Multipartite entanglement detection via projective tensor norms

We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class $S_1$ to the Euclidean space $\ell_2$, which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that previously studied entanglement criteria, such as the realignment and the SIC POVM criteria, can be viewed inside this framework. This allows us to answer in the positive two conjectures of Shang, Asadian, Zhu, and G\"uhne by deriving systematic relations between the performance of these two criteria.