Causal and compositional structure in quantum theory

Thesis

Quantum theory challenges classical causal explanations for physical phenomena, calling for a quantum generalisation of causal structure. This thesis advances the study of causal decompositions, a research programme that aims to bridge two prominent, yet conceptually distinct types of structural assumption found across the quantum causality literature. The first is in terms of the global dependency relations between input and output systems of quantum transformations, which in the unitary case we shall refer to as causal structure; the second is in terms of the compositional structure of quantum circuits or networks. Concretely, a causal decomposition is a circuit representation of a unitary transformation in which no-influence relations that hold between its inputs and outputs are made manifest in the absence of paths through the circuit connecting them. Applications include translating between different quantum causal modelling frameworks; equipping a framework based on unitary causal structure with a local, bottom-up description via generalised autonomous causal mechanisms; and relating notions of compatibility between quantum transformations and relativistic causality.

Whether causal decompositions always exist remains an open problem. The contributions of this thesis are threefold.

First, we develop an order-theoretic account of circuit syntax and demonstrate that every circuit with a given input-output connectivity can be rewritten into a one of a canonical form given by the concept lattice construction from lattice theory. This systematises the search for causal decompositions by providing the shape of the quantum circuit one should look for.

Second, we establish a combinatorial condition that fully characterises when causal decompositions into unitary quantum circuits exist without requiring the generalisation to routed quantum circuits prompted by earlier research. This solves a well-defined subproblem of the broader quest, and our proof, an interplay of order theory and operator algebra, lays a foundation for future extensions.

Third, we detail the application to relativistic causality. Within a multipartite framework generalising existing literature, we emphasise that the mere lack of superluminal signalling through a quantum channel is not sufficient to conclude its realisability by dynamics that are locally consistent with relativity theory. We conjecture that, on the other hand, a lack of superluminal causal influences through its unitary dilations is sufficient—at least up to arbitrarily small errors. We demonstrate this in various special cases and show that the general statement reduces to the existence of a weak version of causal decompositions.

Authors: van der Lugt, T

• DOI: 10.5287/ora-vq7opd7ap
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: quantum causation; causal decompositions; lattice theory; operator algebra
• Subjects: Quantum theory