In the analysis of experiments designed to reveal violation of Bell-type inequalities, it is usually assumed that any hidden variables associated with the nth particle pair would be independent of measurement choices and outcomes for the first (n-1) pairs. Models which violate this assumption exploit what we call the memory loophole. We focus on the strongest type of violation, which uses the 2-sided memory loophole, in which the hidden variables for pair n can depend on the previous measurement choices and outcomes in both wings of the experiment. We show that the 2- sided memory loophole allows a systematic violation of the CHSH inequality when the data are analysed in the standard way, but cannot produce a violation if a CHSH expression depending linearly on the data is used. In the first case, the maximal CHSH violation becomes small as the number of particle pairs tested becomes large. Hence, although in principle the memory loophole implies a slight flaw in existing analyses of Bell experiments, the data still strongly confirm quantum mechanics against local hidden variables. We consider also a related loophole, the simultaneous measurement loophole, which applies if all measurements on each side are carried out simultaneously. We show that this can increase the probability of violating the linearised CHSH inequality as well as other Bell-type inequalities. Notes: Jonathan Barrett, DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA UK Daniel Collins & Sandu Popescu, H.H. Wills Physics Laboratory, University of Bristol, Tyndall Ave, Bristol, BS8 1TL UK Lucien Hardy, Centre for Quantum Computation, Clarendon Laboratory, Parks Road, Oxford, OX1 3PU UK 11 Pages
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