If one seeks to test quantum theory against many alternatives in a landscape of possible physical theories, then it is crucial to be able to analyze experimental data in a theory-agnostic way. This can be achieved using the framework of Generalized Probabilistic Theories (GPTs). Here, we implement GPT tomography on a three-level system corresponding to a single photon shared among three modes. This scheme achieves a GPT characterization of each of the preparations and measurements implemented in the experiment without requiring any prior characterization of either. Assuming that the sets of realized preparations and measurements are tomographically complete, our analysis identifies the most likely dimension of the GPT vector space describing the three-level system to be nine, in agreement with the value predicted by quantum theory. Relative to this dimension, we infer the scope of GPTs that are consistent with our experimental data by identifying polytopes that provide inner and outer bounds for the state and effect spaces of the true GPT. From these, we are able to determine quantitative bounds on possible deviations from quantum theory. In particular, we bound the degree to which the no-restriction hypothesis might be violated for our three-level system.
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