The binary-outcome detection loophole

The detection loophole problem arises when quantum devices fail to provide an output for some runs. If treating these devices in a device-independent manner, failure to include the unsuccessful runs in the output statistics can lead to an adversary falsifying security i.e. Bell inequality violation. If the devices fail with too high frequency, known as the detection threshold, then no security is possible, as the full statistics cannot violate a Bell inequality. In this work we provide an intuitive local hidden-variable strategy that the devices may use to falsify any two-party, binary-outcome no-signalling distribution up to a threshold of 2(mA + mB − 8)/(mAmB − 16), where mA, mB refer to the number of available inputs choices to the two parties. This value is the largest analytically predicted lower bound for no-signalling distributions. We strongly conjecture it gives the true detection threshold for mA = mB, and for computationally tractable scenarios we provide the Bell inequality which verifies this. We also prove that a non-trivial detection threshold remains, even when allowing one party an arbitrary number of input choices