{"paper":{"title":"Highly entangled tensors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.OC","authors_text":"Harm Derksen, Visu Makam","submitted_at":"2018-03-26T18:50:50Z","abstract_excerpt":"A geometric measure for the entanglement of a unit length tensor $T \\in (\\mathbb{C}^n)^{\\otimes k}$ is given by $- 2 \\log_2 ||T||_\\sigma$, where $||.||_\\sigma$ denotes the spectral norm. A simple induction gives an upper bound of $(k-1) \\log_2(n)$ for the entanglement. We show the existence of tensors with entanglement larger than $k \\log_2(n) - \\log_2(k) - o(\\log_2(k))$. Friedland and Kemp have similar results in the case of symmetric tensors. Our techniques give improvements in this case."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.09788","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}