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In quantum key distribution, measurement-device-independent and decoy-statetechniques enable the two cooperative agents to establish a shared secret keyusing imperfect measurement devices and weak Poissonian sources, respectively.Investigations so far are not comprehensive as they restrict to less than orequal to four decoy states. Moreover, many of them involves pure numericalstudies. Here I report a general security proof that works for any fixed numberof decoy states and any fixed raw key length. The two key ideas involved here.The first one is the repeated application of the inversion formula forVandermonde matrix to obtain various bounds on certain yields and error rates.The second one is the use of a recently proven generalization of the McDiarmidinequality. These techniques rise the best provably secure key rate of themeasurement-device-independent version of the BB84 scheme by at least 1.25times and increase the workable distance between the two cooperative agentsfrom slightly less than 60 km to slightly greater than 130 km in case there are$10^{10}$ photon pulse pair sent without a quantum repeater.

Security Of Finite-Key-Length Measurement-Device-Independent Quantum Key Distribution Using Arbitrary Number Of Decoys