Contextuality without access to a tomographically complete set

Abstract: The non-classicality of single quantum systems can be formalised using the notion of contextuality. But can contextuality be convincingly demonstrated in an experiment, without reference to the quantum formalism? The operational approach to contextuality due to Spekkens requires finding operationally equivalent preparation procedures. Previously these have been obtained by demanding indistinguishability under a set of measurements taken to be tomographically complete. In the language of generalised probability theories, this requires the ability to explore all the dimensions of the system's state space. However, if the true tomographically complete set is larger than the set assumed, the extra measurements could break the operational equivalences and hence eliminate the putative contextuality. Such extra dimensions could arise in post-quantum theories, but even if quantum theory is exact there can be unexpected degrees of freedoms due to imperfections in an experiment. Here we design tests of contextuality that are immune to this effect for a given number of extra measurements in the tomographically complete set, even if nothing is known about their statistics. This allows contextuality to be demonstrated with weaker assumptions.