{"paper":{"title":"Trace-to-Hilbert-Schmidt Speed Ratio in Quantum Dynamics: Universal Bounds and Effective Rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Hossein Rangani Jahromi","submitted_at":"2026-07-05T20:25:43Z","abstract_excerpt":"We study the ratio between the trace speed and the Hilbert-Schmidt speed for differentiable finite-dimensional quantum states, $\\mathcal R(\\phi)=\\|\\partial_\\phi\\rho(\\phi)\\|_1/\\|\\partial_\\phi\\rho(\\phi)\\|_2$. Because $\\partial_\\phi\\rho(\\phi)$ is always Hermitian and traceless, this ratio is constrained more strongly than for a generic operator. For any nonzero tangent operator $X=\\partial_\\phi\\rho$ of rank $r$, we prove the sharp bounds $\\sqrt{2}\\le \\|X\\|_1/\\|X\\|_2\\le \\sqrt r$. The lower bound is attained exactly for rank-two tangents, while the upper bound is attained exactly when all nonzero s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04488","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.04488/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}