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| Main Authors: | , , |
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| Format: | Preprint |
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2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2101.06408 |
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| _version_ | 1866916150706700288 |
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| author | Sharma, Gautam Ghosh, Sibasish Sazim, Sk |
| author_facet | Sharma, Gautam Ghosh, Sibasish Sazim, Sk |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_06408 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Bloch sphere analog of qudits using Heisenberg-Weyl Operators Sharma, Gautam Ghosh, Sibasish Sazim, Sk Quantum Physics 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. |
| title | Bloch sphere analog of qudits using Heisenberg-Weyl Operators |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2101.06408 |