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Positivity of state, trace, and moment polynomials, and applications in quantum information
IACAPAP ArXiv (Online)
16 Dec 2024
Abstract
State, trace, and moment polynomials are polynomial expressions in several
operator or random variables and positive functionals on their products
(states, traces or expectations). While these concepts, and in particular their
positivity and optimization, arose from problems in quantum information theory,
yet they naturally fit under the umbrella of multivariate operator theory. This
survey presents state, trace, and moment polynomials in a concise and unified
way, and highlights their similarities and differences. The focal point is
their positivity and optimization. Sums of squares certificates for
unconstrained and constrained positivity (Positivstellens\"atze) are given, and
parallels with their commutative and freely noncommutative analogs are
discussed. They are used to design a convergent hierarchy of semidefinite
programs for optimization of state, trace, and moment polynomials. Finally,
circling back to the original motivation behind the derived theory, multiple
applications in quantum information theory are outlined.
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Details
- Title
- Positivity of state, trace, and moment polynomials, and applications in quantum information
- Creators
- Felix Huber - University of GdańskVictor Magron - Institut de Mathématiques de ToulouseJurij Volčič - Drexel University
- Publication Details
- IACAPAP ArXiv (Online)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022005687304721