Understanding Contextuality in General Probabilistic Theories

Table of Links

Abstract and 1. Introduction

2. Operational theories, ontological models and contextuality

3. Contextuality for general probabilistic theories

   3.1 GPT systems

   3.2 Operational theory associated to a GPT system

   3.3 Simulations of GPT systems

   3.4 Properties of univalent simulations

4. Hierarchy of contextuality and 4.1 Motivation and the resource theory

   4.2 Contextuality of composite systems

   4.3 Quantifying contextuality via the classical excess

   4.4 Parity oblivious multiplexing success probability with free classical resources as a measure of contextuality

5. Discussion

   5.1 Contextuality and information erasure

   5.2 Relation with previous works on contextuality and GPTs

6. Conclusion, Acknowledgments, and References

A Physicality of the Holevo projection

3 Contextuality for general probabilistic theories

One can often prove contextuality of a theory by studying statistical behaviours of an individual system A (such as a qubit) in prepare-and-measure scenarios. The information that suffices to describe the possible behaviours constitutes of

• a set Ω of possible states that A can be prepared in, according to the theory in question,

• a set of possible measurements that can be applied to A, each consisting of a collection of effects associated with the measurement outcomes, and

• for every state ω ∈ Ω and every effect e the probability of obtaining e when measurement M is applied to system A prepared in state ω.

This information is commonly expressed as a system in a general probabilistic theory— a GPT system.

In the following we omit any mention of dynamics that the system may undergo. This simplification has no consequence on our discussion of preparation and measurement contextuality.

3.1 GPT systems

Let us now discuss the mathematical description of GPT systems [54–57] and the definition of contextuality in this context.

Definition 1. A GPT system A is specified by two non-empty convex sets

Likewise, every classical probabilistic system is a GPT system [54, §4].

Besides finite classical systems, we will also need the notion of a countably infinite one.

Besides individual systems, we will occasionally also need to refer to composite ones. In general, the composite of two GPT systems is not unique – see [58, Section 5] for an in-depth discussion of tensor products of GPT systems. However, among all the possible choices, there is a ‘minimal composite’ of two GPT systems A and B, whose state and effect space merely contain the separable ones. Any meaningful composite of two GPT systems necessarily contains their minimal composite as a subsystem.

Of particular interest is the tensor product of an arbitrary GPT system and a classical GPT system, in which case there is a unique choice of a sensible tensor product corresponding to the minimal composite of GPT systems.