Bloch sphere analog of qudits using Heisenberg-Weyl Operators

• URL: http://arxiv.org/abs/2101.06408v4
• Date: Thu, 1 Feb 2024 12:25:47 GMT
• Title: Bloch sphere analog of qudits using Heisenberg-Weyl Operators
• Authors: Gautam Sharma and Sibasish Ghosh and Sk Sazim
• Abstract summary: We study an analogous Bloch sphere of higher-level quantum systems using the Heisenberg-Weyl operator basis. It is well known that we need eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits)
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• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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