## Abstract

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 II
_{1}
, thus witnessing new infinite-dimensional phenomena in quantum information theory. We show that the set of n× 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 Cqs(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.

Original language | English |
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Journal | Communications in Mathematical Physics |

Volume | 375 |

Issue number | 3 |

Pages (from-to) | 1761-1776 |

Number of pages | 16 |

ISSN | 0010-3616 |

DOIs | |

Publication status | Published - 2020 |