{"as_of":"2026-08-18T12:00:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:184f1a25ffac05f74d14ce10eda0c7b0db5f8bdfbd90b6d1c67e1053fb901813","coverage":[{"denominator":46,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":46,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-06T19:03:28.272894Z","state":"measured"},{"denominator":59,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":59,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-18T06:34:40.430872+00:00","state":"measured"},{"denominator":13,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":13,"source":"paper_references, paper_reference_links","source_observed_at":"2026-08-15T15:44:19.017463Z","state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":1,"source":"pith","source_observed_at":"2026-08-05T02:28:24.338817Z","state":"measured"}],"external_citation_measurements":[{"count":0,"observed_at":"2026-08-05T02:28:24.338817Z","source":"pith"}],"inbound":[{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-06T19:25:47.156225Z","title":"Layer cake representations for quantum divergences,","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2507.06232","last_updated":"2026-06-04T15:02:02Z","snapshot_observed_at":"2026-08-15T17:04:08.518545Z","submitted_at":"2025-07-08T17:59:58Z","title":"Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem","version":4},"reference_index":61,"source":"pdf_text","source_observed_at":"2026-08-06T19:25:47.156225Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2507.06232"},"observation_digest":"sha256:0f868cddd4ca4b4ef1c8d41dc6e541a1b1be3c0714896243315677d6bd1d157b","observation_id":"3d28299f-9000-4238-bf55-9f13aa987fdd","resolution":{"observed_at":"2026-08-06T19:25:47.156225Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-06T18:43:48.407052Z","title":"Layer cake representations for quantum divergences,","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2507.07961","last_updated":"2025-07-10T17:41:50Z","snapshot_observed_at":"2026-08-15T17:03:01.540601Z","submitted_at":"2025-07-10T17:41:50Z","title":"Sharp estimates of quantum covering problems via a novel trace inequality","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-06T18:43:48.407052Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2507.07961"},"observation_digest":"sha256:bca93decec805fc5951711e87427b40eb7f702bffe3c05221493237f1f2d37aa","observation_id":"2fc9c2a8-68a0-479f-9ae6-f3cc278e9691","resolution":{"observed_at":"2026-08-06T18:43:48.407052Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2604.08380","last_updated":"2026-05-14T14:45:21Z","snapshot_observed_at":"2026-07-06T22:57:31.046146Z","submitted_at":"2026-04-09T15:42:35Z","title":"Sufficiency and Petz recovery for positive maps","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-05-10T18:02:19.513490Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2604.08380"},"observation_digest":"sha256:a2ca7d460d26e2a819435eaf41c6e3ab981d0cd874d20ebb9c25a8e24f313578","observation_id":"4949ba7f-19ed-42f2-b8a5-8f8917dfac22","resolution":{"observed_at":"2026-05-10T20:35:47.429200Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2604.08380","last_updated":"2026-05-14T14:45:21Z","snapshot_observed_at":"2026-07-06T22:57:31.046146Z","submitted_at":"2026-04-09T15:42:35Z","title":"Sufficiency and Petz recovery for positive maps","version":2},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-05-15T06:45:03.519810Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2604.08380"},"observation_digest":"sha256:591284422305409766d3cb877870982a9af654a6009510a1e46b04b448b6ab42","observation_id":"9088ebf7-72d2-4dc7-a6ec-01ff7dedb541","resolution":{"observed_at":"2026-05-15T06:45:10.251750Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2604.14617","last_updated":"2026-05-12T15:10:54Z","snapshot_observed_at":"2026-08-15T17:02:47.337323Z","submitted_at":"2026-04-16T04:57:25Z","title":"Optimal Trace Inequalities for Single-Shot Quantum Information","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-05-10T11:53:52.233992Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2604.14617"},"observation_digest":"sha256:46650fcb1b55b05cec3671cda388a6bba1d2f1be73c5c19fbf5ab1c3c5c3f49e","observation_id":"bfbb0c32-4ce3-42bd-bd18-f02ee97c5411","resolution":{"observed_at":"2026-05-10T11:55:20.552526Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2604.14617","last_updated":"2026-05-12T15:10:54Z","snapshot_observed_at":"2026-08-15T17:02:47.337323Z","submitted_at":"2026-04-16T04:57:25Z","title":"Optimal Trace Inequalities for Single-Shot Quantum Information","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-05-13T07:55:19.205772Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2604.14617"},"observation_digest":"sha256:e9c0b1e089fca6a3c6a01812dffe2aee68c8e16feffc85e60baa7e67454dea59","observation_id":"43757e14-23c3-4ba3-a0be-3375ff9ac95a","resolution":{"observed_at":"2026-05-13T07:57:31.461131Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2605.31379","last_updated":"2026-05-29T14:48:00Z","snapshot_observed_at":"2026-08-15T08:30:58.257330Z","submitted_at":"2026-05-29T14:48:00Z","title":"R\\'enyi divergences and binary state discrimination error exponents for fermionic quasi-free states","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-06-28T22:16:31.903127Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2605.31379"},"observation_digest":"sha256:f333528dfa1591b2bbf292f5a8d9ee04e76753af564cbc59cedfb1de0939a58b","observation_id":"e0a2299d-0e20-460c-9ff9-5061fbe78e46","resolution":{"observed_at":"2026-07-01T19:36:09.042863Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2606.06246","last_updated":"2026-06-04T14:49:12Z","snapshot_observed_at":"2026-08-18T03:48:38.287230Z","submitted_at":"2026-06-04T14:49:12Z","title":"Multiple Quantum Hypothesis Testing: One-Shot Pairwise Bounds and Sharp Asymptotics","version":1},"reference_index":74,"source":"pdf_text","source_observed_at":"2026-06-28T01:25:38.145414Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2606.06246"},"observation_digest":"sha256:3aa3cdb52ba00a836dbcd75569167deef05542bf68d49107dc8e7b4be1acc050","observation_id":"526cc3cc-506f-4c8b-99c0-fa356c3106b4","resolution":{"observed_at":"2026-07-02T13:16:58.966020Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2607.01712","last_updated":"2026-07-30T03:01:28Z","snapshot_observed_at":"2026-08-15T15:33:05.453568Z","submitted_at":"2026-07-02T05:07:07Z","title":"Quantum Noncommutativity Uniquely Determines Relative Entropy","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-07-03T12:50:47.787535Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2607.01712"},"observation_digest":"sha256:e618c363300513f970767e08f2a59443a80f463fef99e7f8e6e80ee9713072cc","observation_id":"7a03a59d-dd55-412d-a6c2-576661beb22e","resolution":{"observed_at":"2026-07-03T12:58:07.532144Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-15T15:44:19.017463Z","title":null,"venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2607.01712","last_updated":"2026-07-30T03:01:28Z","snapshot_observed_at":"2026-08-15T15:33:05.453568Z","submitted_at":"2026-07-02T05:07:07Z","title":"Quantum Noncommutativity Uniquely Determines Relative Entropy","version":2},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-15T15:44:19.017463Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2607.01712"},"observation_digest":"sha256:ea4f30e57e326b6ce16324f096a1b295500d5c933c6fad203fe95400e26f716e","observation_id":"17969673-03b5-42bb-aa7c-a14dd17c0bd0","resolution":{"observed_at":"2026-08-15T15:44:19.017463Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-07-12T01:24:32.061557Z","title":"Layer Cake Representations for Quantum Divergences","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2607.03579","last_updated":"2026-07-03T19:44:10Z","snapshot_observed_at":"2026-08-17T16:36:19.397486Z","submitted_at":"2026-07-03T19:44:10Z","title":"Device-independent Quantum Key Distribution in the commuting operator framework","version":1},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-07-12T01:24:32.061557Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2607.03579"},"observation_digest":"sha256:82f821b4c40b0614be3be8193a9647b90ef7e5f302135a3bc9b0c914067d8f37","observation_id":"ca614b9d-a02f-43ab-ad5c-30be3e04903b","resolution":{"observed_at":"2026-07-12T01:24:32.061557Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2607.05195","last_updated":"2026-07-06T15:11:39Z","snapshot_observed_at":"2026-08-16T05:35:52.092896Z","submitted_at":"2026-07-06T15:11:39Z","title":"Integral representations of $f$-divergences for general von Neumann algebras","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-07-08T00:07:38.510338Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2607.05195"},"observation_digest":"sha256:18c07410a98ff14f0623686766aeb417176990eecd39e2656fe7d24c10a05254","observation_id":"bce9c1de-8506-4160-8fd4-1251fdaed6b0","resolution":{"observed_at":"2026-07-08T00:14:22.286021Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"cited_work":{"arxiv_id":"2507.07065","doi":"10.48550/arxiv.2507.07065","metadata_source":"pith","pith_arxiv_id":"2507.07065","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Layer Cake Representations for Quantum Divergences","venue":"quant-ph","work_id":"b8a14a22-aaf5-4a28-a98c-db74bcfd34da","year":2025},"citing_paper":{"arxiv_id":"2607.08760","last_updated":"2026-07-09T17:56:17Z","snapshot_observed_at":"2026-08-15T08:30:14.583316Z","submitted_at":"2026-07-09T17:56:17Z","title":"Hockey stick $f$-divergences","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-07-10T02:22:53.196107Z"},"links":{"cited_paper":"/paper/2507.07065","citing_paper":"/paper/2607.08760"},"observation_digest":"sha256:f4621dbfc741e93f789524f54d999ba2e710362c2b8f55d6d08f06708a78b398","observation_id":"b552c2f3-fbbf-4e55-886c-7a123b9848ac","resolution":{"observed_at":"2026-07-10T02:26:42.819738Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}}],"links":{"evidence":"/evidence","html":"/paper/2507.07065/citation-record","integrity":"/paper/2507.07065/integrity","json":"/paper/2507.07065/citation-record.json","paper":"/paper/2507.07065"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:37.086516Z","title":"α-z-R´ enyi relative entropies,","venue":null,"work_id":"985a8ea9-171a-4dde-9d11-c0b44b8bfb15","year":2015},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:22.847782Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:8f772ae776a7a3ac1b92ae316828984900df7911a2119f13650a78c90b0da8dd","observation_id":"a19979b2-9fd3-4c34-968d-370e9c015b42","resolution":{"observed_at":"2026-08-06T19:03:37.168369Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:36.890661Z","title":"Tomamichel, Quantum Information Processing with Finite Resources","venue":null,"work_id":"886fbc30-058c-4b8d-bd2a-48e2f8df64fd","year":2016},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:22.954981Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:86c79b90da4c083569c4dc8109621822b86f90212ff65b1cac82cb4a168e9027","observation_id":"7a87a4c9-a0e1-4715-8fdd-3ee268e07cdf","resolution":{"observed_at":"2026-08-06T19:03:36.976296Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:36.609443Z","title":"On measures of entropy and information,","venue":null,"work_id":"a8b8d111-f20d-401d-b217-6786a3d88bc1","year":1961},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.031845Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:a9aa3985b581f073dccb665a8c7596dcf00a1bfd34070c5a42c8ad8e47a022be","observation_id":"02172e1a-6f83-4be0-9f21-1c101d49db3b","resolution":{"observed_at":"2026-08-06T19:03:36.749729Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:36.387642Z","title":"On the relative entropy,","venue":null,"work_id":"cede3ef6-1d0c-4aa0-9987-67da7035bfa6","year":1986},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.128517Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:d55d05d553b2b960669bd8349aacfad34e37f0646c818d98234449c213263e0e","observation_id":"bb1d5323-1acb-4b2a-b6a0-1f2916f0fdd5","resolution":{"observed_at":"2026-08-06T19:03:36.497146Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:36.142562Z","title":"The proper formula for relative entropy and its asymptotics in quantum probability,","venue":null,"work_id":"233523d1-e20e-4465-a79b-2ca24900453f","year":1991},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.242808Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:73c88318ebacbed5a20e800bb763259ac1f626d97c19c338c619d6830dcc4820","observation_id":"978ec423-fc80-4809-bfec-f3ca1713e162","resolution":{"observed_at":"2026-08-06T19:03:36.278750Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:35.864829Z","title":"Matsumoto, A New Quantum Version of f -Divergence","venue":null,"work_id":"8dd7ea99-b15c-4428-86d5-9f2539e49bdc","year":2018},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.352092Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:898b1d68718d88b2a6aa62d095df2e29e126d0e40b5bf6df56710199d4ae725e","observation_id":"c4b74c83-bc2b-49d9-a8a6-7b288aaf428c","resolution":{"observed_at":"2026-08-06T19:03:35.979346Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:35.683187Z","title":"Quantum f -divergences in von Neumann algebras. ii. maximal f -divergences,","venue":null,"work_id":"3af32bd8-8184-484e-a62b-6068445a9620","year":2019},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.416348Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:59513f968da4742c93f0bf7e9836641c71683b1d9a52fdcc5a29bdc88d54f314","observation_id":"6fca6ee6-03e4-4e8f-a61c-c5845b5811d3","resolution":{"observed_at":"2026-08-06T19:03:35.776046Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:35.359082Z","title":"Relative entropy of states of von Neumann algebras,","venue":null,"work_id":"e46b48be-d43d-4810-9541-dbf3d2d835fa","year":1976},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.552589Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:427f864b5c7fe2a641ab05b9f504c026dd63867724cae345b4e9d2a0498e4230","observation_id":"bf3757cf-6390-4f25-8946-315764d56b7c","resolution":{"observed_at":"2026-08-06T19:03:35.536437Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:35.094435Z","title":"Relative entropy of states of von Neumann algebras. ii,","venue":null,"work_id":"53fec358-2b76-4c2b-b66f-0a572bf0b362","year":1977},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.628896Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:8e51c011ec6dfe0966c822c7701e66eb83cec30c27caa4dced0cc1786f8a06ff","observation_id":"1ee376d2-548a-4293-912e-0f89c4b56c04","resolution":{"observed_at":"2026-08-06T19:03:35.218761Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:34.864004Z","title":"Quasi-entropies for states of a von Neumann algebra,","venue":null,"work_id":"bf41b055-226e-4534-9058-c29772cc2e00","year":1985},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.736594Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:3933d7dccc5a18dcf77e535a1fd9f82328988a545a0c6a68f7feeefce331c79a","observation_id":"6de6a0bc-1d92-4d66-988a-110c1853ae2a","resolution":{"observed_at":"2026-08-06T19:03:34.983306Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:34.665022Z","title":"Quasi-entropies for finite quantum systems,","venue":null,"work_id":"053d7ff5-dd14-4b1f-bf57-f952a2f96a27","year":1986},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.847076Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:baef6fa5a62fd99e5fff848cb457d2933e03df2f55ca4b5533378de4b88d86f1","observation_id":"24ac3bbd-6e71-4d4a-8c6c-21569f5951ca","resolution":{"observed_at":"2026-08-06T19:03:34.763754Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:23.945711Z","title":"Ohya and D","venue":null,"work_id":null,"year":1993},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:23.945711Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:ccc31d4c385e6616ffe4968da931d9cdb686d0a22c1dbf7c0d59b00c60dc249e","observation_id":"7bb969b4-26a8-45f4-bad8-f1502769b623","resolution":{"observed_at":"2026-08-06T19:03:23.945711Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:34.318184Z","title":"Quantumf -divergences and error correction,","venue":null,"work_id":"eeb915ac-3f0d-4765-b13b-6ab91c619104","year":2011},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.065103Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:6787b1762bca6911953d1349f412c24e617a0d5dccce1542933cb9821ef796e3","observation_id":"5d2d67a5-ae2b-4239-a6bc-17e959201c0c","resolution":{"observed_at":"2026-08-06T19:03:34.464302Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:34.097763Z","title":"Quantum f -divergences in von Neumann algebras. i. standard f -divergences,","venue":null,"work_id":"a6a8022c-f27a-4044-98f0-4db56283ee12","year":2018},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.149864Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:091d95d68d57217cfa67db25b904f79211a82c2bcec5c98724e4d1e1a0c134d8","observation_id":"ecb88e3e-6dcd-4974-ae7f-7bbc406cb91c","resolution":{"observed_at":"2026-08-06T19:03:34.187632Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:33.808250Z","title":"Functional calculus for sesquilinear forms and the purification map,","venue":null,"work_id":"92cccf8f-7c67-4848-bbb6-f26aa4dc27c6","year":1975},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.261366Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:18b7446d93df4ff7413b4fcd2a5514d8eb0900387fb2a9f82307c5b2e3218833","observation_id":"288c6a93-284b-496f-90de-2375c8b49503","resolution":{"observed_at":"2026-08-06T19:03:33.914803Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:33.530772Z","title":"Means of positive linear operators,","venue":null,"work_id":"d86adc8d-db87-43d1-9f03-ec90ccc134cb","year":1980},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.336235Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:a1a4faf7a87e607c30bf91360c0ea57d2b8b676a4c19748670e5309180f3fac4","observation_id":"2bc2d165-6b2c-40e6-9fd0-3274e2b697c7","resolution":{"observed_at":"2026-08-06T19:03:33.696610Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:33.294623Z","title":"C ∗-algebraic generalization of relative entropy and entropy,","venue":null,"work_id":"2227c085-062b-4ee1-b0ee-5b4e3dbc9c88","year":1982},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.394748Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:9422f0df545c7d8d20d522717a06d55297da56ca2b3210a4e1b74bc7d114f932","observation_id":"b4356b75-15e2-4f1e-bffb-801f302b4718","resolution":{"observed_at":"2026-08-06T19:03:33.398271Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:24.524318Z","title":"On quantum R´ enyi entropies: A new generalization and some properties,","venue":null,"work_id":null,"year":2013},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.524318Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:62abdccc917be792e6718999dbda40f3712a57debf52bb98fea0c1cc9dc6e28d","observation_id":"359c011c-9b68-4f48-b5eb-f70087997480","resolution":{"observed_at":"2026-08-06T19:03:24.524318Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:24.593752Z","title":"Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched R´ enyi relative entropy,","venue":null,"work_id":null,"year":2014},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.593752Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:2082f82045849ed33be9073698ca6949c8ceef2d27500963c1a185d831a08527","observation_id":"c708633c-2202-488b-bc8c-a8a22a8b398f","resolution":{"observed_at":"2026-08-06T19:03:24.593752Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:24.734768Z","title":"Error exponent and strong converse for quantum soft covering,","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.734768Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:1676389d8ce0ba2382a9c8b942641129202c4cbfee200c48dc657da1d989f062","observation_id":"71165628-b7cd-47e2-8781-5302f0bb8f21","resolution":{"observed_at":"2026-08-06T19:03:24.734768Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2304.12055","last_updated":"2023-05-04T10:41:04Z","snapshot_observed_at":"2026-08-16T15:38:12.027036Z","submitted_at":"2023-04-24T12:47:37Z","title":"Tight One-Shot Analysis for Convex Splitting with Applications in Quantum Information Theory","version":2},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2304.12055","snapshot_observed_at":"2026-08-06T19:03:24.829066Z","title":"Tight one-shot analysis for convex splitting with applications in quantum information theory,","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.829066Z"},"links":{"cited_paper":"/paper/2304.12055","citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:832f5f578e80778ded3fae1f10fddd3bf0a3787a597d02efe2b81f7f1e6e3185","observation_id":"c0b401c5-f94d-4e00-9794-05f9f8515bbe","resolution":{"observed_at":"2026-08-06T19:03:24.829066Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:33.039945Z","title":"R´ enyi relative entropies and noncommutativeLp-spaces,","venue":null,"work_id":"689dda8c-ad1d-4bf7-ab27-4c65a51aed66","year":2018},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:24.915959Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:a90b010c32302c8481903333123f451a2be8297a6778b39abc8b18625e1f828f","observation_id":"15f03484-8a47-4c3b-b770-5410075afb64","resolution":{"observed_at":"2026-08-06T19:03:33.160536Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:32.791060Z","title":"R´ enyi divergences as weighted non-commutative vector-valued Lp-spaces,","venue":null,"work_id":"940f935d-2bf9-483a-899a-5475d1a03eed","year":2018},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.019685Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:18a10ed5cb3266286d7a7554ac36a37b6485b3a435a77dea1ed680baf865ddef","observation_id":"00f2122f-846e-4236-9285-a88bc9bb06ce","resolution":{"observed_at":"2026-08-06T19:03:32.880496Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:32.560246Z","title":"Hiai, Quantum f -Divergences in von Neumann Algebras: Reversibility of Quantum Operations","venue":null,"work_id":"feaeab7b-5ec6-44c4-bb7e-eb92c615e7bd","year":2021},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.083649Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:ee5b75bf16702c9bb6e6b4832c1d943da31d994fbc1ff591f2f5c2e0bc484286","observation_id":"c34e1f84-95b2-4919-a5fe-8cbeb2eb1b3c","resolution":{"observed_at":"2026-08-06T19:03:32.653115Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:32.318711Z","title":"Applications of the complex interpolation method to a von Neumann algebra: Non-commutative Lp-spaces,","venue":null,"work_id":"92c36e1a-aaf1-4eeb-acad-13cda9ee4a7a","year":1984},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.142090Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:896bedfa40ba7a8e449fc6fa519d305135ced3c9954d9d728968abf6a41cba6e","observation_id":"6ad0facd-b580-4395-a526-2138b88b7beb","resolution":{"observed_at":"2026-08-06T19:03:32.426653Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:25.266140Z","title":"Quantum r´ enyi andf -divergences from integral representations,","venue":null,"work_id":null,"year":2024},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.266140Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:4874a76212304db4446d1acec74bdef1409d0be7761dd6196679b43e7119a08d","observation_id":"c3ae7f4f-67b4-419d-bb2c-a987d87984f2","resolution":{"observed_at":"2026-08-06T19:03:25.266140Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:32.094193Z","title":"f -divergence inequalities,","venue":null,"work_id":"01b1e779-8713-428e-b783-6349ac04fd0a","year":2016},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.421805Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:099214ff64509c599f3c028cee29593fdbf628c128527f27261a2ee9eb249dac","observation_id":"5f16f61e-fe78-4d0e-b46e-f08d3593f960","resolution":{"observed_at":"2026-08-06T19:03:32.179093Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:31.878671Z","title":"Lieb and M","venue":null,"work_id":"6ff84af8-76dd-44f7-9218-0d79b83ef5ea","year":2001},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.568700Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:9dc643ecb1cfa3bd9d2d4bc4f7a2f0cb08a8314b7e4f7e044ce4bf8cf902f759","observation_id":"c45b177a-ca15-4179-bd91-4542411e3606","resolution":{"observed_at":"2026-08-06T19:03:31.994285Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2501.03799","last_updated":"2025-01-07T14:06:56Z","snapshot_observed_at":"2026-08-16T12:59:29.967378Z","submitted_at":"2025-01-07T14:06:56Z","title":"Some properties and applications of the new quantum $f$-divergences","version":1},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2501.03799","snapshot_observed_at":"2026-08-06T19:03:25.719216Z","title":"Some properties and applications of the new quantumf -divergences,","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.719216Z"},"links":{"cited_paper":"/paper/2501.03799","citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:35f5eb604cf9ceeb6fd7dfdff73bebf49ccbc10644e0820c212346628c36967f","observation_id":"839d8461-b86b-4ba3-9df8-0fbfc1bbddea","resolution":{"observed_at":"2026-08-06T19:03:25.719216Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2507.06232","last_updated":"2026-06-04T15:02:02Z","snapshot_observed_at":"2026-08-15T17:04:08.518545Z","submitted_at":"2025-07-08T17:59:58Z","title":"Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem","version":4},"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":"2507.06232","snapshot_observed_at":"2026-08-06T19:03:25.822779Z","title":"Error exponents for quantum packing problems via an operator layer cake theorem,","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.822779Z"},"links":{"cited_paper":"/paper/2507.06232","citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:421ac4459da12c7f7411a802708ec6ed212582335eebb0ab2484d8310cece80c","observation_id":"dace4324-1bde-4039-8bf7-1203b89dbf36","resolution":{"observed_at":"2026-08-06T19:03:25.822779Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:31.646547Z","title":"Conditional expectation in an operator algebra. IV. entropy and information,","venue":null,"work_id":"96b6a38c-beb9-495a-a94a-80ea2cee9ccc","year":1962},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:25.931627Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:1c9ee8492727e3d6779d88e98550a7fcf290c1ee2cdd8d18b1b959d1b27d5f55","observation_id":"913f71cc-7039-4b1a-894b-7b9fbae25f67","resolution":{"observed_at":"2026-08-06T19:03:31.748218Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:31.436928Z","title":"Integral formula for quantum relative entropy implies data processing inequality,","venue":null,"work_id":"63977a15-5acd-4f4e-89a3-1b07261e217d","year":2022},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:26.078307Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:97749eecc290e155583bfdf7a7282a26c5223da6f3fe46e95bfbb4d946c443ea","observation_id":"ce36c567-0ccc-46c4-a55a-fc8f3507f496","resolution":{"observed_at":"2026-08-06T19:03:31.526797Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:31.211195Z","title":"Discriminating states: The quantum Chernoff bound,","venue":null,"work_id":"6f6080b1-cbce-45c2-a82e-239f97fd8284","year":2007},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:26.194264Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:164213bc8e2b2ee6c1a9fa5f66ae827bd85328e14e5aeeeeb1eb0452d1c475b9","observation_id":"9f385343-c9ce-47b8-a2ef-fcc0953b6276","resolution":{"observed_at":"2026-08-06T19:03:31.306958Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:30.989283Z","title":"Asymptotic error rates in quantum hy- pothesis testing,","venue":null,"work_id":"aea64bce-496b-4183-94fb-2b9ae4cacc85","year":2008},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:26.381417Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:e2e6c8f7bfed5dac36dff986b8b052054bad4040ac6d292701ba4c37a97c3f11","observation_id":"dc130bfc-e15b-42f9-9a37-2f7390020ec7","resolution":{"observed_at":"2026-08-06T19:03:31.110100Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:30.793329Z","title":"Convex trace functions and the Wigner-Yanase-Dyson conjecture,","venue":null,"work_id":"95a31922-c05a-45d8-962c-52e2f123aa1a","year":1973},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:26.615564Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:61b4b2ebfcbeb84784872f5c03d7dc4dc5410028d685526222b2415ca366cdff","observation_id":"5bfa136f-01f0-4ebb-be81-3a25a977f32c","resolution":{"observed_at":"2026-08-06T19:03:30.889068Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:30.592240Z","title":"Multivariate trace inequalities,","venue":null,"work_id":"6a34b783-823f-40b6-b586-d8952d24477d","year":2017},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:26.743281Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:4024baa4a4c8b4d0e001516325ca0b86fbd95bdf683636fedc13eddccc32d110","observation_id":"0d3e1faa-36e6-401a-90bd-9fbc8c789a6a","resolution":{"observed_at":"2026-08-06T19:03:30.678268Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:30.287742Z","title":"On the strong converses for the quantum channel capacity theorems,","venue":null,"work_id":"dc9dc564-12e2-4632-a4b2-e97726e64a5e","year":2012},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:26.943279Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:ae02f0ebae6ff363498ea367df5f3ce3e4be37af8b19ba1ec84716941f07b149","observation_id":"d1577284-f5ef-4276-96f2-b96dcb5df95a","resolution":{"observed_at":"2026-08-06T19:03:30.427976Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:30.123918Z","title":"Quantum differential privacy: An information theory perspective,","venue":null,"work_id":"1530a83d-b67f-49d1-98a4-4f30ccbf9871","year":2023},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:27.064471Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:2e7c57ec3ecb115d7f473ba587969ac696221d922619fe6d25799de4d021c1d3","observation_id":"5bc36c67-c0b0-42be-8738-d94b8681fa36","resolution":{"observed_at":"2026-08-06T19:03:30.184976Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:30.008339Z","title":"Sample complexity of locally differentially private quantum hypothesis testing,","venue":null,"work_id":"8b6aebcd-5b3e-47db-a9b8-89d20f54bf1b","year":2024},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:27.228962Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:f49b862fa553e4bc96cf256b637220aee54ab4ee9aae3b4f2f57ad2a7619b589","observation_id":"07443556-6af0-437a-80ab-48a73af0cc3c","resolution":{"observed_at":"2026-08-06T19:03:30.066300Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:29.756933Z","title":"Kato, Perturbation Theory for Linear Operators","venue":null,"work_id":"401f8255-ada2-4a73-a9bd-4bff2d87821f","year":1995},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:27.381303Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:696f53d925358ddf4686a988091b90d85de9d0f41b561dc5103f39c7e57f7870","observation_id":"359df932-5bb9-4407-9c4e-bf5b9d430341","resolution":{"observed_at":"2026-08-06T19:03:29.890516Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:29.549479Z","title":"Hiai and D","venue":null,"work_id":"0727cf39-bd18-4418-8375-f241775b0e64","year":2014},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:27.485612Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:8e1a4241fed0925a98ae5bd46001b34b97f01a0c8ff0d04e42a58c9d2eeda866","observation_id":"3dc78a2b-9629-4586-8944-238158831827","resolution":{"observed_at":"2026-08-06T19:03:29.646110Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:29.307126Z","title":"Polyanskiy, Channel coding: Non-asymptotic fundamental limits","venue":null,"work_id":"d4b6f7a3-dc25-46d1-a567-6e5aa185a779","year":2010},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:27.606899Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:5b0702fdee80c3eb42391a858cc8b58192d39147b775f1c217d376d12dcc6188","observation_id":"75c82a81-aea0-4843-abf8-e57a5efdef82","resolution":{"observed_at":"2026-08-06T19:03:29.413926Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:29.113810Z","title":"Quantum hypothesis testing and the operational interpretation of the quantum r´ enyi relative entropies,","venue":null,"work_id":"3e63f794-3f69-49a5-9b1d-f3095eb6bba3","year":2015},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:27.784163Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:a6dd611d79d1de7c457c52a4706828b0872135eeaa6b38efec94ce39eda32b52","observation_id":"43267af7-dad2-46c4-9bc8-a77d266d32ea","resolution":{"observed_at":"2026-08-06T19:03:29.204874Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:28.888518Z","title":"The Chernoff lower bound for symmetric quantum hypothesis testing,","venue":null,"work_id":"fe2b7f1b-92f5-4fc0-9253-786dd69a8da3","year":2009},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:27.951495Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:f0465735390c2265a6dc42d3ef7721863673a816cb396417dd08f7f70ed4ddad","observation_id":"e300f677-569a-4143-a4ac-70de32230cd3","resolution":{"observed_at":"2026-08-06T19:03:29.011544Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:28.694581Z","title":"Polyanskiy and Y","venue":null,"work_id":"e9b8ea83-1cfe-427d-810d-a0ae6c3e6bef","year":2024},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:28.099061Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:5bfe29cc4d43638bf9bd735faf4115b0b9a18f07d26e676704b135d059857b47","observation_id":"0ab938c9-7cb8-44f4-9604-715340811f3d","resolution":{"observed_at":"2026-08-06T19:03:28.781297Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":"raw_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-06T19:03:28.437921Z","title":"A variational expression for the relative entropy,","venue":null,"work_id":"426d84ad-0db9-4883-92ba-8781d1da4ec0","year":1988},"citing_paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences","version":2},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-08-06T19:03:28.272894Z"},"links":{"citing_paper":"/paper/2507.07065"},"observation_digest":"sha256:1ad4012c817a8f32d623c0a955e2c9f68c207d80c23a4ac33329e468a7e40777","observation_id":"539466af-ac90-45fe-8898-70520af6f225","resolution":{"observed_at":"2026-08-06T19:03:28.553673Z","resolver_source":"raw_fallback","status":"verified_fuzzy"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-18T06:34:40.430872+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"state":"measured"}}],"paper":{"arxiv_id":"2507.07065","last_updated":"2025-07-10T17:53:52Z","latest_version":2,"primary_category":"quant-ph","snapshot_observed_at":"2026-08-15T17:40:32.481640Z","submitted_at":"2025-07-09T17:27:34Z","title":"Layer Cake Representations for Quantum Divergences"},"reference_resolution":{"displayed":46,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":8,"verified_exact":0,"verified_fuzzy":38},"total_outbound_references":46},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-18T06:34:40.430872+00:00","source":"crossref"},{"observed_at":"2026-08-18T06:34:34.496301+00:00","source":"retraction_watch"}],"thesis":"As of 18 August 2026, this Paper Citation Record lists 46 of 46 outbound references and 13 inbound Pith citation observations for arXiv:2507.07065."}