The Road to a New Quantum Formalism - Categories as a Canvas for Quantum Foundations
1st September 2008 to 30th November 2009
When watching television, we don't observe the `low-level' matrices of tiny pixels the screen is made up from, but rather `high-level' Gestalts of each of the figuring entities making up the story that the images convey. These entities and their story is the essence of the images; the matrix of pixels is just a technologically convenient representation. Similarly, in modern computer programming, one does not `speak' in terms of 0s and 1s, but rather relies on high-level concepts about information flow.
The way we reason nowadays about quantum theory is still very `low-level', in terms of matrices of so-called complex numbers. Recent work identified a high-level formal framework for quantum theory, resulting in purely diagrammatic languages for reason and compution, and also corresponding logics, that is, languages which a computer `understands'. By not making explicit the underlying `low-level pixels' there is still plenty of freedom to articulate the ideas behind other important foundational work on quantum theory. Therefore our approach can act as a canvas on which we can paint a variety of theories of physical reality. This would provide us with a true image of nature, rather than its pixels.
Monoidal categories have recently proven to be an excellent high-level framework for reasoning about quantum information and computation. Features are an intuitive purely diagrammatic calculus, which enables pictorial derivation of several protocols as well as computing the quantum Fourier transform, and comprehension of quantum, classical and mixed data types. The corresponding categorical logic enables automation.
Important operational features, not present in the usual quantum formalism, are types and compositionality. The approach moreover reveals those high-level concepts which are key to quantum phenomena, and still leaves a large degree of axiomatic freedom. It therefore has the potential to become a unifying framework for a variety of existing approaches in quantum foundations. Compelling evidence for this is provided by our recent presentations of Spekkens' Toy Theory and C*-algebras within this approach, as well as promising developments towards describing BBLW convex theories, algebraic QFTs and Doering-Isham-Butterfield topoi, to name a few. We expect to obtain important insights on all of these approaches as well as to blend the key features of these within new models and axiom systems for quantum reasoning, and possibly make important steps towards quantum gravity.