The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere

• URL: http://arxiv.org/abs/2601.10943v1
• Date: Fri, 16 Jan 2026 02:15:43 GMT
• Title: The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere
• Authors: Yuan Li, Zhengli Chen, Zhihua Guo, Yongfeng Pang,
• Abstract summary: We study the range of all possible values of $|mathcalE|2+|widetildemathcalE|2$ for quantum channels.<n>We get a concrete description of completely positive maps on infinite dimensional systems preserving pure states.
• Score: 2.474279525068631
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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