Optimization Over Trace Polynomials

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Optimization Over Trace Polynomials. / Klep, I.; Magron, V.; Volčič, J.

I: Annales Henri Poincare, Bind 23, 2022, s. 67–100.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt

Harvard

Klep, I, Magron, V & Volčič, J 2022, 'Optimization Over Trace Polynomials', Annales Henri Poincare, bind 23, s. 67–100. https://doi.org/10.1007/s00023-021-01095-4

APA

Klep, I., Magron, V., & Volčič, J. (2022). Optimization Over Trace Polynomials. Annales Henri Poincare, 23, 67–100. https://doi.org/10.1007/s00023-021-01095-4

Vancouver

Klep I, Magron V, Volčič J. Optimization Over Trace Polynomials. Annales Henri Poincare. 2022;23:67–100. https://doi.org/10.1007/s00023-021-01095-4

Author

Klep, I. ; Magron, V. ; Volčič, J. / Optimization Over Trace Polynomials. I: Annales Henri Poincare. 2022 ; Bind 23. s. 67–100.

Bibtex

@article{6fe7bb5ef8e941dbbc6f2a42698d1b68,
title = "Optimization Over Trace Polynomials",
abstract = "Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascu{\'e}s and Ac{\'i}n scheme (Pironio et al. in New J. Phys. 10(7):073013, 2008) for optimization of noncommutative polynomials. The Gelfand–Naimark–Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.",
author = "I. Klep and V. Magron and J. Vol{\v c}i{\v c}",
year = "2022",
doi = "10.1007/s00023-021-01095-4",
language = "Udefineret/Ukendt",
volume = "23",
pages = "67–100",
journal = "Annales Henri Poincare",
issn = "1424-0637",
publisher = "Springer Basel AG",

}

RIS

TY - JOUR

T1 - Optimization Over Trace Polynomials

AU - Klep, I.

AU - Magron, V.

AU - Volčič, J.

PY - 2022

Y1 - 2022

N2 - Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme (Pironio et al. in New J. Phys. 10(7):073013, 2008) for optimization of noncommutative polynomials. The Gelfand–Naimark–Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.

AB - Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme (Pironio et al. in New J. Phys. 10(7):073013, 2008) for optimization of noncommutative polynomials. The Gelfand–Naimark–Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-85111505876&partnerID=MN8TOARS

U2 - 10.1007/s00023-021-01095-4

DO - 10.1007/s00023-021-01095-4

M3 - Tidsskriftartikel

VL - 23

SP - 67

EP - 100

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

ER -

ID: 284012369