The Density Operators of Qubit Systems in the Multiparticle Spacetime Algebra

TL;DR

This paper introduces a geometric method using multiparticle spacetime algebra to represent density operators of multi-qubit pure states, facilitating analysis of quantum correlations and invariants.

Contribution

It presents a novel approach to express multi-qubit density operators in MSTA and analyzes quantum properties geometrically, including Bell inequalities and state invariants.

Findings

Geometric representation of multi-qubit density operators

Analysis of Bell inequality within MSTA framework

Construction of local unitary invariants for multi-qubit states

Abstract

We provide a method to write down the density operator for any pure state of multi-qubit systems in the multiparticle spacetime algebra (MSTA) introduced by Doran, Gull, and Lasenby. Using the MSTA formulation, we analyze several aspects of quantum mechanics in a geometrical way including the Bell inequality and the dynamics of two coupled qubits. Lastly, we provide a natural way to construct the local unitary invariants in the MSTA. Using these invariants, we analyze the space of the two-qubit and three-qubit pure states with local and non-local degrees of freedom separated.