Experimentally Bounding Deviations From Quantum Theory in the Landscape of Generalized Probabilistic Theories

Many experiments in the field of quantum foundations seek to adjudicate between quantum theory and speculative alternatives to it. This requires one to analyze the experimental data in a manner that does not presume the correctness of the quantum formalism. The mathematical framework of generalized probabilistic theories (GPTs) provides a means of doing so. We present a scheme for determining which GPTs are consistent with a given set of experimental data. It proceeds by performing tomography on the preparations and measurements in a self-consistent manner, i.e., without presuming a prior characterization of either. We illustrate the scheme by analyzing experimental data for a large set of preparations and measurements on the polarization degree of freedom of a single photon. We first test various hypotheses for the dimension of the GPT vector space for this degree of freedom. Our analysis identifies the most plausible hypothesis to be dimension 4, which is the value predicted by quantum theory. Under this hypothesis, we can draw the following additional conclusions from our scheme: (i) that the smallest and largest GPT state spaces that could describe photon polarization are a pair of polytopes, each approximating the shape of the Bloch sphere and having a volume ratio of 0.977±0.001, which provides a quantitative bound on the scope for deviations from the state and effect spaces predicted by quantum theory, and (ii) that the maximal violation of the Clauser, Horne, Shimony, and Holt inequality can be at most 1.3%±0.1 greater than the maximum violation allowed by quantum theory, and the maximal violation of a particular inequality for universal noncontextuality can not differ from the quantum prediction by more than this factor on either side. The only possibility for a greater deviation from the quantum state and effect spaces or for greater degrees of supraquantum nonlocality or contextuality, according to our analysis, is if a future experiment (perhaps following the scheme developed here) discovers that additional dimensions of GPT vector space are required to describe photon polarization, in excess of the four dimensions predicted by quantum theory to be adequate to the task.