Measurement simulability and incompatibility in quantum theory and other operational theories

Abstract

Quantum theory is a particularly important instance of an operational theory.By looking at quantum theory from the perspective of more abstract operational framework one is able to study its properties in a wider context. This allows us to identify some of the physical features characteristic of quantum theory and it helps us to understand what makes quantum theory special among other theories. From the information-theoretic point of view this might give us insight into the foundations behind the advantages of quantum information processing over its classical counterpart.

In this thesis, based on Publications I – VI, we consider the properties of measurements in quantum theory and other operational theories. After having introduced the framework of operational theories, we consider a communication scheme based on an experimental prepare-and-measure scenario and demonstrate this with different communication tasks. This gives us context for how the different communication tasks can be implemented in different theories and how operational theories can be compared to each other, in doing so establishing quantum theory intuitively as an operational theory among other theories.

The main property of measurements we focus on in this work is the simulation of measurements, which consists of manipulating the inputs and outputs of the measurement devices. We study how using this process on existing measurement devices can be used to operationally imitate new devices, and what kind of structure the simulation process induces on measurements. We look at the components of simulability, analyzing and demonstrating them in quantum theory as well as various toy theories. This gives us structural information that differentiates quantum theory from other theories.

We also consider applications of simulability. Firstly, we consider operational restrictions imposed upon measurements. We argue that the restricted set of physical measurements must be closed with respect to the simulation process since the simulation of physical devices must lead toother physically feasible devices. We demonstrate different types of restrictions by classifying them and analysing their structure.

As a second application we see how the simulation of measurements relates to joint measurability, i.e. compatibility of measurements, and how it can be viewed as a generalisation of it. This allows us to present an operational principle previously known to quantum theory, the no-freeinformation principle, according to which any measurement that is compatible with all other measurement must not provide any useful, and therefore free, information about the system. Whilst this principle holds in quantum theory, there are non-classical theories for which it is violated, and so enforcing this principle may be considered a way to exclude some unphysical theories.