interesting property of this formulation of quantum mechanics is that in the semi-classical limit $\hbar\rightarrow 0$, the dynamical equation reduces to the classical equation of motion expressed in terms of Poisson brackets

$$\partial_t f = \left\lbrace H,f\right\rbrace_P = H \left(\overleftarrow{\partial_x}\overrightarrow{\partial_v} -
\overleftarrow{\partial_v}\overrightarrow{\partial_x} \right) f.$$

This illustrates how the algebraic structures of classical and quantum mechanics are related through the continuous changing of the parameter $\hbar$. This is the reason why such an approach to quantum mechanics is known as *deformation quantization* [@Hirshfeld2002].

Let’s now stop this overview and move to the part of this formalism which will be useful for the construction of our new simulation framework.

Wigner distribution function {#ssec:WDF}
----------------------------

The Wigner transform (equation \[eq:Wigner\_transform\]) maps a quantum operator $\hat{A}$ to a classical function in phase space. Wigner used this to associate a real phase space function to a quantum system [@Wigner1932], now called the *Wigner distribution function* (WDF). It is defined as the symbol in phase space associated to the density operator $\hat\rho$:

$$P_W(x,v) := {\mbox{sym}}\left(\hat\rho\right) = \int e^{\frac{i}{\hbar}vy} \left\langle x-\frac{y}{2}
\right|\hat{\rho}\left|x+\frac{y}{2}\right\rangle d^3y.$$

As usual, the density operator can be expressed as the combination of pure state wavefunctions $\psi_n$:

$$\hat\rho = \sum_n \lambda_n \left|\psi_n\right\rangle\left\langle\psi_n\right|, \quad \lambda_n \geq 