Connected components of irreducible maps and 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.

19 pages, 0 figures

References in corpus (4)

• Matrix product states represent ground states faithfully
• Undecidability of the Spectral Gap (short version)
• Entropy and Entanglement in Quantum Ground States
• $C^1$-classification of gapped parent Hamiltonians of quantum spin chains

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• Matrix Product State Description and Gaplessness of the Haldane-Rezayi State
• A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III
• C^1-Classification of gapped parent Hamiltonians of quantum spin chains with local symmetry
• Non-local order parameters for fermion chains via the partial transpose
• Connecting Commuting Normal Matrices