  We discuss the problem of finding the most favorable conditions for closing
the detection loophole in a test of local realism with a Bell inequality. For a
generic non-maximally entangled two-qubit state and two alternative measurement
bases we apply Hardy's proof of non-locality without inequality and derive an
Eberhard-like inequality. For an infinity of non-maximally entangled states we
find that it is possible to refute local realism by requiring perfect detection
efficiency for only one of the two measurements: the test is free from the
detection loophole for any value of the detection efficiency corresponding to
the other measurement. The maximum tolerable noise in a loophole-free test is
also evaluated.
