Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
The matrix rank and its positive versions are robust for small approximations, i.e. they do not decrease under small perturbations. In contrast, the multipartite tensor rank can collapse for arbitrarily small errors, i.e. there may be a gap between rank and border rank, leading to instabilities in t...
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2025-02-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2025-02-26-1649/pdf/ |
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| author | Andreas Klingler Tim Netzer Gemma De les Coves |
| author_facet | Andreas Klingler Tim Netzer Gemma De les Coves |
| author_sort | Andreas Klingler |
| collection | DOAJ |
| description | The matrix rank and its positive versions are robust for small approximations, i.e. they do not decrease under small perturbations. In contrast, the multipartite tensor rank can collapse for arbitrarily small errors, i.e. there may be a gap between rank and border rank, leading to instabilities in the optimization over sets with fixed tensor rank. Can multipartite positive ranks also collapse for small perturbations? In this work, we prove that multipartite positive and invariant tensor decompositions exhibit gaps between rank and border rank, including tensor rank purifications and cyclic separable decompositions. We also prove a correspondence between positive decompositions and membership in certain sets of multipartite probability distributions, and leverage the gaps between rank and border rank to prove that these correlation sets are not closed. It follows that testing membership of probability distributions arising from resources like translational invariant Matrix Product States is impossible in finite time. Overall, this work sheds light on the instability of ranks and the unique behavior of bipartite systems. |
| format | Article |
| id | doaj-art-119545b81aed4bc3bf5cc4935b376a52 |
| institution | DOAJ |
| issn | 2521-327X |
| language | English |
| publishDate | 2025-02-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj-art-119545b81aed4bc3bf5cc4935b376a522025-08-20T03:04:53ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2025-02-019164910.22331/q-2025-02-26-164910.22331/q-2025-02-26-1649Border Ranks of Positive and Invariant Tensor Decompositions: Applications to CorrelationsAndreas KlinglerTim NetzerGemma De les CovesThe matrix rank and its positive versions are robust for small approximations, i.e. they do not decrease under small perturbations. In contrast, the multipartite tensor rank can collapse for arbitrarily small errors, i.e. there may be a gap between rank and border rank, leading to instabilities in the optimization over sets with fixed tensor rank. Can multipartite positive ranks also collapse for small perturbations? In this work, we prove that multipartite positive and invariant tensor decompositions exhibit gaps between rank and border rank, including tensor rank purifications and cyclic separable decompositions. We also prove a correspondence between positive decompositions and membership in certain sets of multipartite probability distributions, and leverage the gaps between rank and border rank to prove that these correlation sets are not closed. It follows that testing membership of probability distributions arising from resources like translational invariant Matrix Product States is impossible in finite time. Overall, this work sheds light on the instability of ranks and the unique behavior of bipartite systems.https://quantum-journal.org/papers/q-2025-02-26-1649/pdf/ |
| spellingShingle | Andreas Klingler Tim Netzer Gemma De les Coves Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations Quantum |
| title | Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations |
| title_full | Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations |
| title_fullStr | Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations |
| title_full_unstemmed | Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations |
| title_short | Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations |
| title_sort | border ranks of positive and invariant tensor decompositions applications to correlations |
| url | https://quantum-journal.org/papers/q-2025-02-26-1649/pdf/ |
| work_keys_str_mv | AT andreasklingler borderranksofpositiveandinvarianttensordecompositionsapplicationstocorrelations AT timnetzer borderranksofpositiveandinvarianttensordecompositionsapplicationstocorrelations AT gemmadelescoves borderranksofpositiveandinvarianttensordecompositionsapplicationstocorrelations |