Complexity of mixed Schatten norms of quantum maps

Abstract: We study the complexity of computing the mixed Schatten $|\Phi|{q\to p}$ norms of linear maps $\Phi$ between matrix spaces. When $\Phi$ is completely positive, we show that $| \Phi |{q \to p}$ 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 $\Phi$, we show that computing $| \Phi |{1 \to p}$ is $\mathsf{NP}$-complete when $p>1$. Moving beyond the completely-positive case and considering $\Phi$ to be difference of entanglement breaking completely-positive trace-preserving maps, we prove that computing $| \Phi |^+{1 \to 1}$ is $\mathsf{NP}$-complete. In contrast, for the completely-bounded (cb) case, we describe a polynomial-time algorithm to compute $|\Phi|{cb,1\to p}$ and $|\Phi|^+{cb,1\to p}$ for any linear map $\Phi$ and $p\geq1$.