Quantum references

Abstract: We present a theory of "quantum references", similar to lenses in classical functional programming, that allow to point to a subsystem of a larger quantum system, and to mutate/measure that part. Mutable classical variables, quantum registers, and wires in quantum circuits are examples of this, but also references to parts of larger quantum datastructures. Quantum references in our setting can also refer to subparts of other references, or combinations of parts from different references, or quantum references seen in a different basis, etc. Our modeling is intended to be well suited for formalization in theorem provers and as a foundation for modeling variables in quantum programs. We study quantum references in greater detail and cover the infinite-dimensional case as well, but also provide a more general treatment not specific to the quantum case. We implemented a large part of our results (including a small quantum Hoare logic and an analysis of quantum teleportation) in the Isabelle/HOL theorem prover.