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

Abstract

We show that there exist factorizable quantum channels in each dimension $\ge 11$ which do not admit a factorization through any finite dimensional von Neumann algebra, and do require ancillas of type II$_1$, thus witnessing new infinite-dimensional phenomena in quantum information theory. We show that the set of n by 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 $n \ge 5$, 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 $\ge 11$, from which we derive the first result above.