Hierarchy of Theories with Indefinite Causal Structures

The Causaloid framework introduced by Hardy suggests a research program aimed at finding a theory of Quantum Gravity. On one side General Relativity while deterministic (once the metric is provided) features dynamic causal structures, on the other side Quantum Theory while having fixed causal structures is probabilistic in nature. It is natural to then expect Quantum Gravity to house both of the radical aspects of GR and QT, and therefore incorporate indefinite causal structure. The Causaloid framework is operational, it is based on the assertion that any physical theory, whatever it does, must correlate recorded data. Imagine a person inside a closed space, having access to a stack of cards with recorded data (procedures, outcomes, locations); and the person is tasked with inferring(aspects of) the underlying physical theory that governs the data. The correlation of recorded data due to the physical theory means the stack of cards is riddled with redundancy. The person in the box distills away the redundancy by compressing the data. We call this physical compression. In this framework there are three levels of compression: 1) Tomographic Compression, 2) CompositionalCompression and 3) Meta Compression. In this work, we present a diagrammatic form for physical compression to facilitate exposition of the Causaloid framework. Further, building upon the work from we study Meta compression and find a hierarchy of theories characterised by Meta Compression for which we provide a general form. We will proceed to populate this 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 broad implications of this work.