Impact of dephased entanglement, mixed input states and varying measurement orientations on the reliability of cryptographic keys generated via the quantum protocol E91.

Photons and optical circuits are among the most promising platforms for implementation of quantum technologies, including quantum computing and quantum cryptography. For entanglement-based quantum key distribution, on-demand sources of highly entangled photon pairs are essential, and semiconductor quantum dots have emerged as prominent candidates to satisfy the necessary conditions of brightness and fidelity. However, in most cases the photon pair emitted by these sources is an anomalous Bell state whose relative phase and purity are non-ideal, and then, their suitability for distributing secret keys must be assessed. This work studies the performance of the quantum key distribution protocol E91, under the influence of three elements: the phase in the input state, the impurity of such a state, and the relative directions of the polarization analyzers. We derive analytical expressions for the minimum quantum bit error rate and the Bell's parameter, and validate them by using quantum computational tools. Our results show that the performance of the quantum transmission is highly impacted by the phase and purity of the entangled input states, and that such an impact may be modulated through the orientation of the polarizers. These findings provide important insight for the scalable implementation of quantum key distribution protocols with realistic entanglement sources.