Spectral theory of spin substitutions

2021/08/19 by Natalie Priebe Frank, Frank, Natalie Priebe, Neil Mañibo +1 · 1 citation
Computer Science · #37A30 #37B10 #42A16 #52C23 #Dynamical Systems (math.DS) #FOS: Mathematics #Quantum Computing Algorithms and Architecture #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2108.08642

openalex publication_date 2021/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce qubit substitutions in ℤ^m, which have non-rectangular domains based on an endomorphism Q of ℤ^m and a set D of coset representatives of ℤ^m/Qℤ^m. We then focus on a specific family of qubit substitutions which we call spin substitutions, whose combinatorial definition requires a finite abelian group G as its spin group. We investigate the spectral theory of the underlying subshift (Σ,ℤ^m). Under certain assumptions, we show that it is measure-theoretically isomorphic to a group extension of an m-dimensional odometer, which induces a complete decomposition of the function space L²(Σ,μ) . This enables one to use group characters in \widehatG to derive substitutive factors and carry out a spectral analysis on specific subspaces. We provide general sufficient criteria for the existence of pure point, absolutely continuous and singular continuous spectral measures, together with some bounds on their spectral multiplicity.

Cited by

• Radius-zero Extended Symmetries and Irregular Fibres of ℤ^d-Substitution Subshifts

Related

• Deformations of the symmetric subspace of qubit chains
• Representations associated to small nilpotent orbits for real Spin groups
• Spin group and almost commutative geometry
• The Spin Group in Superspace
• Representations associated to small nilpotent orbits for complex Spin groups