A note on unitary invariance of Connes spectral distances of quantum states

Abstract

In this paper, we study the properties of Connes spectral distances between quantum states under unitary transformations. We mainly focus on spectral triples with matrix algebras acting on finite dimensional Hilbert spaces. We prove that there are some finite spectral triples in which the Lipschitz seminorms are equal to the operator norms. We also explicitly construct some spectral triples in which the Connes spectral distances between quantum states are exactly the quantum trace distances. These results are helpful for us to better study the relationships among Connes spectral distance, quantum trace distance and other quantum distance measures. These concrete examples are significant for studies of geometric structures of finite spectral triples and mathematical relations of qubits and other quantum states in the framework of noncommutative geometry.

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BibTeX

@misc{lin2026note,
  title = {{A note on unitary invariance of Connes spectral distances of quantum states}},
  author = {Bing-Sheng Lin and Ji-Hong Wang and Zhi-Kang You},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.13014},
  eprint = {2605.13014},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.13014},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A note on unitary invariance of Connes spectral distances of quantum states
AU  - Bing-Sheng Lin
AU  - Ji-Hong Wang
AU  - Zhi-Kang You
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2605.13014
UR  - https://arxiv.org/abs/2605.13014
AB  - In this paper, we study the properties of Connes spectral distances between quantum states under unitary transformations. We mainly focus on spectral triples with matrix algebras acting on finite dimensional Hilbert spaces. We prove that there are some finite spectral triples in which the Lipschitz seminorms are equal to the operator norms. We also explicitly construct some spectral triples in which the Connes spectral distances between quantum states are exactly the quantum trace distances. These results are helpful for us to better study the relationships among Connes spectral distance, quantum trace distance and other quantum distance measures. These concrete examples are significant for studies of geometric structures of finite spectral triples and mathematical relations of qubits and other quantum states in the framework of noncommutative geometry.
ER  -

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