Bloch vectors for qudits and geometry of entanglement

TL;DR

This paper introduces three matrix bases for decomposing qudit density matrices, explores their geometric properties related to entanglement, and finds the Weyl basis most effective for analyzing entanglement in two-qudit states.

Contribution

It compares three bases for qudit state decomposition and demonstrates the Weyl basis's effectiveness in entanglement analysis, providing new insights into quantum state geometry.

Findings

Weyl basis closely linked to entanglement measures

Decomposition simplifies comparison with measurable quantities

Optimal basis choice depends on entanglement properties

Abstract

We present three different matrix bases that can be used to decompose density matrices of d--dimensional quantum systems, so-called qudits: the generalized Gell-Mann matrix basis, the polarization operator basis, and the Weyl operator basis. Such a decomposition can be identified with a vector --the Bloch vector, i.e. a generalization of the well known qubit case-- and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We consider the important case of an isotropic two--qudit state and decompose it according to each basis. Investigating the geometry of entanglement of special parameterized two--qubit and two--qutrit states, in particular we calculate the Hilbert--Schmidt measure of entanglement, we find that the Weyl operator basis is the optimal choice since it is closely connected to the entanglement of the considered states.