  Entanglement witnesses form an effective method to locally detect
entanglement in the laboratory without having the prior knowledge of the full
density matrix. However, separable states can be erroneously indicated as
entangled in such detections in the presence of wrong measurements or loss in
detectors. Measurement-device-independent entanglement witnesses (MDI-EWs)
never detect fake entanglement even under wrong measurements and for a
particular kind of lossy detectors. A crucial assumption in the case of
faithful detection of entanglement employing MDI-EWs is that the preparation
devices producing "quantum inputs" - which are inputs additional to the quantum
state whose entanglement is to be detected - are perfect and there is no noise
during their transmission. Here, we relax these assumptions and provide a
general framework for studying the effect of noise on the quantum inputs,
invoking uniform and non-uniform noise models. We derive sufficient conditions
on the uniform noisy map for retaining the characteristic of MDI-EWs. We find
that in the context of non-uniform and entangling noise, fake entanglement
detection is possible even by MDI-EWs. We also investigate various paradigmatic
models of local noise and find conditions of revealing entanglement in the
class of Werner states.
