Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914094282440704 |
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| author | Oeding, Luke Tan, Ian |
| author_facet | Oeding, Luke Tan, Ian |
| contents | We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all $n$-qubit pure states under the action of the local unitary group. Taking advantage of the double cover $\operatorname{SL}_2(\mathbb{C}) \times \operatorname{SL}_2(\mathbb{C}) \to \operatorname{SO}_4({\mathbb{C}})$, we produce similar algorithms (distinguished by the parity of $n$) that compute normal forms for almost all $n$-qubit pure states under the action of the SLOCC group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12542 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states Oeding, Luke Tan, Ian Quantum Physics Algebraic Geometry Representation Theory 15A69 We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all $n$-qubit pure states under the action of the local unitary group. Taking advantage of the double cover $\operatorname{SL}_2(\mathbb{C}) \times \operatorname{SL}_2(\mathbb{C}) \to \operatorname{SO}_4({\mathbb{C}})$, we produce similar algorithms (distinguished by the parity of $n$) that compute normal forms for almost all $n$-qubit pure states under the action of the SLOCC group. |
| title | Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states |
| topic | Quantum Physics Algebraic Geometry Representation Theory 15A69 |
| url | https://arxiv.org/abs/2402.12542 |