Cayley's First Hyperdeterminant is an Entanglement Measure
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916701861314560 |
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| author | Dobes, Isaac Jing, Naihuan |
| author_facet | Dobes, Isaac Jing, Naihuan |
| contents | Previously, it was shown that both the concurrence and n-tangle on $2n$-qubits can be expressed in terms of Cayley's first hyperdeterminant, indicating that Cayley's first hyperdeterminant captures at least some aspects of a quantum state's entanglement. In this paper, we rigorously prove that Cayley's first hyperdeterminant is an entanglement measure on $2n$-qudits, and thus a legitimate generalization of the concurrence and n-tangle. In particular, we show that the modulus of the hyperdeterminant and its square both vanish on separable $2n$-qudits, are LU invariants, and are non-increasing on average under LOCC. Lastly, we then consider their convex roof extensions and show that they may in fact be viewed as entanglement measures on mixed states of $2n$-qudits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15511 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cayley's First Hyperdeterminant is an Entanglement Measure Dobes, Isaac Jing, Naihuan Quantum Physics Mathematical Physics Previously, it was shown that both the concurrence and n-tangle on $2n$-qubits can be expressed in terms of Cayley's first hyperdeterminant, indicating that Cayley's first hyperdeterminant captures at least some aspects of a quantum state's entanglement. In this paper, we rigorously prove that Cayley's first hyperdeterminant is an entanglement measure on $2n$-qudits, and thus a legitimate generalization of the concurrence and n-tangle. In particular, we show that the modulus of the hyperdeterminant and its square both vanish on separable $2n$-qudits, are LU invariants, and are non-increasing on average under LOCC. Lastly, we then consider their convex roof extensions and show that they may in fact be viewed as entanglement measures on mixed states of $2n$-qudits. |
| title | Cayley's First Hyperdeterminant is an Entanglement Measure |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2504.15511 |