Abstract Quantum Mechanics

We introduce a categorical axiomatization of (in the first instance: finitary, non-relativistic) Quantum Mechanics with the following main features.

• We expose the underlying structural mathematics - or logic in a suitably broad sense (very different to traditional quantum logic) - of both quantum and classical information flow, allowing effective reasoning about entanglement in compound systems, and information protocols and computation schemes based on this.
• We develop AQM as a typed theory: the types correspond to important conceptual distinctions in the physical theory. This allows e.g. a proper compositional treatment of measurements.
• The theory is `structural' and `qualitative', yet quantitative features (scalars, a Born rule) arise from it automatically.
• Notions such as projectors, which are fundamental building blocks in the usual formalism, are decomposed in AQM. This decomposition plays a key role in the analysis of quantum information flow.

We will also discuss some ideas towards introducing a discrete version of relativistic concepts into AQM.
(joint work with Bob Coecke)