the $m$-th Hermite polynomial. Generalizations to higher dimensions are straightforward.

To each state $\rho$ corresponds a *Wigner distribution* $W_{\rho}$, which is defined via its Fourier transform in the way indicated by equation (\[def.Wigner\]): $$\label{def.Wigner}
    \widetilde{W}_\rho(u,v) {\mathrel{\mathop:} = }\iint e^{-i(uq+vp)}W_\rho(q,p) dq dp
    {\mathrel{\mathop:} = }\mathrm{Tr}\big(\rho \exp(-iu\mathbf{Q}-iv\mathbf{P})\big)$$ where $\mathbf{Q}$ and $\mathbf{P}$ are canonically conjugate observables (e.g. electric and magnetic fields) satisfying the commutation relation $[\mathbf{Q}, \mathbf{P}] = i$ (we assume a choice of units such that $\hbar = 1$). It is easily checked that $W_{\rho}$ is real-valued, has integral $\iint_{{\mathbb{R}}^2} W_\rho(q,p)dqdp=1$ and uniform bound ${{\left\lvertW_{\rho}(q,p)\right\rvert}} \leq
\frac{1}{\pi}$.

For any $\phi \in {\mathbb{R}}$, the Wigner distribution allows one to easily recover the probability density $x \mapsto p_{\rho}(x,\phi)$ of $\mathbf{Q} \cos \phi + \mathbf{P} \sin \phi$ by $$\label{eq:prhophi}
    p_{\rho}(x,\phi) =\mathcal{R}[W_{\rho}] (x,\phi),$$ where $\mathcal{R}$ is the Radon transform defined in equation (\[eq.Radon.transform\]) $$\label{eq.Radon.transform}
    \mathcal{R}[W_{\rho}](x,\phi)=\int_{-\infty}^\infty W_{\rho}(
    x\cos\phi - t\sin\phi, \, x \sin\phi + t\cos\phi )dt.$$

Moreover, the correspondence between $\rho$ and $W_{\rho}$ is one to one and isometric with respect to the $\mathbb{L}_{2}$ norms as in equation (\[eq.l2.isometry\]): $$\label{eq.l2.isometry}
    \| W_{\rho}\|_2^2 {\mathrel{\mathop:} = }\iint |W_{\rho}(q,p)
    |^2 dq dp = \frac{1}{2\pi}\| \rho \|_{2}^{2} {\mathrel{\mathop:} = }\frac{1}{2\pi}\sum_{j, k=0}^{\infty} |\rho_{jk}|^2.$$

From now on we denote by $\langle \cdot, \cdot 