between the two coherent orthogonally-polarised beams after recombination in PBS$_1$, but before traversing PBS$_2$, is equivalent to detecting which of the two quantum states, $$|\Phi_0 \rangle=|\Phi (\tau_0) \rangle_x |\Phi (\tau_0) \rangle_y$$ or $$|\Phi_1 \rangle=|\Phi(\tau_0) \rangle_x |\Phi (\tau_0+\tau)
\rangle_y \label{state1}$$ is the output quantum state which describes the coherent pulse leaving PBS$_1$. $(x,y)$ designates the corresponding polarisations. The spectral shape (mode function) $\Phi$ writes $$\Phi(\tau_0+\tau) = \Psi(\Omega) \exp \left[ i (\omega_0+\Omega)
(\tau_0+\tau) \right],  \label{modes_input}$$ where $\omega_0$ is the central frequency of the laser pulse, $\Omega=2\pi f$ is the angular frequency deviation from the the center frequency and $\Psi(\Omega)$ is the spectral shape of the input coherent laser signal.

The minimum probability of error that can be made when distinguishing between two quantum states is related to the trace distance between the states [@fuchs1999]. For two pure state, $\Phi_0$ and $\Phi_1$, the (minimum) probability of error is [@helstrom1976; @ou1996; @englert1996] $$\label{average_error}
P_{\mathrm{error}}=\frac{1}{2}\left(1-\sqrt{1-|\langle
\Phi_0|\Phi_1\rangle|^2} \right).$$ For $\Phi_0=\Phi_1$, $P_{\mathrm{error}}=0.5$. On the contrary, to be successful in distinguishing two quantum states with low probability of error ($P_{\mathrm{error}} \sim 0$) requires $|\langle \Phi_0|\Phi_1\rangle| \sim 0$, i.e., the two states should be close to orthogonal.

The coherent broadband states considered here can be generally described as single-mode 