Bloch vectors for qudits

TL;DR

This paper introduces three matrix bases for decomposing density matrices of qudits, facilitating easier calculations and experimental comparisons, with a focus on isotropic states and entanglement witnesses.

Contribution

It presents three novel bases for qudit state decomposition, a new method using standard matrices, and an application to entanglement witnesses for qutrits.

Findings

Decomposition of density matrices using three bases

Application to isotropic two-qudit states

Representation of entanglement witness for qutrits

Abstract

We present three different matrix bases that can be used to decompose density matrices of $d$--dimensional quantum systems, so-called qudits: the \emph{generalized Gell-Mann matrix basis}, the \emph{polarization operator basis}, and the \emph{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 present a new method to decompose density matrices via so--called standard matrices, consider the important case of an isotropic two--qudit state and decompose it according to each basis. In case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin 1 measurements, which is appropriate for an experimental realization.