Computational Complexity of Tensor Nuclear Norm

The main result of this paper is that the weak membership problem in the unit ball of a given norm is NP-hard if and only if the weak membership problem in the unit ball of the dual norm is NP-hard. Equivalently, the approximation of a given norm is polynomial time if and only if the approximation of the dual norm is polynomial time. Using the NP-hardness of the approximation of spectral norm of tensors we prove that the approximation of nuclear norm of tensors is NP-hard. In addition, we show that bipartite separability of a density matrix is equivalent its corresponding 4-tensor having unit nuclear norm, relating these results to quantum information theory.