Computational Complexity of Tensor Nuclear Norm

The main result of this paper shows 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 the spectral norm of tensors we show that the approximation of the nuclear norm is NP-hard. We relate our results to bipartite separable states in quantum mechanics.

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