Non-closure of Quantum Correlation Matrices and Factorizable Channels that Require Infinite Dimensional Ancilla (With an Appendix by Narutaka Ozawa)

We show that there exist factorizable quantum channels in each dimension >= 11 which do not admit a factorization through any finite dimensional von Neumann algebra, and do require ancillas of type II1, thus witnessing new infinite-dimensional phenomena in quantum information theory. We show that the set of n x n matrices of correlations arising as second-order moments of projections in finite dimensional von Neumann algebras with a distinguished trace is non-closed, for all n >= 5, and we use this to give a simplified proof of the recent result of Dykema, Paulsen and Prakash that the set of synchronous quantum correlations C-q(s)(5, 2) is non-closed. Using a trick originating in work of Regev, Slofstra and Vidick, we further show that the set of correlation matrices arising from second-order moments of unitaries in finite dimensional von Neumann algebras with a distinguished trace is non-closed in each dimension >= 11, from which we derive the first result above.