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Positivity for matrix systems: a case study from quantum mechanics

Summary: Positivity for matrix systems: a case study
from quantum mechanics
Claudio Altafini
SISSA-ISAS, International School for Advanced Studies
via Beirut 2-4, 34014 Trieste, Italy
Abstract. We discuss an example from quantum physics of "positive system" in
which the state (a density operator) is a square matrix constrained to be positive
semidefinite (plus Hermitian and of unit trace). The positivity constraint is captured
by the notion of complete positivity of the corresponding flow. The infinitesimal
generators of all possible admissible ODEs can be characterized explicitly in terms
of cones of matrices. Correspondingly, it is possible to determine all linear time-
varying systems and bilinear control systems that preserve positivity of the state
1 Introduction
For a matrix system, i.e., a system of ODEs having as state space a set of
square matrices, the idea of state that can assume only positive values nat-
urally generalizes to operator positivity i.e., to positive (semi)definiteness of
the square matrix representing the state variables. Such a concept is not new
in applied mathematics and for example it has long been used in model-


Source: Altafini, Claudio - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics