In this post, I derive an identity showing the sense in which information about coherence over long distances in phase space for a quantum state
is encoded in its quasicharacteristic function
, the (symplectic) Fourier transform of its Wigner function. In particular I show
(1) 
where
and
are coherent states,
is the mean phase space position of the two states, “
” denotes the convolution, and
is the (Gaussian) quasicharacteristic function of the ground state of the Harmonic oscillator.
Definitions
The quasicharacteristic function for a quantum state
of a single degree of freedom is defined as
![Rendered by QuickLaTeX.com \[\mathcal{F}_\rho(\xi) := \mathrm{Tr}[\rho D_\xi] = \langle\rho,D_\xi\rangle_{\mathrm{HS}},\]](https://blog.jessriedel.com/wp-content/ql-cache/quicklatex.com-02f0b073e91f2af4d275c0371210ab01_l3.svg)
where
is the Weyl phase-space displacement operator,
are coordinates on “reciprocal” (i.e., Fourier transformed) phase space,
is the phase-space location operator,
and
are the position and momentum operators, “
” denotes the Hilbert-Schmidt inner product on operators,
, and “
” denotes the symplectic form,
. (Throughout this post I use the notation established in Sec. 2 of my recent paper with Felipe Hernández.) It has variously been called the quantum characteristic function, the chord function, the Wigner characteristic function, the Weyl function, and the moment-generating function. It is the quantum analog of the classical characteristic function.
Importantly, the quasicharacteristic function obeys
and
, just like the classical characteristic function, and provides a definition of the Wigner function where the linear symplectic symmetry of phase space is manifest:
(2) 
where
is the phase-space coordinate and
is the position-space representation of the quantum state. This first line says that
and
are related by the symplectic Fourier transform. (This just means the inner product “
” in the regular Fourier transform is replaced with the symplectic form, and has the simple effect of exchanging the reciprocal variables,
, simplifying many expressions.)… [continue reading]