Complexity of mixed Schatten norms of quantum maps
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911394109063168 |
|---|---|
| author | Kochanowski, Jan Fawzi, Omar Rouzé, Cambyse |
| author_facet | Kochanowski, Jan Fawzi, Omar Rouzé, Cambyse |
| contents | We study the complexity of computing the mixed Schatten $\|Φ\|_{q\to p}$ norms of linear maps $Φ$ between matrix spaces. When $Φ$ is completely positive, we show that $\| Φ\|_{q \to p}$ can be computed efficiently when $q \geq p$. The regime $q \geq p$ is known as the non-hypercontractive regime and is also known to be easy for the mixed vector norms $\ell_{q} \to \ell_{p}$ [Boyd, 1974]. However, even for entanglement-breaking completely-positive trace-preserving maps $Φ$, we show that computing $\| Φ\|_{1 \to p}$ is $\mathsf{NP}$-complete when $p>1$. Moving beyond the completely-positive case and considering $Φ$ to be difference of entanglement breaking completely-positive trace-preserving maps, we prove that computing $\| Φ\|^+_{1 \to 1}$ is $\mathsf{NP}$-complete. In contrast, for the completely-bounded (cb) case, we describe a polynomial-time algorithm to compute $\|Φ\|_{cb,1\to p}$ and $\|Φ\|^+_{cb,1\to p}$ for any linear map $Φ$ and $p\geq1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complexity of mixed Schatten norms of quantum maps Kochanowski, Jan Fawzi, Omar Rouzé, Cambyse Quantum Physics Mathematical Physics We study the complexity of computing the mixed Schatten $\|Φ\|_{q\to p}$ norms of linear maps $Φ$ between matrix spaces. When $Φ$ is completely positive, we show that $\| Φ\|_{q \to p}$ can be computed efficiently when $q \geq p$. The regime $q \geq p$ is known as the non-hypercontractive regime and is also known to be easy for the mixed vector norms $\ell_{q} \to \ell_{p}$ [Boyd, 1974]. However, even for entanglement-breaking completely-positive trace-preserving maps $Φ$, we show that computing $\| Φ\|_{1 \to p}$ is $\mathsf{NP}$-complete when $p>1$. Moving beyond the completely-positive case and considering $Φ$ to be difference of entanglement breaking completely-positive trace-preserving maps, we prove that computing $\| Φ\|^+_{1 \to 1}$ is $\mathsf{NP}$-complete. In contrast, for the completely-bounded (cb) case, we describe a polynomial-time algorithm to compute $\|Φ\|_{cb,1\to p}$ and $\|Φ\|^+_{cb,1\to p}$ for any linear map $Φ$ and $p\geq1$. |
| title | Complexity of mixed Schatten norms of quantum maps |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2507.08358 |