Minimum Detection Efficiencies for a Loophole-Free Bell-type Test

TL;DR

This paper identifies optimal conditions for closing the detection loophole in Bell tests, showing that perfect detection efficiency in one measurement suffices for a loophole-free test across many non-maximally entangled states.

Contribution

It derives a new Eberhard-like inequality and demonstrates that only one measurement needs perfect detection efficiency to close the loophole for numerous entangled states.

Findings

Perfect detection efficiency in one measurement suffices for loophole closure.

Loophole-free tests tolerate a certain level of noise.

Applicable to a wide range of non-maximally entangled states.

Abstract

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.