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

• URL: http://arxiv.org/abs/2412.12342v1
• Date: Mon, 16 Dec 2024 20:33:42 GMT
• Title: Positivity of state, trace, and moment polynomials, and applications in quantum information
• Authors: Felix Huber, Victor Magron, Jurij Volčič,
• Abstract summary: State, trace, and moment expressions are expressions in several operator or random variables and positive functionals on their products. They arose from problems in quantum information theory, yet they naturally fit under the umbrella of the theory. This survey presents state, trace, and moment expressions in a concise and unified way.
• Score: 4.369550829556578
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• 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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