Bell's Inequality, Random Sequence, and Quantum Key Distribution

At first glance Ekert 91 quantum key distribution protocol can be secure even if legitimate users adopt whatever devices for the protocol as long as the devices give a result that violates Bell's inequality. However, it is not the case if they ignore non-detection events because Eve can make use of detection-loophole, as Larrson showed. What we show is that even when the legitimate users take into account non-detection events Eve can successfully eavesdrop if the manufacturer design the QKD system appropriately. A loophole utilized here is that of `free-choice' (or `real randomness'). We show how (local) quantum key distribution devices with pseudo-random sequence generator installed in them can apparently violate Bell's inequality. We briefly discuss a debate on Bell's inequality violation that is involved with a question on randomness