Abstract
Generalized contextuality refers to our inability of explaining measurement statistics using a context-independent 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.
- Received 27 January 2020
- Revised 24 April 2020
- Accepted 11 January 2021
DOI:https://doi.org/10.1103/PRXQuantum.2.010330
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
Imagine a theory whose account for the trajectory of an apple falling from a tree depends on whether it is Gala or Pink Lady. We know, however, that it is impossible to tell the apple’s variety from its trajectory. One such theory thus has a seemingly unnecessary and nonsensical parameter, which makes its description of a physical phenomenon contextual. While classical theories are immune to contextuality, it is known that, upon attempting to provide a classical-like interpretation of quantum mechanics, it becomes contextual and hence counterintuitive. But what structural property of a theory like quantum mechanics makes it prone to contextuality? Is it possible to construct a theory that goes beyond classical and quantum theories while maintaining noncontextuality?
In this work, we determine the geometrical characteristics that give rise to either noncontextuality or contextuality of a physical theory. In doing so, we consider the framework of general probabilistic theories that provide a unified tool for constructing theories. This subsumes both classical and quantum mechanics. We then use some algebra to establish the precise conditions that allow us to replace the elements of an arbitrary theory with probability distributions, hence obtaining a classical-like theory. A failure of these conditions implies contextuality and nonclassicality. It then follows that any theory that reproduces quantum-mechanical predictions must be contextual.
The techniques and results obtained in this work can be extended and applied to quantum-information processing tasks that aim for nonclassical advantages. In particular, they may help us reveal where the quantumness of quantum computations is hiding.
