Positivity of state, trace, and moment polynomials, and applications in quantum information

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 (Positivstellensätze) 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.

28 pages, 1 table

References in corpus (14)

• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions
• Beyond Bell's Theorem: Correlation Scenarios
• Uncertainty relations from graph theory
• Dimension-free entanglement detection in multipartite Werner states
• Positive maps and trace polynomials from the symmetric group
• A convergent inflation hierarchy for quantum causal structures
• Bounding the joint numerical range of Pauli strings by graph parameters
• The inflation hierarchy and the polarization hierarchy are complete for the quantum bilocal scenario
• State polynomials: positivity, optimization and nonlinear Bell inequalities
• Positive univariate trace polynomials
• Uncertainty relations from state polynomial optimization
• Noncommutative Nullstellensätze and Perfect Games
• Globally trace-positive noncommutative polynomials and the unbounded tracial moment problem