Abstract
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.
- Received 27 January 2020
- Revised 28 July 2020
- Accepted 11 January 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.010331
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Quantum theory is famously counterintuitive, admitting of many phenomena that do not obviously have counterparts in classical physics. Nonetheless, some of these initially surprising features can in fact be given a very natural account in terms of an underlying classical theory, called a noncontextual model. Proving that a given quantum experiment does not admit of a noncontextual model formally establishes that it is strongly nonclassical, a feature that is increasingly being found to be associated with quantum advantages for information processing.
The framework of generalized probabilistic theories (GPTs) provides a means of summarizing the scope of possible statistics that can be achieved in any physical theory, including alternatives to quantum theory. It also provides the means of doing so for any subtheory of quantum theory, that is, any closed subset of quantum states and measurements, and therefore for any experiment. These summaries are termed GPTs and for prepare-measure experiments are represented by a pair of geometric objects: a state space and an effect space.
The question we address here is: which GPTs may be said to exhibit strong nonclassicality? More precisely, we ask: which GPTs correspond to operational theories that do not admit of a noncontextual model? It turns out that, for prepare-measure experiments, the answer has an elegant geometric characterization in terms of whether or not it is possible to find a simplex and a hypercube dual to this simplex that respectively contain the state and effect spaces of the GPT. This result opens up new avenues for experimentally witnessing strong nonclassicality, including the possibility of doing so on data from experiments that were not specifically designed to test noncontextuality.
