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

TL;DR

This paper demonstrates the existence of infinite-dimensional quantum channels and correlation matrices that cannot be approximated by finite-dimensional models, revealing new phenomena in quantum information theory.

Contribution

It proves the non-closure of certain sets of quantum correlations and constructs factorizable channels requiring infinite-dimensional ancillas, advancing understanding of quantum channel structures.

Findings

Existence of factorizable channels in dimensions ≥11 needing infinite-dimensional ancillas

Non-closure of correlation matrices from finite-dimensional von Neumann algebras for n ≥ 5

Non-closure of correlation matrices from unitaries in dimensions ≥11

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.