Structure of Quantum Theory

Textbook quantum superposition refers to the feature that certain linear combinations of Hilbert space rays, each representing a valid state, are themselves valid states. Recently, superposition has been at the heart of experimental proposals for demonstrating indefinite causal order within quantum theory as well as non-classicality of gravity. However, unlike Bell nonlocality, an operational description of superposition has not been extensively explored. In addition, whether such an understanding can provide insights into finding information-theoretic principles to reconstruct quantum theory has not been explored. In this work we explore both and answer the latter in the positive while making connections between superposition, entanglement and preparational uncertainty.
We present an operational notion of superposition that is based on constraints on observed measurement statistics. By remaining theory-independent, our definition extends the notion of superposition to frameworks of theories where states are not necessarily represented by rays in a Hilbert space. Specifically, we formalise superposition within the framework of Generalised Probabilistic Theories (GPTs). Far from being a purely descriptive exercise, an operational notion of superposition offers insights into the structure of quantum theory by analysing its relationship with other non-classical features. We use this notion to formulate superposition principles that capture structural features of the quantum state space related to superposition, and we explore the presence of superposition in specific GPTs, including Boxworld, Spekkens' toy model, and regular polygons. We show that the existence of superposition is not automatically implied from the non-classicality of a theory, and that preparational uncertainty and entanglement can be viewed as special forms of superposition. Finally, we show that the quantum tensor product is the largest composition for quantum systems that satisfies all our superposition principles.