Bilkent University, Turkey, cihan.okay@bilkent.edu.tr
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. Homotopy theory has proved to be useful in detecting contextual LCSs [1] and extending earlier results such as Arkhipov’s graph-theoretic characterization of contextuality. In the present work we extend the homotopical approach to a generalized cohomology theory that can be used to classify quantum solutions of a LCS. For this we introduce a topological version of quantum solutions which relies on classifying spaces [2] 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 known as commutative K-theory. This procedure is in close analogy with the classification of symmetry-protected topological phases via generalized cohomology theories as in the work of Kitaev et al. We apply our machinery to prove various results about LCSs. This talk is based on the arxiv preprint [3].
[1] Homotopical approach to quantum contextuality. Cihan Okay and Robert Raussendorf, Quantum 4: 217 (2020).
[2] Classifying space for quantum contextuality. Cihan Okay and Daniel Sheinbaum, Annales Henri Poincaré. Vol. 22. No. 2. (2021).
[3] Commutative d-torsion K-theory and its applications. Cihan Okay, arXiv:2006.07542 (2020).