Non-closure of quantum correlation matrices and factorizable channels that require infinite dimensional ancilla

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.

16 pages. An appendix by Narutaka Ozawa has been added. To appear in Comm. Math. Phys

References in corpus (2)

• Entanglement of assistance and multipartite state distillation
• An asymptotic property of factorizable completely positive maps and the Connes embedding problem

Cited by in corpus (8)

• Self-testing of quantum systems: a review
• Duality for optimal couplings in free probability
• Schur multipliers and mixed unitary maps
• Extreme Points and Factorizability for New Classes of Unital Quantum Channels
• Separation of quantum, spatial quantum, and approximate quantum correlations
• Faithfulness and sensitivity for ancilla-assisted process tomography
• A bound on the quantum value of all compiled nonlocal games
• Absolutely dilatable bimodule maps