Causal types for higher-order quantum theory

Thesis

The emerging study of higher-order causal structures in the setting of quantum theory has made crucial developments in our understanding of potential impacts of quantum gravity on how processes can compose, but has historically been treated in a somewhat ad-hoc way that is specific to quantum theory. This thesis is part of an ongoing movement to put the composition of higher-order causal structures front and centre via categorical frameworks, which helps in the generation of theory-independent definitions for causal structures. This thesis builds on Kissinger and Uijlen's Caus[-] construction to add more expressive causal structures and obtain a tight correspondence with a formal logic precisely describing consistency of composite causal structures via proof-nets.

More specifically, the resulting theory features a number of monoidal structures to describe different causal relations between local systems (which may have some internal causal structure themselves), including non-signalling, one-way signalling, bidirectional signalling, compatibility with a directed graph, classical (probabilistic) choice, and combinations of these. We define causal consistency of a closed system of black boxes with fixed connections to mean that the composite can be realised with unit probability regardless of the implementations within the boxes, and we investigate a number of logics whose proofs can be translated into causally consistent setups. Completeness is given by proof-nets of causal logic, a new logic extending pomset with directed axiom links to describe first-order systems (the degenerate systems which can only transmit information in one direction) in which the proof-net criterion guarantees causal consistency via the absence of any cycle of information flow into which a paradox could be encoded.

After a detour into Measurement-Based Quantum Computing to present a new algorithm for extraction of an ancilla-free circuit from a Pauli Flow, the thesis ends by showing how measurement patterns and parameterised quantum circuits can be formally attributed with causal structures in the Caus[-] framework.

Authors: Simmons, W

• DOI: 10.5287/ora-qr9z52v97
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: linear logic; pomset logic; higher-order processes; category theory; quantum theory; probability theory; causality
• Subjects: Closed categories (Mathematics); Quantum computing; Causality (Physics); Categories (Mathematics); Quantum theory; Logic