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

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]