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Journal of Modern Optics

Generalized Husimi functions: analyticity and information content

D. M. Appleby

Published Apr 15, 1999• 2 citations
Publisher

Abstract

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.

DOI
10.1080/095003499149566

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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.
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 local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.
Abstract This paper considers a Husimi representation of quantum mechanics in which the (stationary) state of a system or ensemble is described by a Husimi function and an observable is described by a phase space function or distribution such that the expectation value of the observable is given by an integral over phase space of the product of that function or distribution and the Husimi function. The density matrix, Wigner function, and Husimi function are considered to be alternative ways of describing the state of a system or ensemble, and methods of recovering the Wigner function or density matrix from the Husimi function are discussed. The classical limits of the Wigner and Husimi functions and of the relationship between them are considered. © 1993 John Wiley & Sons, Inc.
Abstract It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call thecomplex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.