Complexity of mixed Schatten norms of quantum maps
Jan Kochanowski, Omar Fawzi, Cambyse Rouzé
TL;DR
This work maps the computational landscape of mixed Schatten norms and their cb variants for quantum maps, revealing a sharp contrast between regimes: non-hypercontractive regions admit efficient algorithms (via convex optimization and quantum Boyd-type methods), while key CP and EB-difference settings become NP-hard. Central to the hardness results is a 2-Out-of-4-SAT reduction that encodes SAT solutions into optimal pure-state outputs of carefully crafted channels. In the cb framework, the paper derives convex-optimization reformulations that render $cb,1\to p$ norms tractable for arbitrary maps, including non-CP ones, and shows non-hypercontractive cb norms of CP maps collapse to their non-cb counterparts. Overall, the results delineate when quantum norm computations are feasible and when they inherit classical NP-hardness, with practical implications for channel distinguishability and open-system dynamics.
Abstract
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 $p\geq1$.