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PreprintarXiv licencearXiv

From Majorana to Bloch: two ways to generalize the Bloch sphere representation

  • Louis Hanotel
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Abstract

While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.

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Cite

BibTeXRIS
BibTeX
@misc{hanotel2026from,
  title = {{From Majorana to Bloch: two ways to generalize the Bloch sphere representation}},
  author = {Louis Hanotel},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11868},
  eprint = {2610.11868},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11868},
}
RIS
TY  - UNPB
TI  - From Majorana to Bloch: two ways to generalize the Bloch sphere representation
AU  - Louis Hanotel
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11868
UR  - https://arxiv.org/abs/2610.11868
AB  - While the Bloch sphere provides a canonical geometric representation for qubits, its generalization to pure qudit states (d>2) is not unique. This representation can be extended following two different approaches: increasing the number of points, as in the Majorana representation, or using a single point but in a space of larger dimension, as in the generalized Bloch representation. In this work, we compare these representations of pure qudit states by finding explicit analytical relations between them. In particular, a general formula of the Bloch vector in terms of the Majorana points of a given state is provided. We also analyze some suitable extensions of these representations in the case of pure permutation-symmetric two- and three-qubit states, deriving the corresponding relation between them. These analytical results allow us to study, from a geometric point of view, quantum correlations of special classes of two-qubit states.
ER  -