Massless Majorana bispinors and two-qubit entangled state

TL;DR

This paper demonstrates the equivalence between Weyl and Majorana bispinors, showing that Majorana bispinors correspond to maximally entangled two-qubit states and proposing a topological quantum computation model using Majorana operators.

Contribution

It establishes the unitary equivalence of Weyl and Majorana bispinors, links bispinor types to entangled states, and introduces a topological quantum computation framework with Majorana zero modes.

Findings

Weyl and Majorana bispinors are unitarily equivalent.

Majorana bispinors correspond to Bell states (maximally entangled).

A topological quantum computation model with Majorana operators is proposed.

Abstract

This is a pedagogical paper, where bispinors solutions to the four-dimensional massless Dirac equation are considered in relativistic quantum mechanics and in quantum computation, taking advantage of the common mathematical description of four dimensional spaces. First, Weyl and massless Majorana bispinors are shown to be unitary equivalent, closing a gap in the literature regarding their equivalence. A discrepancy in the number of linearly independent solutions reported in the literature is also addressed. Then, it is shown that Weyl bispinors are algebraically equivalent to two-qubit direct product states, and that the massless Majorana bispinors are algebraically equivalent to maximally entangled sates (Bell states), with the transformations relating the two bispinors types acting as entangling gates in quantum computation. Different types of entangling gates are presented, highlighting a set that fulfills the required properties for Majorana zero mode operators in topological quantum computation. Based on this set, a general topological quantum computation model with four Majorana operators is presented, which exhibits all the required technical and physical properties to obtain entanglement of two logical qubits from topological operations.