Abstract
We present formulas that tighten the upper bound for the phase-flip errors in the decoy-state method by using four intensities. Our result compressed the bound to about a quarter of the known result for measurement-device-independent quantum key distribution. Based on this, we find that the key rate is improved by more than 100% given weak coherent state sources, and even more than 200% with heralded single-photon sources.
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