In a large medical trial, if one obtained a ridiculously small p-value like 10^-12, one would typically move from a plain hypothesis test to trying to estimate the parameters of the effect. For example, one might try to estimate the optimal dosage of a drug or the optimal length of a course of treatment. Tests of Bell and noncontextuality inequalities are hypotheses tests, and typical p-values are much lower than this, e.g. 12-sigma effects are not unheard of and a 7-sigma violation already corresponds to a p-value of about 10^-12. Why then, in quantum foundations, are we still obsessed with proposing and testing new inequalities rather than trying to estimate the parameters of the effect from the experimental data? Here, we will try to do this for preparation contextuality, but will also make some related comments on recent loophole-free Bell inequality tests.
We introduce two measures of preparation contextuality: the maximal
overlap and the preparation contextuality fraction. The latter is
linearly related to the degree of violation of a preparation
noncontextuality inequality, so can be estimated from experimental data.
Although the measures are different in general, they can be equal for
proofs of preparation contextuality that have sufficient symmetry, such
as the timelike analogue of the CHSH scenario. We give the value of
these measures for this scenario. Using our result, we can consider
pairty-epsilon multiplexing, Alice must try to communicate two bits to
Bob so that he can choose to determine either of them with high
probability, but where Alice must ensure that Bob cannot guess the
parity of the bits with probability greater than 1/2 + epsilon, and
determine the range of epislon for which there is still an advantage in
preparation contextual theories. If time permits, I will make some
brief comments on how to robustify experimental tests of this result.
joint work with Eric Freda and David Schmid