Jan Kochanowski

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 $\NP$-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 $\NP$-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 $p\geq1$.

We show that approximating the trace norm contraction coefficient of a quantum channel within a constant factor is NP-hard. Equivalently, this shows that determining the optimal success probability for encoding a bit in a quantum system undergoing noise is NP-hard. This contrasts with the classical analogue of this problem that can clearly be solved efficiently. Our hardness results also hold for deciding if the contraction coefficient is equal to 1. As a consequence, we show that deciding if a non-commutative graph has an independence number of at least 2 is NP-hard. In addition, we establish a converging hierarchy of semidefinite programming upper bounds on the contraction coefficient.