Extensions and Restrictions of Generalized Probabilistic Theories

Jonathan Steinberg

Zhaoqi Wu, Chuanxi Zhu, Shunlong Luo +1 more

Serge Massar

Farid Shähandeh

Generalized contextuality refers to our inability of explaining measurement statistics using a context-independent probabilistic model. On the other hand, measurement statistics can also be modeled most generally within the framework of general probabilistic theories. Here, starting from a construction of general probabilistic theories based on a Gleason-type theorem, we show that for any such theory 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. In other words, requiring any two of these properties for a general probabilistic theory implies the violation of the third one. We extensively discuss general probabilistic theories for which (ii) and (iii) are met while (i) is violated, exploring the role of the no-restriction hypothesis as a sufficient condition that enforces ontological contextuality on nonsimplicial general probabilistic theories. 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.

Waldemar Kłobus

Peter Janotta, Raymond Lal

The framework of generalized probabilistic theories is a popular approach for studying the physical foundations of quantum theory. The standard framework assumes the no-restriction hypothesis, in which the state space of a physical theory determines the set of measurements. However, this assumption is not physically motivated. We generalize the framework to account for systems that do not obey the no-restriction hypothesis. We then show how our framework can be used to describe certain classes of probabilistic theories, for example, those which include intrinsic noise. Relaxing the restriction hypothesis also allows us to introduce a ``self-dualization'' procedure, which yields a class of theories that share many features of quantum theory. We then characterize joint states, generalizing the maximal tensor product. We show how this tensor product can be used to describe the convex closure of the Spekkens toy theory, and in doing so we obtain an analysis of why it is local in terms of the geometry of its state space. We show that the unrestricted version of the Spekkens toy theory is the theory known as ``boxworld'' that allows maximal nonlocal correlations.