## 1 Introduction  
Quantum Key Distribution (QKD) is an application of quantum cryptography--the process of exploiting quantum effects to establish secure communications between two authorized users, [PERSON] and [PERSON], in the presence of an unwanted third party, [PERSON]. QKD protocols promise unconditionally secure communications through exchange of encoded photons, which, due to their quantum nature, are altered in a detectable way if they are observed by an unauthorized eavesdropper [1]. In conventional QKD protocols such as BB84 [2], the original QKD protocol, a photon is randomly encoded in one of a pre-determined set of quantum states by [PERSON]. [PERSON] transmits the photon to [PERSON] who obtains partial information on [PERSON]'s encoding by performing a randomly selected projective measurement on the photon. Subsequently [PERSON] reveals partial information on her encoding over an open classical channel. Along with the measurement results, this additional information allows [PERSON] to identify the original encoding state of a subset of the transmitted photons with certainty. The encoding of this subset constitutes the raw shared secret key that [PERSON] and [PERSON] use for encrypted communications [3]. As [PERSON] does not have access to the results of [PERSON]'s measurements, she lacks the necessary information to infer the secret bits. Since 1984, other protocols have been proposed to address security vulnerabilities in implementations of BB84 [4; 5; 6; 7; 8]. One such protocol is Measurement-Device-Independent QKD (MDI-QKD), depicted in Figure 1, which addresses the vulnerability of [PERSON]'s measurement to malicious signals introduced on the quantum channel by [PERSON]. MDI-QKD moves the act of photon measurement from [PERSON] to an untrusted third party, [PERSON] (shown in red). [PERSON] and [PERSON] each send one polarization-encoded photon to [PERSON], who subjects them to two-photon interference in a symmetric 50:50 beam splitter, followed by polarization-sensitive detection via a pair of polarizing beam splitters and four single-photon detectors, each corresponding to a polarization and beam splitter output port (\(d_{\text{V}}\), \(d_{\text{H}}\), \(c_{\text{V}}\), \(c_{\text{H}}\)). [PERSON] publicly announces which detectors registered photons. Subsequently, [PERSON] and [PERSON] publicly announced which polarization basis (horizontal-vertical or antidiagonal-diagonal) they used for encoding. From this information and their private knowledge of their own photon encodings, [PERSON] and [PERSON] can infer one bit of shared secret key in 25% of the exchanges under ideal circumstances. Figure 2 demonstrates these steps explicitly for two key attempts--one in the horizontal-vertical basis and one in the antidiagonal-diagonal basis. The blue and green arrows denote communication on a sensitive quantum channel, while red arrows denote communication on an open classical channel visible to [PERSON]. Similarly, Table 1 displays all possible same-basis input states, [PERSON]'s potential detection states for each input, and the shared bit value associated with a given input state. When [PERSON] and [PERSON] choose opposite encoding bases for a trial, that trial is discarded and no key is generated. The same-basis inputs \(|V\rangle|V\rangle\) and \(|H\rangle|H\rangle\) cannot be used to generate a key bit because [PERSON] can infer the bit from his measurement and knowledge of the encoding basis. There are also some detected states for antidiagonal-diagonal basis encodings that can occur for multiple input states, making them unusable for key generation. Figure 1: Diagram of MDI-QKD instrumentation. Despite its immunity from eavesdropping attacks on the detector, MDI-QKD is susceptible to physical non-idealities that can limit the key generation rate by introducing quantum bit errors, which are physically indistinguishable from perturbations caused by [PERSON]'s measurements [9].