Fourth-order interference and the extended phase-space formalism for finite quantum systems

Finite quantum systems where the position $x$ and the momentum $p$ take values in $\mathbb{Z}\left(d\right)$ are considered. An extended phase space $x-p-X-P$, where $X$ is position increment and $P$ is momentum increment, is introduced. The extended Wigner function $\mathcal{W}\left(x,p,X,P\right)$ and the extended Weyl function $\tilde{\mathcal{W}}\left(x^{\prime },p^{\prime },X^{\prime },P^{\prime }\right)$ are defined and their properties are studied. They quantify fourth-order interference. Several examples are discussed.

Figure 1

The Wigner function $W_{\rho }\left(x,p\right)$ as a function of $x,p$ for the pure state of Eq. (30).

Figure 2

The Wigner function $W_{\rho }\left(x,p\right)$ as a function of $x,p$ for the mixed state of Eq. (29).

Figure 3

The absolute value of the Weyl function $\left|{\tilde{W}}_{\rho }\left(X,P\right)\right|$ as a function of $X,P$ for the pure state of Eq. (30).

Figure 4

The absolute value of the Weyl function $\left|{\tilde{W}}_{\rho }\left(X,P\right)\right|$ as a function of $X,P$ for the mixed state of Eq. (29).

Figure 5

The Wigner function $W_{\rho }\left(x,p\right)$ as a function of $x,p$ for the Legendre state of Eq. (11).

Figure 6

The extended Wigner function ${\mathcal{W}}_{\rho }\left(x,p,0,0\right)$ as a function of $x,p$ for the pure state of Eq. (30).

Figure 7

The extended Wigner function ${\mathcal{W}}_{\rho }\left(x,p,0,0\right)$ as a function of $x,p$ for the mixed state of Eq. (29).

Figure 8

The extended Wigner function ${\mathcal{W}}_{\rho }\left(0,0,X,P\right)$ as a function of $X,P$ for the pure state of Eq. (30) and also for the mixed state of Eq. (29).

Figure 9

The extended Wigner function ${\mathcal{W}}_{\rho }\left(x,p,0,0\right)$ as a function of $x,p$ for the Legendre state of Eq. (11).

Figure 10

The extended Wigner function ${\mathcal{W}}_{\rho }\left(0,0,X,P\right)$ as a function of $X,P$ for the Legendre state of Eq. (11).