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

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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.

OriginalsprogEngelsk
TidsskriftCommunications in Mathematical Physics
Vol/bind375
Udgave nummer3
Sider (fra-til) 1761-1776
Antal sider16
ISSN0010-3616
DOI
StatusUdgivet - 2020

ID: 223821779