Hierarchy of Theories with Indefinite Causal Structures: A Second Look at the Causaloid Framework*

The Causaloid framework¹ is useful to study Theories with Indefinite Causality; since Quantum Gravity is expected to marry the radical aspects of General Relativity (dynamic causality) and Quantum Theory (probabilistic-ness). To operationally study physical theories one finds the minimum set of quantities required to perform any calculation through physical compression. In this framework, there are three levels of compression: 1) Tomographic Compression, 2) Compositional Compression and 3) Meta Compression.

We present a diagrammatic representation of the Causaloid framework to facilitate exposition and study Meta compression. We show that there is a hierarchy of theories with respect to Meta compression and characterise its general form. Next, we populate the hierarchy. The theory of circuits forms the simplest case, which we express diagrammatically through Duotensors, following which we construct Triotensors using hyper₃wires (hyperedges connecting three operations) for the next rung in the hierarchy. Finally, we discuss the implications for the field of Indefinite Causality.

[1] Journal of Physics A: Mathematical and Theoretical, 40(12), 3081