Page 88 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 88
COMPUTATIONAL ASPECTS OF COMMUTATIVE AND NONCOMMUTATIVE
POSITIVE POLYNOMIALS (MS-77)
Non-commutative polynomial in quantum physics
Antonio Acin, antonio.acin@icfo.eu
ICFO, Spain
Quantum physics is a natural source of problems involving the optimization of polynomials
over non-commuting variables. The talk first introduces some of the most relevant problems,
explains how they can be tackled using semi-definite programming hierarchies and concludes
with some of the open problems.
References
[1] S. Pironio, M. Navascues, A. Acin, Convergent relaxations of polynomial optimization
problems with non-commuting variables, SIAM J. Optim. 20 (5), 2157 (2010).
[2] E. Wolfe, A. Pozas-Kerstjens, M. Grinberg, D. Rosset, A. Acín, M. Navascues, Quantum
Inflation: A General Approach to Quantum Causal Compatibility, to appear in Phys. Rev.
X.
[3] A. Pozas-Kerstjens, R. Rabelo, L. Rudnicki, R. Chaves, D. Cavalcanti, M. Navascues, A.
Acín, Bounding the sets of classical and quantum correlations in networks, Phys. Rev.
Lett. 123, 140503 (2019).
[4] F. Baccari, C. Gogolin, P. Wittek, A. Acín, Verifying the output of quantum optimizers
with ground-state energy lower bounds, Phys. Rev. Research 2, 043163 (2020).
SOS relaxations for detecting quamtum entanglement
Abhishek Bhardwaj, Abhishek.Bhardwaj@laas.fr
CNRS, France
Positive maps which are not completely positive (PNCP maps) are of importance in quantum
information theory, in particular they can be used to identify entanglement in quantum states.
PNCP maps can naturally be associated to non-negative polynomials which are not sums of
squares (SOS). An algorithm for constructing such maps is given in “There are many more
positive maps than completely positive maps" by Klep et al.
In this talk we will present a summary of their construction and discuss theoterical and
numerical issues that arise in practice.
Sum-of-Squares proofs of logarithmic Sobolev inequalities on finite
Markov chains
Hamza Fawzi, hf323@cam.ac.uk
University of Cambridge, United Kingdom
Logarithmic Sobolev inequalities play an important role in understanding the mixing times
of Markov chains on finite state spaces. It is typically not easy to determine, or indeed ap-
proximate, the optimal constant for which such inequalities hold. In this paper, we describe a
86
POSITIVE POLYNOMIALS (MS-77)
Non-commutative polynomial in quantum physics
Antonio Acin, antonio.acin@icfo.eu
ICFO, Spain
Quantum physics is a natural source of problems involving the optimization of polynomials
over non-commuting variables. The talk first introduces some of the most relevant problems,
explains how they can be tackled using semi-definite programming hierarchies and concludes
with some of the open problems.
References
[1] S. Pironio, M. Navascues, A. Acin, Convergent relaxations of polynomial optimization
problems with non-commuting variables, SIAM J. Optim. 20 (5), 2157 (2010).
[2] E. Wolfe, A. Pozas-Kerstjens, M. Grinberg, D. Rosset, A. Acín, M. Navascues, Quantum
Inflation: A General Approach to Quantum Causal Compatibility, to appear in Phys. Rev.
X.
[3] A. Pozas-Kerstjens, R. Rabelo, L. Rudnicki, R. Chaves, D. Cavalcanti, M. Navascues, A.
Acín, Bounding the sets of classical and quantum correlations in networks, Phys. Rev.
Lett. 123, 140503 (2019).
[4] F. Baccari, C. Gogolin, P. Wittek, A. Acín, Verifying the output of quantum optimizers
with ground-state energy lower bounds, Phys. Rev. Research 2, 043163 (2020).
SOS relaxations for detecting quamtum entanglement
Abhishek Bhardwaj, Abhishek.Bhardwaj@laas.fr
CNRS, France
Positive maps which are not completely positive (PNCP maps) are of importance in quantum
information theory, in particular they can be used to identify entanglement in quantum states.
PNCP maps can naturally be associated to non-negative polynomials which are not sums of
squares (SOS). An algorithm for constructing such maps is given in “There are many more
positive maps than completely positive maps" by Klep et al.
In this talk we will present a summary of their construction and discuss theoterical and
numerical issues that arise in practice.
Sum-of-Squares proofs of logarithmic Sobolev inequalities on finite
Markov chains
Hamza Fawzi, hf323@cam.ac.uk
University of Cambridge, United Kingdom
Logarithmic Sobolev inequalities play an important role in understanding the mixing times
of Markov chains on finite state spaces. It is typically not easy to determine, or indeed ap-
proximate, the optimal constant for which such inequalities hold. In this paper, we describe a
86
