Speaker
Description
The framework of higher-order quantum operations (process matrices/supermaps) on finite-dimensional systems is primarily motivated by providing an information-theoretic abstraction of a spacetime environment. In doing so, it provides a way to extract operational causal structure from quantum circuit configurations, and a way to formalise quantum (indefinite) causal structures, which have been proposed as a salient information-theoretic feature of quantum gravity. However, there are strong reasons to push the theory of higher-order operations to infinite-dimensional and supra-quantum settings. Quantum gravity, when built on quantum field theory rather than quantum computation, is likely to involve infinite-dimensional quantum systems. And just as studying generalised theories places quantum theory along a spectrum of non-local correlations, studying higher-order processes on generalised theories places it along a spectrum of non-causal correlations. In this talk I will explain how abstract categorical methods give a way to develop these desirable generalisations via a theory-independent axiom for higher-order processes in terms of “locally-applicable transformations” [1], together with its application to infinite-dimensional systems in the unitary setting [2]. I will then describe recent results [3] giving a concrete representation of locally-applicable transformations for a broad class of theories—including classical theory, quantum theory, real quantum theory, and boxworld—revealing a surprising stability within the theory of higher-order processes beyond finite-dimensional quantum theory. The talk will emphasise the conceptual takeaways, and present the infinite-dimensional and supra-quantum cases in familiar linear-algebraic, functional-analytic, and circuit-diagrammatic terms, remaining accessible to those without prior exposure to category theory. [1] https://arxiv.org/abs/2205.09844 [2] https://arxiv.org/abs/2207.09180 [3] https://arxiv.org/abs/2602.23865