The Hilbert-Schmidt norms of quantum channels and matrix integrals over the unit sphere
Yuan Li, Zhengli Chen, Zhihua Guo, Yongfeng Pang
TL;DR
The paper determines sharp bounds on the sum $\|\mathcal{E}\|_2^2+\|\widetilde{\mathcal{E}}\|_2^2$ for quantum channels $\mathcal{E}$ and their complements, proving that $\tfrac{n+n^2}{d} \le \|\mathcal{E}\|_2^2+\|\widetilde{\mathcal{E}}\|_2^2 \le n^2+n$ with exact characterizations of the extremal cases. It also classifies completely positive maps preserving all pure states in both finite and infinite dimensions, showing these maps are either isometric conjugations or depolarizing-type channels. A key methodological thread is the use of matrix integrals over the unit sphere with Haar measure and their equivalences, together with Schur-Weyl duality, to obtain closed-form integral identities. The results culminate in an application to random isometric channels and broadcasting maps, demonstrating that the same interval is realized by a natural family of channels and that the depolarizing channel fits into this family. Overall, the work connects operator-norm questions for quantum channels to explicit sphere-integral identities and representation-theoretic tools, with implications for channel capacities and entanglement-preserving maps.
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.