We complete the characterization of synchronous quantum conditional probability densities in terms of traces on C*-algebras. Building on work of Slofstra and recent work on the graph isomorphism game, this then allows us to construct two graphs which are qa-isomorphic but not q-isomorphic and a graph for which the q-independence number and qa-independence numbers are different.

This talk is based on joint work with Se-Jin Kim and Chris Schafhauser.