A transformational property of the Husimi function and its relation to the wigner function and symplectic tomograms

We consider the Husimi Q-functions, which are quantum quasiprobability distributions in the phase space, and investigate their transformation properties under a scale transformation (q, p) → (λq, λp). We prove a theorem that under this transformation, the Husimi function of a physical state is transformed into a function that is also a Husimi function of some physical state. Therefore, the scale transformation defines a positive map of density operators. We investigate the relation of Husimi functions to Wigner functions and symplectic tomograms and establish how they transform under the scale transformation. As an example, we consider the harmonic oscillator and show how its states transform under the scale transformation.

[1]  H. Korsch,et al.  On the zeros of the Husimi distribution , 1997 .

[2]  Y. Kano,et al.  A NEW PHASE-SPACE DISTRIBUTION FUNCTION IN THE STATISTICAL THEORY OF THE ELECTROMAGNETIC FIELD , 1965 .

[3]  Kevin Cahill,et al.  Ordered Expansions in Boson Amplitude Operators , 1969 .

[4]  V. Man'ko,et al.  Eigenfunctions of quadratic Hamiltonians in the Wigner representation , 1984 .

[5]  H. Fan,et al.  Entangled Husimi operator as a pure state density matrix of two-mode squeezed coherent state☆ , 2006 .

[6]  A. Ibort,et al.  An introduction to the tomographic picture of quantum mechanics , 2009, 0904.4439.

[7]  Beck,et al.  Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. , 1993, Physical review letters.

[8]  V. I. Tatarskii,et al.  The Wigner representation of quantum mechanics , 1983 .

[9]  H. Fan.  Newton-Leibniz integration for ket-bra operators in quantum mechanics (V) : Deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism , 2008 .

[10]  P. M. Mathews,et al.  STOCHASTIC DYNAMICS OF QUANTUM-MECHANICAL SYSTEMS , 1961 .

[11]  E. Sudarshan.  Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams , 1963 .

[12]  D. Lalović,et al.  An interesting property of the Husimi function and its implications , 1994 .

[13]  E. Wigner,et al.  Quantum-mechanical distribution functions: Conditions for uniqueness , 1981 .

[14]  R. G. Lerner,et al.  Encyclopedia of Physics , 1990 .

[15]  Takahashi,et al.  Chaos and Husimi distribution function in quantum mechanics. , 1985, Physical review letters.

[16]  Girish S. Agarwal,et al.  CALCULUS FOR FUNCTIONS OF NONCOMMUTING OPERATORS AND GENERAL PHASE-SPACE METHODS IN QUANTUM MECHANICS. I. MAPPING THEOREMS AND ORDERING OF FUNCTIONS OF NONCOMMUTING OPERATORS. , 1970 .

[17]  M. Scully,et al.  Distribution functions in physics: Fundamentals , 1984 .

[18]  Distribution of Husimi zeros in polygonal billiards. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  Pauli problem and related mathematical problems , 2008 .

[20]  S. S. Mizrahi.  Quantum mechanics in the Gaussian wave-packet phase space representation , 1984 .

[21]  When does a given function in phase space belong to the class of Husimi distributions , 1993 .

[22]  S. Luo.  Fisher Information Matrix of Husimi Distribution , 2001 .

[23]  L. Ballentine,et al.  Probabilistic and Statistical Aspects of Quantum Theory , 1982 .

[24]  G. Ferri,et al.  Note on semiclassical uncertainty relations , 2009 .

[25]  P. Roy.  Quantum statistical properties of Gazeau–Klauder coherent state of the anharmonic oscillator , 2003 .

[26]  The Pauli equation for probability distributions , 2000, quant-ph/0005058.

[27]  E. Wigner.  On the quantum correction for thermodynamic equilibrium , 1932 .

[28]  Generalized husimi functions: Analyticity and information content , 1998, quant-ph/9805054.

[29]  Stefano Mancini,et al.  Wigner function and probability distribution for shifted and squeezed quadratures , 1995 .