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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.04488","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"quant-ph","submitted_at":"2026-07-05T20:25:43Z","cross_cats_sorted":[],"title_canon_sha256":"5a6153a17e21e070ad61a284ce51f062699831e3d45fcbeeb9d9bc894c20426b","abstract_canon_sha256":"5f819b6d154050c47527c05dae075f87f450049f16387673fb2291fcce541de7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:19:21.765915Z","signature_b64":"nIwMQ1l23pMXkvbpMxDrnRapnQpughURa09nt30sbglYl6WhRZnzlp73EpIl9WZ/1PCSKt+CISGpylHjBa9qDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6180e8d7b824577c6d4d61e151a3d1888ac9963e3d46d0625d379013fe811cc9","last_reissued_at":"2026-07-07T02:19:21.765289Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:19:21.765289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\sqrt{2}\\le \\|X\\|_1/\\|X\\|_2\\le \\sqrt r$. 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