{\left\langle\psi\right|} \hat{N}_{\text{O}}^{1\sigma} {\left|\psi\right\rangle}\right]^{M-Q_{1\sigma}} {\left\langle\psi\right|} 
\hat{N}_{\text{O}}^{1\sigma} {\left|\psi\right\rangle}^{Q_{1\sigma}} 
\label{stp}$$ and the joint probability distribution: $$\begin{aligned}
\nonumber
P (Q_{1\sigma},Q_{2\sigma'})=\sum_{k=\textrm{Max}[M-Q_{1\sigma},M-Q_{2\sigma'}]}^{(M-Q_{1\sigma})+(M-Q_{2\sigma'})}
A^{2M-Q_{1\sigma}-Q_{2\sigma'}-k} ~B^{Q_{1\sigma}-M+k} ~C^{k-M+Q_{2\sigma'}}\times \\
\times {\left\langle\psi\right|} \hat{N}_{\text{O}}^{1\sigma} \hat{N}_{\text{O}}^{2\sigma'} {\left|\psi\right\rangle}^{M-k}  f(M,Q_{1\sigma},Q_{2\sigma'},k)
\label{FCSt}\end{aligned}$$ where $
A=1- {\left\langle\psi\right|}\hat{N}_{\text{O}}^{1\sigma} {\left|\psi\right\rangle} -{\left\langle\psi\right|}
\hat{N}_{\text{O}}^{2\sigma'} {\left|\psi\right\rangle}
+ {\left\langle\psi\right|} \hat{N}_{\text{O}}^{1\sigma} \hat{N}_{\text{O}}^{2\sigma'} {\left|\psi\right\rangle}
$, $
B={\left\langle\psi\right|} \hat{N}_{\text{O}}^{1\sigma} (1- \hat{N}_{\text{O}}^{2\sigma'}){\left|\psi\right\rangle}
$, $
C={\left\langle\psi\right|} (1- \hat{N}_{\text{O}}^{1\sigma})\hat{N}_{\text{O}}^{2\sigma'} {\left|\psi\right\rangle} 
$ and $
f(M,Q_{1\sigma},Q_{2\sigma'},k) =M!/[(k-M+Q_{2\sigma'})!(2M-k-Q_{1\sigma}-Q_{2\sigma'})!] 
$. In doing so we have written the expressions for the probability distributions in terms of the expectation values of “outgoing” number operators. For $Q_{1\sigma}=Q_{2\sigma'}=M$, Eq.(\[FCSt\]) reduces to $$P(Q_{1\sigma}=M,Q_{2\sigma'}=M)={\left\langle\psi\right|} \hat{N}_{\text{O}}^{1\sigma} \hat{N}_{\text{O}}^{2\sigma'} {\left|\psi\right\rangle}^M .$$

When both spin species are counted in one of the terminals the characteristic function is different from the one given in Eq.(\[chi\_s\]). In particular, the characteristic function for counting both spins in terminal 1 reads: $$\begin{aligned}
\nonumber
\chi_E (\lambda_1,\lambda_{2\sigma'})=1+
\left( e^{-i\lambda_1} -1 \right) \langle \left( \hat{N}_{\text{O}}^{1\uparrow} +
\hat{N}_{\text{O}}^{1\downarrow}\right) \rangle +
\left( e^{-i\lambda_{2\sigma'}} -1 \right) \langle \hat{N}_{\text{O}}^{2\sigma'} \rangle +
\\\nonumber+
\left( e^{-i\lambda_1} -1 \right) \left( e^{-i\lambda_{2\sigma'}} -1 \right)
\langle \left( \hat{N}_{\text{O}}^{1\uparrow}+
\hat{N}_{\text{O}}^{1\downarrow}\right) \hat{N}_{\text{O}}^{2\sigma'} \rangle +
\left( e^{-i\lambda_1} -1 \right)^2
\langle \hat{N}_{\text{O}}^{1\uparrow} \hat{N}_{\text{O}}^{1\downarrow} \rangle +
\\
\left( e^{-i\lambda_1} -1 \right)^2 \left( e^{-i\lambda_{2\sigma'}} -1 \right)
\langle \hat{N}_{\text{O}}^{1\uparrow} \hat{N}_{\text{O}}^{1\downarrow} 
\hat{N}_{\text{O}}^{2\sigma'} \rangle .
\label{chi_s1}\end{aligned}$$ where we have set $\lambda_{1\uparrow}=\lambda_{1\downarrow}\equiv \lambda_1$. The expression for the joint 