Quantum extensions of ordinary maps
A quantum homomorphism between ordinary graphs can be viewed as a strategy for the graph homomorphism game, or as a quantum family of graph homomorphisms, within the framework of noncommutative mathematics. This talk is about quantum continuous functions between ordinary compact Hausdorff spaces, which are quantum in the same sense. Specifically, we investigate the existence of a quantum extension of an ordinary continuous function. A functor echoing the quantum monad of Abramsky, Barbosa, de Silva, and Zapata yields an existence criterion.
A loop is defined to be quantum nullhomotopic if and only if it can be extended to a quantum continuous function on the disk. There are loops in the real projective plane and the figure of eight that are quantum nullhomotopic, but not nullhomotopic in the ordinary sense. However, the canonical loop is not quantum nullhomotopic, demonstrating that the notion is not trivial. The index Hilbert space is required to be finite-dimensional: if an infinite-dimensional Hilbert space is permitted, then every loop becomes quantum nullhomotopic by an application of Kuiper's theorem.