The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866909992091648000 |
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| author | Li, Yuan Chen, Zhengli Guo, Zhihua Pang, Yongfeng |
| author_facet | Li, Yuan Chen, Zhengli Guo, Zhihua Pang, Yongfeng |
| contents | The dynamics of quantum systems are generally described by a family of quantum channels (linear, completely positive and trace preserving maps). In this note, we mainly study the range of all possible values of $\|\mathcal{E}\|_2^2+\|\widetilde{\mathcal{E}}\|_2^2$ for quantum channels $\mathcal{E}$ and give the equivalent characterizations for quantum channels that achieve these maximum and minimum values, respectively, where $\|\mathcal{E}\|_2$ is the Hilbert-Schmidt norm of $\mathcal{E}$ and $\widetilde{\mathcal{E}}$ is a complementary channel of $\mathcal{E}.$ Also, we get a concrete description of completely positive maps on infinite dimensional systems preserving pure states. Moreover, the equivalency of several matrix integrals over the unit sphere is demonstrated and some extensions of these matrix integrals are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_10943 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere Li, Yuan Chen, Zhengli Guo, Zhihua Pang, Yongfeng Quantum Physics Mathematical Physics The dynamics of quantum systems are generally described by a family of quantum channels (linear, completely positive and trace preserving maps). In this note, we mainly study the range of all possible values of $\|\mathcal{E}\|_2^2+\|\widetilde{\mathcal{E}}\|_2^2$ for quantum channels $\mathcal{E}$ and give the equivalent characterizations for quantum channels that achieve these maximum and minimum values, respectively, where $\|\mathcal{E}\|_2$ is the Hilbert-Schmidt norm of $\mathcal{E}$ and $\widetilde{\mathcal{E}}$ is a complementary channel of $\mathcal{E}.$ Also, we get a concrete description of completely positive maps on infinite dimensional systems preserving pure states. Moreover, the equivalency of several matrix integrals over the unit sphere is demonstrated and some extensions of these matrix integrals are obtained. |
| title | The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2601.10943 |