Braided QM and Majorana qubits at third root of unity: a color Heisenberg-Lie (super)algebra framework

TL;DR

This work develops a framework of mixed-bracket color Lie (super)algebras graded by abelian groups such as ${\mathbb Z}_3^2$ and ${\mathbb Z}_2^p\times{\mathbb Z}_3^2$, constructing associated para-oscillators and two parastatistics families: parabosons and parafermions. Parafermionic partners yield roots-of-unity truncations in multi-particle spectra and realize braided Majorana qubits, while parabosons exhibit no spectral truncation but imprint distinguishable two-particle probability densities, establishing observable signatures. The paper also discusses braiding and two complementing routes to multi-particle states—via graded Hopf-algebra coproducts and via symmetrized state constructions—connecting the color-framework to braided Majorana qubits and Gentile-type parastatistics. Explicit minimal models, including a ${\mathbb Z}_3\times{\mathbb Z}_3$ parabosonic model and $s=3,6$ colored parafermionic models, illustrate the physical consequences and hint at links to representations at roots of unity in quantum groups and potential topological quantum computation applications.

Abstract

We introduce color Heisenberg-Lie (super)algebras graded by the abelian groups $Z_3^2$, $Z_2^p\times Z_3^2$ for $p=1,2,3$, and investigate the properties of their associated multi-particle quantum paraoscillators. In the Rittenberg-Wyler's color Lie (super)algebras framework the above abelian groups are the simplest ones which induce mixed brackets interpolating commutators and anticommutators. These mixed brackets allow to accommodate two types of parastatistics: one based on the permutation group (beyond bosons and fermions in any space dimension) and an anyonic parastatistics based on the braid group. In both such cases the two broad classes of paraparticles are given by parabosons and parafermions. Mixed-bracket parafermions are created by nilpotent operators; they satisfy a generalized Pauli exclusion principle leading to roots-of-unity truncations in their multi-particle energy spectrum (braided Majorana qubits and their Gentile-type parastatistics are recovered in this color Lie superalgebra setting). Mixed-bracket parabosons do not admit truncations of the spectrum; the minimal detectable signature of their parastatistics is encoded in the measurable probability density of two indistinguishable parabosonic oscillators in a given energy eigenstate.