The Binary-Outcome Detection Loophole

Abstract: The detection loophole problem arises when quantum devices fail to provide an output for some of the experimental runs. These failures allow for the possibility of a local hidden-variable description of the resulting statistics; even if the correlations obtained from successful runs appear non-local. This is of particular significance in device-independent quantum cryptography, where verifiable non-locality is a necessary requirement for the security proof. For every scenario characterised by the amount of devices, along with the number of inputs and outputs available to them, there is a detection threshold - if the efficiency of the device falls below this no verification is possible. In this work I present an intuitive local hidden-variable construction for all no-signalling distributions with two parties and binary outcomes. This provides a lower bound on the detection threshold for quantum measurements in the same scenario tighter than known previously. When both parties have the same number of inputs into their device, this construction is shown optimal for small input numbers. I finish with some interesting conjectures.