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

Magdalena Musat; Mikael Rørdam

doi:10.1007/s00220-019-03449-w

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

Magdalena Musat^*, Mikael Rørdam

^*Corresponding author for this work

Department of Mathematical Sciences

6 Citations (Scopus)

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
Journal	Communications in Mathematical Physics
Number of pages	16
ISSN	0010-3616
DOIs	https://doi.org/10.1007/s00220-019-03449-w
Publication status	Published - 1 May 2020

Access to Document

10.1007/s00220-019-03449-w

Cite this

@article{71d10ac5f69c4ef1b515390b744122e2,

title = "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 ≥ 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. ",

author = "Magdalena Musat and Mikael R{\o}rdam",

year = "2020",

month = may,

day = "1",

doi = "10.1007/s00220-019-03449-w",

language = "English",

journal = "Communications in Mathematical Physics",

issn = "0010-3616",

publisher = "Springer",

}

TY - JOUR

T1 - Non-closure of Quantum Correlation Matrices and Factorizable Channels that Require Infinite Dimensional Ancilla (With an Appendix by Narutaka Ozawa)

AU - Musat, Magdalena

AU - Rørdam, Mikael

PY - 2020/5/1

Y1 - 2020/5/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85065142457&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03449-w

DO - 10.1007/s00220-019-03449-w

M3 - Journal article

AN - SCOPUS:85065142457

SN - 0010-3616

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

ER -

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

Abstract

Access to Document

Other files and links

Fingerprint

Cite this