Connected components of irreducible maps and 1D quantum phases

TL;DR

This paper explores the topological structure of completely positive maps in quantum theory, classifies their connected components, and applies these results to analyze 1D quantum phases via Matrix Product States.

Contribution

It provides a complete classification of connected components of irreducible CP maps with fixed Kraus rank and spectrum, extending quantum phase analysis without site blocking.

Findings

Primitive CP maps of fixed Kraus rank are path-connected

Connected components of irreducible CP maps are classified by a multiplicity index

Results enable analysis of ergodic components and translational symmetry breaking in 1D quantum phases

Abstract

We investigate elementary topological properties of sets of completely positive (CP) maps that arise in quantum Perron-Frobenius theory. We prove that the set of primitive CP maps of fixed Kraus rank is path-connected and we provide a complete classification of the connected components of irreducible CP maps at given Kraus rank and fixed peripheral spectrum in terms of a multiplicity index. These findings are then applied to analyse 1D quantum phases by studying equivalence classes of translational invariant Matrix Product States that correspond to the connected components of the respective CP maps. Our results extend the previously obtained picture in that they do not require blocking of physical sites, they lead to analytic paths and they allow to decompose into ergodic components and to study the breaking of translational symmetry.