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.
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.
We propose an uncertainty-complementarity balance relation and build quantitative connections among non-locality, complementarity, and uncertainty. Our balance relation, which is formulated in a theory-independent manner, states that for two measurements performed sequentially, the complementarity demonstrated in the first measurement in terms of disturbance is no greater than the uncertainty of the first measurement. Quantum theory respects our balance relation, from which the Tsirelson bound can be derived, up to an inessential assumption. In the simplest Bell scenario, we show that the bound of Clauser-Horne-Shimony-Holt inequality for a general non-local theory can be expressed as a function of the balance strength, a constance for the given theory. As an application, we derive the balance strength as well as the nonlocal bound of Popescu-Rohrlich box. Our results shed light on quantitative connections among three fundamental concepts, i.e., uncertainty, complementarity and non-locality.
In quantum theory, it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values and that there is no joint measurement of them. They are called preparation uncertainty and measurement uncertainty, respectively, and research has unveiled that they are not independent from but related with each other in a quantitative way. This study aims to reveal whether similar relations to quantum ones hold also in generalized probabilistic theories (GPTs). In particular, a certain class of GPTs is considered, which can be characterized by transitivity and self-duality and regarded as extensions of quantum theory. It is proved that there are close connections expressed quantitatively between two types of uncertainty on a pair observables also in those theories: if preparation uncertainty exists, then measurement uncertainty also exists, and they are described by similar inequalities. Our results manifest that their correspondences are not specific to quantum theory but more universal ones.
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 causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
The causal structure of any experiment implies restrictions on the observable correlations between measurement outcomes, which are different for experiments exploiting classical, quantum, or post-quantum resources. In the study of Bell nonlocality, these differences have been explored in great detail for more and more involved causal structures. Here, we go in the opposite direction and identify the simplest causal structure which exhibits a separation between classical, quantum, and post-quantum correlations. It arises in the so-called Instrumental scenario, known from classical causal models. We derive inequalities for this scenario and show that they are closely related to well-known Bell inequalities, such as the Clauser-Horne-Shimony-Holt inequality, which enables us to easily identify their classical, quantum, and post-quantum bounds as well as strategies violating the first two. The relations that we uncover imply that the quantum or post-quantum advantages witnessed by the violation of our Instrumental inequalities are not fundamentally different from those witnessed by the violations of standard inequalities in the usual Bell scenario. However, non-classical tests in the Instrumental scenario require fewer input choices than their Bell scenario counterpart, which may have potential implications for device-independent protocols.
To make precise the sense in which the operational predictions of quantum theory conflict with a classical worldview, it is necessary to articulate a notion of classicality within an operational framework.A widely applicable notion of classicality of this sort is whether or not the predictions of a given operational theory can be explained by a generalized-noncontextual ontological model.We here explore what notion of classicality this implies for the generalized probabilistic theory (GPT) that arises from a given operational theory, focusing on prepare-measure scenarios.We first show that, when mapping an operational theory to a GPT by quotienting relative to operational equivalences, the constraint of explainability by a generalized-noncontextual ontological model is mapped to the constraint of explainability by an ontological model.We then show that, under the additional assumption that the ontic state space is of finite cardinality, this constraint on the GPT can be expressed as a geometric condition which we term simplex embeddability.Whereas the traditional notion of classicality for a GPT is that its state space be a simplex and its effect space be the dual of this simplex, simplex embeddability merely requires that its state space be embeddable in a simplex and its effect space in the dual of that simplex.We argue that simplex embeddability constitutes an intuitive and freestanding notion of classicality for GPTs.Our result also has applications to witnessing nonclassicality in prepare-measure experiments.