Tensors, entanglement, separability, and their complexity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915596356026368 |
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| author | Friedland, Shmuel |
| author_facet | Friedland, Shmuel |
| contents | 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 $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_21639 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensors, entanglement, separability, and their complexity Friedland, Shmuel Quantum Physics 05C50, 15A69, 15A75, 68Q04, 68Q17, 68W25, 81P16, 81P40, 90C05, 90C08 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 $n$. |
| title | Tensors, entanglement, separability, and their complexity |
| topic | Quantum Physics 05C50, 15A69, 15A75, 68Q04, 68Q17, 68W25, 81P16, 81P40, 90C05, 90C08 |
| url | https://arxiv.org/abs/2509.21639 |