## 1 Introduction
In addition to detector inefficiencies, dark count rates [10] and polarization errors, MDI-QKD is also susceptible to bit errors caused by timing differences [7] and photon distinguishability in pulse envelope, bandwidth and wavelength [11]. Prior research has addressed polarization and timing errors [10], as well as the feasibility of implementing the protocol utilizing weak coherent pulses from independent laser sources [11], but significant assumptions about photon source characteristics remain unexplored. In this paper, we analyze the impact of error tolerances in photon polarization encoding, timing, wavelength and bandwidth on the quantum bit error rate (QBER) of an otherwise ideal MDI-QKD implementation using single photon sources with Gaussian pulse envelopes. In Section 2 we specify the photon temporal wave functions and apply the formalism of quantum optics to calculate probabilities for [PERSON]'s measurement outcomes as a function of [PERSON]'s and [PERSON]'s encoding choices and the tolerances of their photon sources. In Section 3 we describe our Monte Carlo simulation of the MDI-QKD protocol based on these probabilities to determine the associated Quantum Bit Error Rate (QBER). In Section 4 we present the results of our simulations and discuss the sensitivity of MDI-QKD protocol based on these probabilities. \begin{table}
\begin{tabular}{|c|c|c|c|c|} \hline \hline [PERSON] & [PERSON] & Possible Detected States & Probability of & Shared \\ Polarization & Polarization & & generating a usable & Bit \\  & & & detection state & Value \\ \hline \(\uparrow\downarrow\)\(|V\rangle\) & \(\leftrightarrow\)\(|H\rangle\) & Usable: \(c_{\
u}c_{H}\), \(d_{\
u}d_{H}\), \(c_{\
u}d_{H}\), \(c_{H}d_{\
u}\) & 1 \\ \hline \(\leftrightarrow\)\(|H\rangle\) & \(\uparrow\)\(|V\rangle\) & Unusable: _none_ & 4 \(\times\) 25\% = 100\%\) & 0 \\ \hline \(\uparrow\)\(|V\rangle\) & \(\uparrow\)\(|V\rangle\) & Unusable: \(c_{\
u}c_{\
u}\), \(d_{\
u}d_{\
u}\) & Usable: _none_ & 0\% & — \\ \hline \(\leftrightarrow\)\(|H\rangle\) & \(\leftrightarrow\)\(|H\rangle\) & Unusable: \(c_{\mu}c_{H}\), \(d_{\mu}d_{H}\) & Usable: _none_ & 0\% & — \\ \hline \(\
diamondsuit\)\(|A\rangle\) & \(\
diamondsuit\)\(|A\rangle\) & Usable: \(c_{\mu}c_{\
u}\), \(d_{\mu}d_{\
u}\) & 2 \(\times\) 25\% = 50\%\) & 0 \\ \hline \(\
diamondsuit\)\(|D\rangle\) & \(\
diamondsuit\)\(|D\rangle\) & Unusable: \(c_{\mu}c_{H}\), \(d_{\mu}d_{H}\), \(c_{\
u}c_{\
u}\), \(d_{\
u}d_{\
u}\) & 2 \(\times\) 25\% = 50\%\) & 1 \\ \hline \(\
diamondsuit\)\(|A\rangle\) & \(\
diamondsuit\)\(|D\rangle\) & Usable: \(c_{\mu}d_{\
u}\), \(c_{\
u}d_{H}\) & 2 \(\times\) 25\% = 50\%\) & 0 \\ \hline \(\
diamondsuit\)\(|D\rangle\) & \(\
diamondsuit\)\(|A\rangle\) & Unusable: \(c_{\mu}c_{\mu}\), \(d_{\mu}d_{H}\), \(c_{\
u}c_{\
u}\), \(d_{\
u}d_{\
u}\) & 2 \(\times\) 25\% = 50\%\) & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Nominal probability of generating a key bit and the shared secret bit value for all eight same-basis photon polarization encodings and associated measurement outcomes. Figure 2: Representative sequences of events for the MDI-QKD protocol for two particular input states. to various contributions to photon distinguishability. We conclude the paper in Section 5 with a discussion of the implications of our results for studying practical implementations of MDI-QKD and identify possible extensions to our work.