  Quantum key distribution (QKD) protocols with threshold detectors are driving
high-performance QKD demonstrations. The corresponding security proofs usually
assume that all physical detectors have the same detection efficiency. However,
the efficiencies of the detectors used in practice might show a mismatch
depending on the manufacturing and setup of these detectors. A mismatch can
also be induced as the different spatial-temporal modes of an incoming signal
might couple differently to a detector. Here we develop a method that allows to
provide security proofs without the usual assumption. Our method can take the
detection-efficiency mismatch into account without having to restrict the
attack strategy of the adversary. Especially, we do not rely on any
photon-number cut-off of incoming signals such that our security proof is
directly applicable to practical situations. We illustrate our method for a
receiver that is designed for polarization encoding and is sensitive to a
number of spatial-temporal modes. In our detector model, the absence of quantum
interference between any pair of spatial-temporal modes is assumed. For a QKD
protocol with this detector model, we can perform a security proof with
characterized efficiency mismatch and without photon-number cut-off assumption.
Our method also shows that in the absence of efficiency mismatch in our
detector model, the key rate increases if the loss due to detection
inefficiency is assumed to be outside of the adversary's control, as compared
to the view where for a security proof this loss is attributed to the action of
the adversary.
