Fidelity of time-bin entangled multi-photon states from a quantum emitter

We devise a mathematical framework for assessing the fidelity of multi-photon entangled states generated by a single solid-state quantum emitter, such as a quantum dot or a nitrogen-vacancy center. Within this formalism, we theoretically study the role of imperfections present in real systems on the generation of time-bin encoded Greenberger-Horne-Zeilinger and one-dimensional cluster states. We consider both fundamental limitations, such as the effect of phonon-induced dephasing, interaction with the nuclear spin bath, and second-order emissions, as well as technological imperfections, such as branching effects, non-perfect filtering, and photon losses. In a companion paper, we consider a particular physical implementation based on a quantum dot emitter embedded in a photonic crystal waveguide and apply our theoretical formalism to assess the fidelities achievable with current technologies.