Tensors, entanglement, separability, and their complexity

One of the most challenging problems in quantum physics is to quantify the entanglement of $d$-partite states and their separability. We show here that these problems are best addressed using tensors. The geometric measure of entanglement of a pure state is one of most natural ways to quantify the entanglement, which is simply related to the spectral norm of a tensor state. On the other hand, the logarithm of the nuclear norm of the state and density tensors can be considered as its ``energy''. We first show that the most geometric measure entangled $d$-partite state has the minimum spectral norm and maximum nuclear norm. Second, we introduce the notion of Hermitian and density tensors, and the subspace of bi-symmetric Hermitian tensors, which correspond to Bosons. We show that separable density tensors, and strongly separable bi-symmetric density tensors are characterized by the value (equal to one) of their corresponding nuclear norms. In general, these characterizations are NP-hard to verify. Third, we show that the above quantities are computed in polynomial time when we restrict our attentions to Bosons: symmetric $d$-qubits, or more generally to symmetric $d$-qunits in $C^n$, and the corresponding bi-symmetric Hermtian density tensors, for a fixed value of $n$.