Tensors, entanglement, separability, and their complexity

arXiv:2509.21639v3 Announce Type: replace Abstract: The aim of this paper is to show how to characterize the entanglement and separability of d-partite states, and to obtain both known and new results using the modern theory of 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 subspaces of bi-symmetric and bi-skew-symmetric Hermitian tensors, which correspond to Bosons and Fermions respectively. 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, the main result of this paper to show that the above quantities are computed in polynomial time when we restrict our attention to Bosons: symmetric $d$-qubits, or more generally to symmetric $d$-qunits in $\mathbb{C}^n$, and the corresponding bi-symmetric Hermtian density tensors, for a fixed value of $n$.

Abstract:The aim of this paper is to show how to characterize the entanglement and separability of d-partite states, and to obtain both known and new results using the modern theory of 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 subspaces of bi-symmetric and bi-skew-symmetric Hermitian tensors, which correspond to Bosons and Fermions respectively. 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, the main result of this paper to show that the above quantities are computed in polynomial time when we restrict our attention to Bosons: symmetric $d$-qubits, or more generally to symmetric $d$-qunits in $\mathbb{C}^n$, and the corresponding bi-symmetric Hermtian density tensors, for a fixed value of $n$.

52 pages. To appear in Communications in Mathematical Physics

arXiv:2509.21639 [quant-ph]

(or arXiv:2509.21639v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2509.21639 Focus to learn more arXiv-issued DOI via DataCite

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Tensors, entanglement, separability, and their complexity

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