In quantum key distribution, measurement-device-independent and decoy-state techniques enable the two cooperative agents to establish a shared secret key using imperfect measurement devices and weak Poissonian sources, respectively. Investigations so far are not comprehensive as they restrict to less than or equal to four decoy states. Moreover, many of them involve pure numerical studies. Here I report a general security proof that works for any fixed number of decoy states and any fixed raw key length. There are two key ideas involved here. The first one is the repeated application of the inversion formula for the Vandermonde matrix to obtain various bounds on certain yields and error rates. The second one is the use of a recently proven generalization of the McDiarmid inequality. These techniques raise the best provably secure key rate of the measurement-device-independent version of the Bennett-Brassard 1984 scheme by at least 1.25 times and increase the workable distance between the two cooperative agents from slightly less than 60 km to slightly greater than 130 km in case where there are ${10}^{10}$ photon pulse pairs sent without a quantum repeater.

• Received 23 March 2020
• Accepted 23 June 2020

DOI:https://doi.org/10.1103/PhysRevA.102.012611