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.
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.
The principle of tomographic locality states that the operational state of a multipartite system can be fully characterized by the statistics obtained from measurements that are local to the individual subsystems. This property holds in quantum theory and features prominently in axiomatic reconstructions of the theory, where it serves to rule out a wide class of alternatives. For instance, quantum theory with Hilbert spaces defined over the real field (rather than the complex field) is an example of a theory that is ruled out in this fashion. Given its foundational importance, it is worthwhile to subject this principle to a direct experimental test. Specifically, we consider an experiment on the polarization degrees of freedom of a pair of photonic modes in a prepare-and-measure scenario and analyze the resulting data within the framework of generalized probabilistic theories. The signature of a failure of tomographic locality is that there are pairs of states on the bipartite system that can only be distinguished by the statistics they yield for non-separable measurements. In the full quantum setting, we find no evidence of a violation of tomographic locality. As a test of our analysis method, we also verify that if we restrict attention to those states and measurements that lie within the fragment described by quantum theory over the real field, then a clear signature of the failure of tomographic locality is observed.
We exploit the interplay between uncertainty, disturbance, nonlocality, and self-duality within the framework of generalized probabilistic theories (GPTs). We first introduce an operational uncertainty-disturbance relation that reveals the discrepancies of GPTs in uncertainty and disturbance exhibited by one single measurement. We then apply the relation to Bell's scenarios and derive upper bounds for the basic spatial Clauser-Horne-Shimony-Holt and the temporal Leggett-Garg inequalities with and without the assumption of strong self-duality, respectively. It turns out that self-duality, though being thought of as one pillar of the axiomatization of quantum theory and implying additional constraints, is insufficient to explain the quantum violations. These insights may serve as a seed for a universal understanding of these basic properties in GPTs.
The formalism of generalized probabilistic theories (GPTs) was originally developed as a way to characterize the landscape of conceivable physical theories. Thus, the GPT describing a given physical theory necessarily includes all physically possible processes. We here consider the question of how to provide a GPT-like characterization of a particular experimental setup within a given physical theory. We show that the resulting characterization is not generally a GPT in and of itself, rather, it is described by a more general mathematical object that we introduce and term an accessible GPT fragment. We then introduce an equivalence relation, termed cone equivalence, between accessible GPT fragments (and, as a special case, between standard GPTs). We give a number of examples of experimental scenarios that are best described using accessible GPT fragments, and where moreover, cone-equivalence arises naturally. We then prove that an accessible GPT fragment admits of a classical explanation if and only if every other fragment that is cone equivalent to it also admits of a classical explanation. Finally, we leverage this result to prove several fundamental results regarding the experimental requirements for witnessing the failure of generalized noncontextuality. In particular, we prove that neither incompatibility among measurements nor the assumption of freedom of choice is necessary for witnessing failures of generalized noncontextuality, and moreover, that such failures can be witnessed even when using arbitrarily inefficient detectors.
Generalized contextuality refers to our inability of explaining measurement statistics using a contextindependent probabilistic and ontological model.On the other hand, measurement statistics can also be modeled using the framework of general probabilistic theories (GPTs).Here, starting from a construction of GPTs based on a Gleason-type theorem, we fully characterize these structures with respect to their permission and rejection of generalized (non)contextual ontological models.It follows that in any GPT construction the three insistence of (i) the no-restriction hypothesis, (ii) the ontological noncontextuality, and (iii) multiple nonrefinable measurements for any fixed number of outcomes are incompatible.Hence, any GPT satisfying the no-restriction hypothesis is ontologically noncontextual if and only if it is simplicial.We give a detailed discussion of GPTs for which the no-restriction hypothesis is violated, and show that they can always be considered as subtheories (subGPTs) of GPTs satisfying the hypothesis.It is shown that subGPTs are ontologically noncontextual if and only if they are subtheories of simplicial GPTs of the same dimensionality.Finally, we establish as a corollary the necessary and sufficient condition for a single resourceful measurement or state to promote an ontologically noncontextual (i.e., classical) general probabilistic theory to an ontologically contextual (i.e., nonclassical) one under the no-restriction hypothesis.
Testing quantum mechanics against alternative theories must be done in a way that does not make assumptions about the validity of any particular theory, in particular, without presuming the correctness of quantum theory. Here, the authors set bounds on possible deviations from quantum theory using the formalism of Generalized Probabilistic Theories by experimentally probing a three-level system consisting of a photon shared among three modes. They find no evidence for the breakdown of quantum theory.
For a collection of states indexed by [n]=1,…,n, let F(H) denote the optimal unnormalized minimum discrimination error for each nonempty subfamily H⊆[n]. We call F the discrimination profile and characterize exactly which profiles can arise in a general probabilistic theory (GPT). For fixed n, the GPT region Gn is a bounded rational polytope, giving a hierarchy Cn⊆Qn⊆Gn of classical, quantum, and GPT profiles, analogous to the local--quantum--no-signalling hierarchy in Bell theory.
For n=3 we determine the classical and GPT polytopes explicitly. Within G3, the classical region is characterized by β:=F(123)−F(12)−F(13)−F(23)≤0, while the GPT maximum is 1. For qubit triples, the maximal value is attained by the trine, βtr=33/2−2, and we prove the dimension-independent quantum bound βQ≤2/3, with a slight further improvement. Hence 0=βC<33/2−2≤βQ≤2/3<1=βGPT. We conjecture βQ=βtr and prove this for arbitrary pure-state triples, together with further supporting evidence.
For arbitrary n, every classical facet admitting a GPT violation already admits a quantum violation in dimension at most 3. A four-state qubit example shows that quantum success profiles need not be submodular, unlike for n=3. Finally, we exhibit a four-dimensional quantum triple whose discrete profile is classical for all subfamilies but becomes nonclassical when unequal priors are allowed.