we obtain $\rho=1$. Fig. 3 shows the mode overlap of the signal in the corresponding output port for a delay of $\tau=100$ as. The mode overlap has a minimum for $\omega_0 \tau-\Gamma=\pi$, where the two mode functions becomes easily distinguishable, as shown in Fig. 2 (a). The effect of the polarisation projection, a key ingredient of the WVA scheme, can be understood as a change of the mode overlap ([*mode distinguishability*]{}) between states with different delay $\tau$.

However, an enhanced mode distinguishability in this output port is accompanied by a corresponding increase of the insertion loss, as it can be seen in Fig. 3. The insertion loss, $P_{\mathrm{out}}(\tau)/P_{\mathrm{in}}=1/2\, \left[
1+\gamma \cos (\omega_0 \tau-\Gamma)\right]$, is the largest when the modes are close to orthogonal ($\rho \sim 0$). Both effects indeed compensate, as it should be, since WVA implements unitary transformations, and the trace distance between quantum states is preserved under unitary transformations. The quantum overlap between the states reads $$\begin{aligned}
& & |\langle \Phi_u(\tau_0)|\Phi_u(\tau_0+\tau)\rangle|^2=|\langle
\Phi_v(\tau_0)|\Phi_v(\tau_0+\tau)\rangle|^2 \nonumber \\
& & =\exp \left[-\frac{N}{2} \left( 1-\gamma \cos \omega_0
\tau\right) \right],\end{aligned}$$ so $$\begin{aligned}
& & |\langle \Phi_0|\Phi_1 \rangle|^2 \nonumber \\
& & =|\langle \Phi_u(\tau_0)|\Phi_u(\tau_0+\tau)\rangle_u|^2
|\langle \Phi_v(\tau_0)|\Phi_v(\tau_0+\tau)\rangle_v|^2 \nonumber \\
& & =\exp \left[-N \left( 1-\gamma \cos 