Jordan-Wigner transformation and qubits with nontrivial exchange rule

TL;DR

This paper develops a consistent algebraic framework for fermionic qubits with nontrivial exchange rules, addressing subtle issues related to antisymmetry and super-selection principles in local fermionic mode models.

Contribution

It introduces a novel algebraic construction for super-indexed fermionic qubits with nontrivial exchange rules, expanding the theoretical understanding of fermionic qubit models.

Findings

Constructed a consistent algebraic model for super-indexed fermionic qubits.

Clarified the relation between the model and super-space constructions.

Addressed super-selection principles in the context of fermionic qubits.

Abstract

Well-known (spinless) fermionic qubits may need more subtle consideration in comparison with usual (spinful) fermions. Taking into account a model with local fermionic modes, formally only the 'occupied' states |1> could be relevant for antisymmetry with respect to particles interchange, but 'vacuum' state |0> is not. Introduction of exchange rule for such fermionic qubits indexed by some 'positions' may look questionable due to general super-selection principle. However, a consistent algebraic construction of such 'super-indexed' qubits is presented in this work. Considered method has some relation with construction of super-spaces, but it has some differences with standard definition of supersymmety sometimes used for generalizations of qubit model.