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

TL;DR

This paper introduces a new algebraic framework using color Heisenberg-Lie superalgebras graded by specific abelian groups to model braided Majorana qubits and parafermions, revealing novel multi-particle quantum statistics and spectrum truncations.

Contribution

It develops a color Lie superalgebra approach to describe braided Majorana qubits and parafermions with mixed brackets, unifying parastatistics and spectrum truncation phenomena.

Findings

Nilpotent operators create mixed-bracket parafermions with spectrum truncation.

Spectrum of parabosons remains untruncated, with measurable probability densities.

Roots-of-unity truncations lead to braided Majorana qubits.

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.