Bloch sphere analog of qudits using Heisenberg-Weyl Operators

TL;DR

This paper develops a Bloch sphere analog for higher-dimensional quantum systems (qudits) using Heisenberg-Weyl operators, providing a new parametrization that reveals geometric and physical properties of these states.

Contribution

It introduces a novel real-valued Bloch vector parametrization for qudits, extending the Bloch sphere concept beyond qubits and analyzing its geometric structure and applications.

Findings

Qutrit Bloch sphere has a non-solid, non-convex structure.

The parametrization separates weight and angular parameters, with weights forming a 4D unit sphere.

Application to MUBs, unital maps, and state metrics demonstrates practical utility.

Abstract

We study an analogous Bloch sphere representation of higher-level quantum systems using the Heisenberg-Weyl operator basis. We introduce a parametrization method that will allow us to identify a real-valued Bloch vector for an arbitrary density operator. Before going into arbitrary $d$-level ($d\geq 3$) quantum systems (qudits), we start our analysis with three-level ones (qutrits). It is well known that we need at least eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits). However, using our method we can divide these parameters into four weight, and four angular parameters, and find that the weight parameters are inducing a unit sphere in four-dimension. And, the four angular parameters determine whether a Bloch vector is physical. Therefore, unlike its qubit counterpart, the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this construction allows us to define different properties of qutrits in terms of Bloch vector components. We also examine the two and three-dimensional sections of the sphere, which reveal a non-convex yet closed structure for physical qutrit states. Further, we apply our representation to derive mutually unbiased bases (MUBs), characterize unital maps for qutrits, and assess ensembles using the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to qudits, showcasing its potential applicability beyond the qutrit scenario.