{"paper":{"title":"Closed-form Bayesian quantum estimation of Gaussian states","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A restriction to polynomial operators in quadratures reduces Bayesian quantum estimation of Gaussian states to closed-form linear problems.","cross_cats":[],"primary_cat":"quant-ph","authors_text":"Edward Gandar, Jes\\'us Rubio","submitted_at":"2026-05-16T12:56:59Z","abstract_excerpt":"Bayesian quantum estimation provides a robust framework for quantum technologies, especially in scenarios with limited data and minimal prior information. Yet, its application to continuous-variable Gaussian systems has remained limited and largely numerical due to the complexity of the underlying parameter integrals. Here, we introduce a variational framework reducing the optimisation over measurements and estimators to a finite-dimensional linear problem and admitting closed-form solutions. This is achieved by restricting the analysis to operators polynomial in the canonical quadratures, lea"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We introduce a variational framework reducing the optimisation over measurements and estimators to a finite-dimensional linear problem and admitting closed-form solutions. This is achieved by restricting the analysis to operators polynomial in the canonical quadratures, leading to solutions with a geometric interpretation as orthogonal projections of the global optimum.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Restricting the measurement and estimator search to operators that are polynomials in the canonical quadratures is sufficient to produce solutions that are either optimal or near-optimal for the original unbounded problem.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A variational framework yields closed-form Bayesian estimators for Gaussian quantum states via polynomial quadrature operators and a global optimality condition.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A restriction to polynomial operators in quadratures reduces Bayesian quantum estimation of Gaussian states to closed-form linear problems.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5734653484824daddfdd866159111b26ba1d17f83038e598b4ef7820ced09fd8"},"source":{"id":"2605.16978","kind":"arxiv","version":1},"verdict":{"id":"0a26cff2-3442-43e0-b854-51c709cdf13f","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:42:43.892739Z","strongest_claim":"We introduce a variational framework reducing the optimisation over measurements and estimators to a finite-dimensional linear problem and admitting closed-form solutions. This is achieved by restricting the analysis to operators polynomial in the canonical quadratures, leading to solutions with a geometric interpretation as orthogonal projections of the global optimum.","one_line_summary":"A variational framework yields closed-form Bayesian estimators for Gaussian quantum states via polynomial quadrature operators and a global optimality condition.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Restricting the measurement and estimator search to operators that are polynomials in the canonical quadratures is sufficient to produce solutions that are either optimal or near-optimal for the original unbounded problem.","pith_extraction_headline":"A restriction to polynomial operators in quadratures reduces Bayesian quantum estimation of Gaussian states to closed-form linear problems."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16978/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T21:01:19.068804Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:50:51.115708Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T19:51:56.388383Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T19:50:06.733667Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.217267Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.304572Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"7f6363a49fa97b252a14e7dd9292afb0704c3d0fbe491fc9a0ce14d809d037af"},"references":{"count":106,"sample":[{"doi":"","year":2026,"title":"Closed-form Bayesian quantum estimation of Gaussian states","work_id":"a2ebf80b-4a76-406e-a1b6-c179420fb90b","ref_index":1,"cited_arxiv_id":"2605.16978","is_internal_anchor":true},{"doi":"","year":null,"title":"(24) can also be proven using the explicitly variational approach of Personick [19, 20], as shown in Appendix A","work_id":"66b15624-5ffa-4056-a0c1-b11daefe64d3","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"This condition is sufficient wheneverSbelongs to the chosen polynomial subspaceV","work_id":"47a23b93-02eb-4aff-a776-49df3a064bd6","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"As shown in Appendix C, the MSL associated with an arbitrary Hermitian operatorXcan always be written as L(X) =L(S) +∥X− S∥ 2 ρ0 ,(35) whereL(X)is defined as in Eq","work_id":"f39e510d-caa7-466b-bc51-81b7b6e4e65b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Improved estimator based on posterior mean So far we have performed a constrained optimisation of the MSL by restricting the operatorM 1 to an operator subspace V. 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