On maps which preserve semipositivity and quantifier elimination theory for real numbers

Author:

Pastuszak Grzegorz1,Skowyrski Adam1,Jamiołkowski Andrzej2

Affiliation:

1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

2. Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Toruń, Poland

Abstract

Assume that [Formula: see text] is a superoperator which preserves hermiticity. We give an algorithm determining whether [Formula: see text] preserves semipositivity (we call [Formula: see text] positive in this case). Our approach to the problem has a model-theoretic nature, namely, we apply techniques of quantifier elimination theory for real numbers. An approach based on these techniques seems to be the only one that allows to decide whether an arbitrary hermiticity-preserving [Formula: see text] is positive. Before we go to detailed analysis of the problem, we argue that quantifier elimination for real numbers (and also for complex numbers) can play a significant role in quantum information theory and other areas as well.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,General Mathematics

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