## 2 Two-Photon Interference in the Polarizing Beam Splitter  
The MDI-QKD protocol relies on quantum interference of two indistinguishable photons incident on a 50:50 beam splitter [10; 12]. This interference, known as the Hong-Ou-Mandel (HOM) effect, suppresses the probability that two identical photons simultaneously entering the beam splitter through different ports will be detected exiting the beam splitter through different ports; they must always exit the beam splitter together [13]. Distinguishable photons under the same circumstances show no such correlations. Ideally, photons employed in MDI-QKD are identical in every physical characteristic except (potentially) their polarization angles. Under these conditions, two identically polarized photons will always exhibit HOM interference; however, non-idealities in the photon sources or beam splitter characteristics give rise to partial distinguishability and a non-zero probability of unexpected coincidence events--observation of ostensibly indistinguishable photons exiting different beam splitter ports [11; 14; 15]. We apply the formalism of quantum optics following the development of Loudon [16] to relate the probabilities of coincidence events, and ultimately the performance limits of MDI-QKD systems, to photon source tolerances. [PERSON] and [PERSON] independently prepare single photon pulses characterized by temporal wave functions \(\xi_{\mathrm{a,b}}(t)\) and polarization angles \(\theta_{\mathrm{a,b}}\). The input state at the beam splitter is then the product state  
\[ \begin{array}{rcl}|\xi_{\mathrm{a}},\theta_{\mathrm{a}}\rangle\otimes|\xi_{ \mathrm{b}},\theta_{\mathrm{b}}\rangle&=&\int\mathrm{d}t_{1}\xi_{\mathrm{a}} (t_{1})\left[\hat{a}_{\mathrm{H}}^{\dagger}(t_{1})\cos\theta_{\mathrm{a}}+ \hat{a}_{\mathrm{V}}^{\dagger}(t_{1})\sin\theta_{\mathrm{a}}\right]\\ &&\int\mathrm{d}t_{2}\xi_{\mathrm{b}}(t_{2})\left[\hat{b}_{\mathrm{H}}^{ \dagger}(t_{2})\cos\theta_{\mathrm{b}}+\hat{b}_{\mathrm{V}}^{\dagger}(t_{2} )\sin\theta_{\mathrm{b}}\right]|0\rangle,\end{array}  \tag{1}\]  
where \(\hat{a}_{\mathrm{H,V}}^{\dagger}(t)\) and \(\hat{b}_{\mathrm{H,V}}^{\dagger}(t)\) are the creation operator densities for horizontally and vertically polarized photons at the \(a\) and \(b\) beam splitter inputs, and \(|0\rangle\) is the 4-mode vacuum. Normalization is achieved by requiring  
\[\int\mathrm{d}t\xi_{\mathrm{a}}^{*}(t)\xi_{\mathrm{a}}(t)=\int\mathrm{d}t\xi_ {\mathrm{b}}^{*}(t)\xi_{\mathrm{b}}(t)=1. \tag{2}\]  
The beam splitter output state \(|\psi_{\mathrm{out}}\rangle\) can be constructed from the input state using the beam splitter transformation,  
\[\hat{a}_{\mathrm{H,V}}^{\dagger}(t)\longrightarrow T\hat{c}_{ \mathrm{H,V}}^{\dagger}(t)+R\hat{d}_{\mathrm{H,V}}^{\dagger}(t)\] \[\hat{b}_{\mathrm{H,V}}^{\dagger}(t)\longrightarrow R\hat{c}_{ \mathrm{H,V}}^{\dagger}(t)+T\hat{d}_{\mathrm{H,V}}^{\dagger}(t), \tag{3}\]  
where \(T\) and \(R\) are the transmission and reflection amplitudes of the symmetric beam splitter, subject to the unitarity conditions  
\[ \begin{array}{rcl}|T|^{2}+|R|^{2}=1\\ TR^{*}=-T^{*}R,\end{array}  \tag{4}\]  
and \(\hat{c}_{\mathrm{H,V}}^{\dagger}(t)\) and \(\hat{d}_{\mathrm{H,V}}^{\dagger}(t)\) are the creation operator densities for horizontally and vertically polarized photons at the \(c\) and \(d\) beam splitter outputs. Applying (3) to (1) we find  
\[\begin{array}{rcl}|\psi_{\mathrm{out}}\rangle&=&\int\mathrm{d}t_{1}\xi_{ \mathrm{a}}(t_{1})\left[\left(T\hat{c}_{\mathrm{H}}^{\dagger}(t_{1})+R\hat{d }_{\mathrm{H}}^{\dagger}(t_{1})\right)\cos\theta_{\mathrm{a}}+\left(T\hat{c} _{\mathrm{V}}^{\dagger}(t_{1})+R\hat{d}_{\mathrm{V}}^{\dagger}(t_{1})\right) \sin\theta  
The right hand side of (5) can be written as a summation of terms each contributing amplitudes to one of the 10 mutually exclusive experimental outcomes:  
\[