A Tailored Fidelity Estimation and Purification Method for Entangled Quantum Networks

We present a method to conduct both quantum state reconstruction and entanglement purification simultaneously that is advantageous in several respects over previous work in this direction, showing that the number of Bell pairs necessary to boot a quantum network can be significantly reduced compared to an existing method. The existing method requires at least 10⁵ Bell pairs for the state reconstruction phase to estimate that the state is of fidelity 0.99 within the error range of 10^-2, whereas our approach only requires around 2,841 to be certain with 99.7\% of confidence that the estimated fidelity lies within [0.99-0.01, 0.99+0.01]. In addition, in our approach we can start with a lower fidelity Bell pair and purify it multiple times, estimating at the same time the resultant fidelity with guarantee of 99.7\% that the fidelity estimate lies within a certain range. Moreover, the existing method cannot correct both bit-flip and phase-flip errors at the same time and can only correct one of these, whereas our approach can correct both bit-flip and phase-flip errors simultaneously. This research produces numerical estimates for the number of Bell pairs actually needed to guarantee a certain threshold fidelity F. The research can support the functioning real-world quantum networking by providing the information of the time needed for the bootstrapping of a quantum network to finish.