Page 90 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 90
COMPUTATIONAL ASPECTS OF COMMUTATIVE AND NONCOMMUTATIVE
POSITIVE POLYNOMIALS (MS-77)
group on tensor product spaces into that of matrix multiplication. As a result, we extend the
polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize
the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct
permutation polynomials and tensor polynomial identities on tensor product spaces. We give
connections to concepts in quantum information theory and invariant theory.
Optimization over trace polynomials
Victor Magron, vmagron@laas.fr
LAAS CNRS & Institute of Mathematics from Toulouse, France
Coauthors: Igor Klep, Jurij Volcˇicˇ
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 [New J. Phys., 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.
Quantum Isomorphism of Graphs: an Overview
Laura Mancˇinska, mancinska@math.ku.dk
University of Copenhagen, Denmark
In this talk I will introduce a quantum version of graph isomorphism. Our point of departure
will be an interactive protocol (nonlocal game) where two provers try to convince a verifier that
two graphs are isomorphic. Allowing provers to take advantage of shared quantum resources
will then allow us to define quantum isomorphism as the ability of quantum players to win the
corresponding game with certainty. We will see that quantum isomorphism can be naturally re-
formulated in the languages of quantum groups and counting complexity. In particular, we will
see that two graphs are quantum isomorphic if and only if they have the same homomorphism
counts from all planar graphs.
88
POSITIVE POLYNOMIALS (MS-77)
group on tensor product spaces into that of matrix multiplication. As a result, we extend the
polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize
the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct
permutation polynomials and tensor polynomial identities on tensor product spaces. We give
connections to concepts in quantum information theory and invariant theory.
Optimization over trace polynomials
Victor Magron, vmagron@laas.fr
LAAS CNRS & Institute of Mathematics from Toulouse, France
Coauthors: Igor Klep, Jurij Volcˇicˇ
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 [New J. Phys., 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.
Quantum Isomorphism of Graphs: an Overview
Laura Mancˇinska, mancinska@math.ku.dk
University of Copenhagen, Denmark
In this talk I will introduce a quantum version of graph isomorphism. Our point of departure
will be an interactive protocol (nonlocal game) where two provers try to convince a verifier that
two graphs are isomorphic. Allowing provers to take advantage of shared quantum resources
will then allow us to define quantum isomorphism as the ability of quantum players to win the
corresponding game with certainty. We will see that quantum isomorphism can be naturally re-
formulated in the languages of quantum groups and counting complexity. In particular, we will
see that two graphs are quantum isomorphic if and only if they have the same homomorphism
counts from all planar graphs.
88
