ASQC IV | Stable homotopy and quantum contextuality

University of British Columbia, Algebraic Structures in Quantum Computation IV, Jun 25 2020

Title: Stable homotopy and quantum contextuality

Abstract: Linear constraint systems (LCS) provide instances of Kochen-Specker type contextuality proofs generalizing the well-known example of Peres-Mermin square. A LCS is specified by a linear system of equations. Solutions in the unitary group are called quantum solutions. In the contextual case no solution exists in the group of scalar matrices. Homotopy theory has proved to be useful in detecting contextual LCS and extending earlier results such as Arkhipov's graph theoretic characterization of contextuality. In the present work we extend the homotopical approach to classify quantum solutions in terms of homotopy classes of maps. For this we introduce a topological version of quantum solutions which uses classifying spaces tailored for contextuality. These classifying spaces can be ``stabilized'' in a way similar to the stabilization of vector bundles to obtain topological K-theory. This brings in a stable notion of contextuality detected by a generalized cohomology theory that is a commutative variant of topological K-theory. This procedure is in close analogy with the classification of symmetry-protected topological phases via generalized cohomology theories. We apply our machinery to prove various results about LCS.

(This talk is based on arXiv:2006.07542)