{"paper":{"title":"The Sample Complexity of Fidelity Estimation to a Known Rank-$r$ Reference State Is $\\widetilde{\\Theta}(r^2/\\varepsilon^2)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Gye Jin Lee, Sunghyeon Jo","submitted_at":"2026-08-03T06:43:01Z","abstract_excerpt":"We settle the sample complexity of estimating the root Uhlmann fidelity $F(\\rho,\\sigma)=\\operatorname{tr}\\sqrt{\\sqrt{\\sigma}\\rho\\sqrt{\\sigma}}$ between an unknown state $\\rho$ and a known rank-$r$ reference state $\\sigma$. Writing $S(r,\\varepsilon)$ for the sample complexity at additive error $\\varepsilon$, we resolve the open problem posed by Wang by closing, up to logarithmic factors, the gap between the previously known bounds $\\Omega(r/\\varepsilon^2)$ and $O(r^2/\\varepsilon^2)$. We prove $S(r,\\varepsilon)=\\widetilde{\\Theta}(r^2/\\varepsilon^2)$ for all $0<\\varepsilon\\le\\varepsilon_0$, where"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01770","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/2608.01770/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"}