describe this state by its Wigner function $W(x_1,x_2,y_1,y_2)$ where index 1 (resp. 2) refers to the first (resp. second) quadrature of Alice or Bob’s mode. Alternatively, since we restrict ourselves to protocols where both Alice and Bob perform a heterodyne detection, we can also consider the convenient characterization in terms of the $Q$-function $Q(x_1,x_2,y_1,y_2)$ of the state $\rho$. This $Q$-function corresponds to the probability distribution sampled from when measuring the state with heterodyne detection [@leo97] and is given by: $$Q(x_1, x_2, y_1,y_2) = \frac{1}{\pi^2} \langle \alpha_A, \alpha_B |\rho|\alpha_A, \alpha_B \rangle,$$ where $|\alpha_{A (B)}\rangle$ is a coherent state centered on $\alpha_{A(B)} = x_{A (B)} + i p_{A (B)}$ in Alice’s (Bob’s) phase space. Interestingly, if we denote $W_0$ the Wigner function of the vacuum, then $Q$ simply corresponds to the convolution of $W$ and $W_0$: $Q= W \star W_0$.

The relation between the two probability distributions $P$ and $Q$ can also be made explicit: if Alice measures one mode of the two-mode squeezed vacuum with a heterodyne detection and obtains outcomes $x_1$, $x_2$, she projects the second mode on the coherent state $|\gamma (x_1 - i x_2)\rangle$ where the factor $\gamma=\sqrt{2(V-1)/(V+1)}$ depends on the variance $V$ of the two-mode squeezed vacuum 