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.
Highlights
In what precise sense does quantum theory necessitate a departure from a classical worldview? this is one of the central questions in the foundations of quantum theory, there is no consensus on its answer
Note that Eq (4) and the assumption of tomography guarantee that every operationally equivalent pair of preparations in the operational theory is mapped to the same generalized probabilistic theory (GPT) state (GPT effect) vector, and that each GPT vector is a representation of an operational equivalence class of operational procedures
As we show in Appendix B, an ontological model of a GPT is equivalent to a positive quasiprobability representation of that GPT
Summary
In what precise sense does quantum theory necessitate a departure from a classical worldview? this is one of the central questions in the foundations of quantum theory, there is no consensus on its answer. The two most stringent notions of nonclassicality proposed to date are: the failure to admit of a locally causal ontological model (Bell’s theorem) [1,2] and the failure to admit of a generalized-noncontextual ontological model [3] Both of these are operationally meaningful notions of nonclassicality, in the sense that one can determine in principle whether a given set of operational statistics admits of a classical explanation by their lights, regardless of its consistency with quantum theory [4]. Given that Einstein made significant use of this principle when he developed the theory of relativity [20], it is seen to have impressive credentials in physics and is a natural constraint to impose on ontological models From this perspective, the impossibility of finding generalized-noncontextual ontological models is best understood as a failing of the framework of ontological models itself, and as a type of nonclassicality.
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