are vectors of $n$ real numbers, one for each open channel. In terms of incoming (outgoing) creation operator the number operators can be expressed as follows $$\hat{N}_I^{j\sigma}= \hat{a}_{j\sigma}^{\dagger}\hat{a}_{j\sigma}; \qquad
\hat{N}_O^{j\sigma}=\hat{\phi}_{j\sigma}^{\dagger}\hat{\phi}_{j\sigma} \;\; .$$ Eq.(\[chiE\]) can also be recasted in the form [@levitov93]: $$\chi_E(\vec{\lambda_{\uparrow}},\vec{\lambda_{\downarrow}})=
\det (\mathbb{I}-n_E+n_E~ S^\dagger \Lambda^\dagger~ S~ \Lambda) ,
\label{chiEdet}$$ where $\mathbb{I}$ is the unit matrix, $n_E$ is the diagonal matrix of Fermi distribution functions $f_j(E)$ for particles in the reservoir $j$ and defined as $(n_E)_{j\sigma,j\sigma}=f_j(E)$, whereas $\Lambda$ is a diagonal matrix defined as: $(\Lambda)_{j\sigma,j\sigma}=\exp (i\lambda_{j\sigma})$. For long measurement times $t$ the total characteristic function $\chi$ is the product of contributions from different energies, so that $$\chi(\vec{\lambda_{\uparrow}},\vec{\lambda_{\downarrow}})=e^{\frac{t}{h}\int
dE~\log{\chi_E(\vec{\lambda_{\uparrow}},\vec{\lambda_{\downarrow}})}} \; .
\label{chi}$$ At zero temperature, the statistical average over the Fermi distribution function in Eq.(\[chiE\]) simplifies to the expectation value calculated on the state ${\left|\psi\right\rangle}$ containing two electrons of both spin species for each channel of a given lead up to the energy corresponding to the chemical potential of such lead. Furthermore, in the limit of a small bias voltage $V$ applied between the reservoirs, the argument of the integral is energy-independent so that Eq.(\[chi\]) can be approximated to $$\chi(\vec{\lambda_{\uparrow}},\vec{\lambda_{\downarrow}})\simeq
\left[ \chi_0(\vec{\lambda_{\uparrow}},\vec{\lambda_{\downarrow}})\right]^M
\label{approx}$$ where only the zero-energy characteristic function appears and 