  There always exists an entanglement witness for every entangled quantum
state. Negativity of the expectation value of an entanglement witness operator
guarantees entanglement of the corresponding state, given that the measurement
devices involved are perfect, i.e., the performed measurements actually
constitute the witness operator for the state under consideration. In a
realistic situation, there are two possible ways of measurements to drive the
process away from the ideal one. Firstly, wrong measurements may be performed,
and secondly, while the measurement operators are implemented correctly, the
detection process is noisy. Entanglement witnesses are prone to both of these
imperfections. The concept of measurement-device-independent entanglement
witnesses was introduced to remove the first problem. We analyze the "detection
loophole" in the context of measurement-device-independent entanglement
witnesses, which deal with the second problem of imprecise measurements. We
obtain an upper bound on the entanglement witness function in the
measurement-device-independent entanglement witness scenario, below which
entanglement is guaranteed for given non-ideal detector efficiencies, that can
involve both lost events and dark counts.
