Reliable experimental demonstrations of violations of local realism are highly desirable for fundamental tests of quantum mechanics. One can quantify the violation witnessed by an experiment in terms of a statistical $p$ value, which can be defined as the maximum probability according to local realism of a violation at least as high as that witnessed. Thus, high violation corresponds to small $p$ value. We propose a prediction-based-ratio (PBR) analysis protocol whose $p$ values are valid even if the prepared quantum state varies arbitrarily and local realistic models can depend on previous measurement settings and outcomes. It is therefore not subject to the memory loophole [J. Barrett et al., Phys. Rev. A 66, 042111 (2002)]. If the prepared state does not vary in time, the $p$ values are asymptotically optimal. For comparison, we consider protocols derived from the number of standard deviations of violation of a Bell inequality and from martingale theory [R. Gill, e-print arXiv:quant-ph/0110137]. We find that the $p$ values of the former can be too small and are therefore not statistically valid, while those derived from the latter are suboptimal. PBR $p$ values do not require a predetermined Bell inequality and can be used to compare results from different tests of local realism independent of experimental details.

• Received 18 August 2011

DOI:https://doi.org/10.1103/PhysRevA.84.062118