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

Abstract

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.

Measuring microwave quantum states: Tomogram and moments

“…Indeed, using the heterodyne detection scheme gives rise to the Husimi function Q(q,p) which also contains the full information about the quantum state [4]. Since both the tomogram and the Husimi function are extracted from experimental data, the relation between them (see, e.g., [16]) can be considered as a crosscheck of the experiment's accuracy. Closely related to the Husimi function are ordered moments (â † ) nâm and â k (â † ) l , whereâ † andâ are photon creation and annihilation operators, respectively.…”

“…(Color online) Snapshots of the normally ordered moments | (â † ) nâm | of the odd coherent state(16) with α = 0.5 at eight successive times of damped evolution (58) with γ = 0.1.…”

Harmonic Oscillator and Related States With Large Value of Uncertainty

“…The main ideas of the method and the results obtained with its help are presented in Ref. [22][23][24][25][26][27].…”

“…(58) It was proved in [22,23] that if Q(q, p) is a Hushimi function of a quantum state and λ < 1, then the quantity…”

Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems

“…Within this formalism one can relate operators to their symbols using dequantizers and can reconstruct operators from their symbols using quantizers. The relations between different phase-space representations can be also determined in this framework [29][30][31][32].…”