are formulated in the language of selfadjoint operators acting on Hilbert spaces. To every quantum system one can associate a complex Hilbert space $\mathcal{H}$ whose vectors represent the wave functions of the system. These vectors are identified to projection operators, or pure states. In general, a state is a mixture of pure states described by a compact operator $\rho$ on $\mathcal{H}$ having the following properties:

1.  Selfadjoint: $\rho=\rho^{*}$, where $\rho^{*}$ is the adjoint of $\rho$.

2.  Positive: $\rho\geq 0$, or equivalently $\langle\psi,
        \rho\psi\rangle\geq 0$ for all $\psi\in\mathcal{H}$.

3.  Trace one: ${\text{tr}}(\rho)=1$.

When $\mathcal{H}$ is separable, endowed with a countable orthonormal basis, the operator $\rho$ is identified to a *density matrix* $[\rho_{m,n}]_{m,n\in{\mathbb{N}}}$.

The positivity property implies that all the eigenvalues of $\rho$ are nonegative and by the trace property, they sum up to one. In the case of the finite dimensional Hilbert space $\mathbb{C}^{d}$, the density matrix is simply a positive semi-definite $d \times d$ matrix of trace one. Our setup from now on will be $\mathcal{H} = L^{2}({\mathbb{R}})$, in which case we employ the orthonormal Fock basis made of the Hermite functions $$\label{eq:2}
    h_m(x) {\mathrel{\mathop:} = }(2^m m! \sqrt\pi)^{-\frac12} H_m(x) e^{-\frac{x^2}{2}}$$ where $H_m(x) {\mathrel{\mathop:} = }(-1)^m e^{x^2} \frac{d^m}{dx^m} e^{-x^2}$ is 