and specific choices of local oscillator frequencies in homodyne detection. In particular, we demonstrate the possibility of a near-ideal measurement of the quadrature of the mechanical oscillator, giving direct access to the position or momentum distribution of the oscillator at a given time. This forms the basis for complete quantum state tomography, albeit in a destructive manner.
In this paper we investigate the coupling properties of pairs of quadrature observables, showing that, apart from the Weyl relation, they share the same coupling properties as the position-momentum pair. In particular, they are complementary. We determine the marginal observables of a covariant phase space observable with respect to an arbitrary rotated reference frame, and observe that these marginal observables are unsharp quadrature observables. The related distributions constitute the Radon transform of a phase space distribution of the covariant phase space observable. Since the quadrature distributions are the Radon transform of the Wigner function of a state, we also exhibit the relation between the quadrature observables and the tomography observable, and show how to construct the phase space observable from the quadrature observables. Finally, we give a method to measure together with a single measurement scheme any complementary pair of quadrature observables.
We study a class of phase-space distribution functions that is generated from a Gaussian convolution of the Wigner distribution function. This class of functions represents the joint count probability in simultaneous measurements of position and momentum. We show that, using these functions, one can determine the expectation value of a certain class of operators accurately, even if measurement data performed only with imperfect detectors are available. As an illustration, we consider the eight-port homodyne detection experiment that performs simultaneous measurements of two quadrature amplitudes of a radiation field.