{"id":"https://openalex.org/W7214319259","doi":"https://doi.org/10.48550/arxiv.2609.30248","title":"Deep thermalization and Hilbert space ergodicity","display_name":"Deep thermalization and Hilbert space ergodicity","publication_year":2026,"publication_date":"2026-09-24","ids":{"openalex":"https://openalex.org/W7214319259","doi":"https://doi.org/10.48550/arxiv.2609.30248"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.30248","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30248","pdf_url":"https://arxiv.org/pdf/2609.30248","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.30248","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5056306081","display_name":"Daniel K. Mark","orcid":"https://orcid.org/0000-0002-5017-5218","observed_orcids":["https://orcid.org/0000-0002-5017-5218"]},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mark, Daniel K.","raw_affiliation_strings":["Center for Theoretical Physics -a Leinweber Institute , Massachusetts Institute of Technology , Cambridge , MA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Theoretical Physics -a Leinweber Institute , Massachusetts Institute of Technology , Cambridge , MA , USA","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5083319852","display_name":"Manuel Endres","orcid":"https://orcid.org/0000-0002-4461-224X","observed_orcids":["https://orcid.org/0000-0002-4461-224X"]},"institutions":[{"id":"https://openalex.org/I122411786","display_name":"California Institute of Technology","ror":"https://ror.org/05dxps055","country_code":"US","type":"education","lineage":["https://openalex.org/I122411786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Endres, Manuel","raw_affiliation_strings":["California Institute of Technology , Pasadena , CA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"California Institute of Technology , Pasadena , CA , USA","institution_ids":["https://openalex.org/I122411786"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029987551","display_name":"Matteo Ippoliti","orcid":"https://orcid.org/0000-0003-4095-0038","observed_orcids":["https://orcid.org/0000-0003-4095-0038"]},"institutions":[{"id":"https://openalex.org/I86519309","display_name":"The University of Texas at Austin","ror":"https://ror.org/00hj54h04","country_code":"US","type":"education","lineage":["https://openalex.org/I86519309"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ippoliti, Matteo","raw_affiliation_strings":["Department of Physics , The University of Texas at Austin , Austin , Texas 78712 , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Physics , The University of Texas at Austin , Austin , Texas 78712 , USA","institution_ids":["https://openalex.org/I86519309"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5081677188","display_name":"Wen Wei Ho","orcid":"https://orcid.org/0000-0001-7702-2900","observed_orcids":["https://orcid.org/0000-0001-7702-2900"]},"institutions":[{"id":"https://openalex.org/I115455960","display_name":"Centre for Quantum Technologies","ror":"https://ror.org/01mgdzc49","country_code":"SG","type":"facility","lineage":["https://openalex.org/I115455960"]},{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Ho, Wen Wei","raw_affiliation_strings":["Centre for Quantum Technologies , National University of Singapore , Singapore 117543","Department of Physics , National University of Singapore , Singapore 117551"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Centre for Quantum Technologies , National University of Singapore , Singapore 117543","institution_ids":["https://openalex.org/I115455960","https://openalex.org/I165932596"]},{"raw_affiliation_string":"Department of Physics , National University of Singapore , Singapore 117551","institution_ids":["https://openalex.org/I165932596"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5053736683","display_name":"Soonwon Choi","orcid":"https://orcid.org/0000-0002-1247-062X","observed_orcids":["https://orcid.org/0000-0002-1247-062X"]},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Choi, Soonwon","raw_affiliation_strings":["Center for Theoretical Physics -a Leinweber Institute , Massachusetts Institute of Technology , Cambridge , MA , USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Theoretical Physics -a Leinweber Institute , Massachusetts Institute of Technology , Cambridge , MA , USA","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11804","display_name":"Quantum many-body systems","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11804","display_name":"Quantum many-body systems","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.00039999998989515007,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quantum-thermalization","display_name":"quantum thermalization","score":0.8659999966621399},{"id":"https://openalex.org/keywords/maximum-entropy","display_name":"maximum entropy","score":0.7649999856948853},{"id":"https://openalex.org/keywords/quantum-information","display_name":"quantum information","score":0.6600000262260437},{"id":"https://openalex.org/keywords/quantum-many-body-dynamics","display_name":"quantum many-body dynamics","score":0.5789999961853027},{"id":"https://openalex.org/keywords/phase-transitions","display_name":"phase transitions","score":0.5600000023841858},{"id":"https://openalex.org/keywords/non-ergodicity","display_name":"ergodicity breaking","score":0.8569999933242798},{"id":"https://openalex.org/keywords/quantum-state-distribution","display_name":"quantum state distribution","score":0.5360000133514404}],"concepts":[{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5752000212669373},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.5619000196456909},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.508400022983551},{"id":"https://openalex.org/C201779956","wikidata":"https://www.wikidata.org/wiki/Q5426803","display_name":"Ergodicity","level":2,"score":0.4733000099658966},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.4652999937534332},{"id":"https://openalex.org/C33332235","wikidata":"https://www.wikidata.org/wiki/Q18362","display_name":"Theoretical physics","level":1,"score":0.43880000710487366},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.4246000051498413},{"id":"https://openalex.org/C190474826","wikidata":"https://www.wikidata.org/wiki/Q2495111","display_name":"Open quantum system","level":3,"score":0.4185999929904938},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.41510000824928284},{"id":"https://openalex.org/C183992945","wikidata":"https://www.wikidata.org/wiki/Q2495574","display_name":"Universality (dynamical systems)","level":2,"score":0.376800000667572},{"id":"https://openalex.org/C15184713","wikidata":"https://www.wikidata.org/wiki/Q7269042","display_name":"Quantum dynamics","level":3,"score":0.366100013256073},{"id":"https://openalex.org/C11255438","wikidata":"https://www.wikidata.org/wiki/Q7269085","display_name":"Quantum process","level":4,"score":0.35679998993873596},{"id":"https://openalex.org/C169699857","wikidata":"https://www.wikidata.org/wiki/Q2122243","display_name":"Quantum information","level":3,"score":0.35409998893737793},{"id":"https://openalex.org/C136269434","wikidata":"https://www.wikidata.org/wiki/Q1502341","display_name":"Quantum chaos","level":4,"score":0.3440000116825104},{"id":"https://openalex.org/C151342819","wikidata":"https://www.wikidata.org/wiki/Q62542","display_name":"Phase space","level":2,"score":0.33869999647140503},{"id":"https://openalex.org/C32848918","wikidata":"https://www.wikidata.org/wiki/Q845789","display_name":"Observable","level":2,"score":0.31859999895095825},{"id":"https://openalex.org/C80891386","wikidata":"https://www.wikidata.org/wiki/Q7269084","display_name":"Quantum phases","level":4,"score":0.31119999289512634},{"id":"https://openalex.org/C200741047","wikidata":"https://www.wikidata.org/wiki/Q1957758","display_name":"Integrable system","level":2,"score":0.31119999289512634},{"id":"https://openalex.org/C139356082","wikidata":"https://www.wikidata.org/wiki/Q7269038","display_name":"Quantum discord","level":4,"score":0.29670000076293945},{"id":"https://openalex.org/C15706264","wikidata":"https://www.wikidata.org/wiki/Q230883","display_name":"Quantum state","level":3,"score":0.29339998960494995},{"id":"https://openalex.org/C34829970","wikidata":"https://www.wikidata.org/wiki/Q845260","display_name":"Quantum statistical mechanics","level":3,"score":0.28850001096725464},{"id":"https://openalex.org/C28855332","wikidata":"https://www.wikidata.org/wiki/Q198099","display_name":"Quantization (signal processing)","level":2,"score":0.2874999940395355},{"id":"https://openalex.org/C190463098","wikidata":"https://www.wikidata.org/wiki/Q2081419","display_name":"Quantum technology","level":4,"score":0.2759999930858612},{"id":"https://openalex.org/C115047598","wikidata":"https://www.wikidata.org/wiki/Q54505","display_name":"Quantum field theory","level":2,"score":0.2752000093460083},{"id":"https://openalex.org/C71191651","wikidata":"https://www.wikidata.org/wiki/Q4218491","display_name":"Quantum nanoscience","level":5,"score":0.274399995803833},{"id":"https://openalex.org/C8492334","wikidata":"https://www.wikidata.org/wiki/Q7269100","display_name":"Quantum spacetime","level":4,"score":0.27309998869895935},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.25999999046325684},{"id":"https://openalex.org/C121040770","wikidata":"https://www.wikidata.org/wiki/Q215675","display_name":"Quantum entanglement","level":3,"score":0.2517000138759613}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.30248","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30248","pdf_url":"https://arxiv.org/pdf/2609.30248","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.30248","doi":"https://doi.org/10.48550/arxiv.2609.30248","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.30248","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.30248","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30248","pdf_url":"https://arxiv.org/pdf/2609.30248","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G149988304","display_name":null,"funder_award_id":"PHY-1734011","funder_id":"https://openalex.org/F4320334437","funder_display_name":"Center for Ultracold Atoms, Massachusetts Institute of Technology"},{"id":"https://openalex.org/G1797871914","display_name":null,"funder_award_id":"W911NF2010136","funder_id":"https://openalex.org/F4320333591","funder_display_name":"Multidisciplinary University Research Initiative"},{"id":"https://openalex.org/G2403330605","display_name":null,"funder_award_id":"PHY-1733907","funder_id":"https://openalex.org/F4320334224","funder_display_name":"Institute for Quantum Information and Matter, California Institute of Technology"},{"id":"https://openalex.org/G3077665131","display_name":"QLCI-CI: NSF Quantum Leap Challenge Institute for Present and Future Quantum Computing","funder_award_id":"2016245","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3209459412","display_name":null,"funder_award_id":"NRFF15-2023-0008","funder_id":"https://openalex.org/F4320320709","funder_display_name":"National Research Foundation Singapore"},{"id":"https://openalex.org/G3266171376","display_name":null,"funder_award_id":"DE-SC0025615","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G3871231613","display_name":"Center: Institute for Quantum Information and Matter: IQIM","funder_award_id":"2317110","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G507237395","display_name":"The Institute for Quantum Information and Matter: Advancing the Entanglement Frontier","funder_award_id":"1733907","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5854887369","display_name":"CAREER: Probing Quantum Matter using Programmable Quantum Simulators","funder_award_id":"2237244","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5884252700","display_name":null,"funder_award_id":"W911NF2010136","funder_id":"https://openalex.org/F4320338281","funder_display_name":"Army Research Office"},{"id":"https://openalex.org/G6020731938","display_name":null,"funder_award_id":"PHY-2317110","funder_id":"https://openalex.org/F4320334224","funder_display_name":"Institute for Quantum Information and Matter, California Institute of Technology"},{"id":"https://openalex.org/G6048591767","display_name":null,"funder_award_id":"NRF-NRFF15-2023-0008","funder_id":"https://openalex.org/F4320320709","funder_display_name":"National Research Foundation Singapore"},{"id":"https://openalex.org/G6512472947","display_name":null,"funder_award_id":"S24Q2d0009","funder_id":"https://openalex.org/F4320320709","funder_display_name":"National Research Foundation Singapore"},{"id":"https://openalex.org/G6906965799","display_name":null,"funder_award_id":"S24Q2d0009","funder_id":"https://openalex.org/F4320324089","funder_display_name":"Centre for Quantum Technologies"},{"id":"https://openalex.org/G7166665977","display_name":"Center for Ultracold Atoms","funder_award_id":"1734011","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7252073159","display_name":null,"funder_award_id":"PHY-2317134","funder_id":"https://openalex.org/F4320334437","funder_display_name":"Center for Ultracold Atoms, Massachusetts Institute of Technology"},{"id":"https://openalex.org/G7454671755","display_name":"Center: Center for Ultracold Atoms","funder_award_id":"2317134","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320320709","display_name":"National Research Foundation Singapore","ror":"https://ror.org/03cpyc314"},{"id":"https://openalex.org/F4320324089","display_name":"Centre for Quantum Technologies","ror":"https://ror.org/01mgdzc49"},{"id":"https://openalex.org/F4320332359","display_name":"Office of Science","ror":"https://ror.org/00mmn6b08"},{"id":"https://openalex.org/F4320333591","display_name":"Multidisciplinary University Research Initiative","ror":null},{"id":"https://openalex.org/F4320334224","display_name":"Institute for Quantum Information and Matter, California Institute of Technology","ror":null},{"id":"https://openalex.org/F4320334437","display_name":"Center for Ultracold Atoms, Massachusetts Institute of Technology","ror":"https://ror.org/053tmcn30"},{"id":"https://openalex.org/F4320337506","display_name":"Advanced Scientific Computing Research","ror":"https://ror.org/0012c7r22"},{"id":"https://openalex.org/F4320338281","display_name":"Army Research Office","ror":"https://ror.org/05epdh915"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214319259.pdf","grobid_xml":"https://content.openalex.org/works/W7214319259.grobid-xml"},"referenced_works_count":27,"referenced_works":["https://openalex.org/W4226185604","https://openalex.org/W4391835629","https://openalex.org/W4401725776","https://openalex.org/W4410288512","https://openalex.org/W4416222423","https://openalex.org/W4416385171","https://openalex.org/W4416596122","https://openalex.org/W7092297281","https://openalex.org/W7118640161","https://openalex.org/W7125822252","https://openalex.org/W7164023358","https://openalex.org/W7164176145","https://openalex.org/W7165030055","https://openalex.org/W7166340074","https://openalex.org/W7166708766","https://openalex.org/W7167625365","https://openalex.org/W7169591413","https://openalex.org/W7170061391","https://openalex.org/W7170599505","https://openalex.org/W7202341376","https://openalex.org/W7204206971","https://openalex.org/W7208708955","https://openalex.org/W7208781077","https://openalex.org/W7212051680","https://openalex.org/W7212154981","https://openalex.org/W7213084098","https://openalex.org/W7214227728"],"related_works":["https://openalex.org/W4392971707","https://openalex.org/W4402705696","https://openalex.org/W4416831475","https://openalex.org/W4416055997","https://openalex.org/W4313330581","https://openalex.org/W4396822049","https://openalex.org/W7223273054","https://openalex.org/W4417200338","https://openalex.org/W2995623481","https://openalex.org/W7164023358"],"abstract_inverted_index":{"Recent":[0],"advances":[1],"in":[2,14,45,80,120,133,140],"quantum":[3,15,39,63,70,81,116,134,141],"simulation":[4],"have":[5],"enabled":[6],"the":[7,34,91],"discovery":[8],"of":[9,12,36,38,69,86,94,114,127],"novel":[10],"forms":[11,93,126],"universality":[13],"many-body":[16],"dynamics.":[17],"In":[18],"this":[19],"Review,":[20],"we":[21],"discuss":[22,103],"deep":[23,121],"thermalization":[24,71,122,135],"and":[25,72,84,106,136,138],"Hilbert":[26],"space":[27],"ergodicity--two":[28],"recently":[29],"discovered":[30],"phenomena":[31],"characterized":[32],"by":[33],"emergence":[35],"distributions":[37],"states":[40],"that":[41,89],"are":[42],"\"maximally":[43],"random\"":[44],"a":[46,51,76],"precise":[47],"sense.":[48],"They":[49],"offer":[50],"new":[52],"perspective":[53],"on":[54],"how":[55],"irreversible":[56],"statistical":[57],"mechanics":[58],"arises":[59],"from":[60],"reversible":[61],"unitary":[62],"dynamics,":[64],"going":[65],"beyond":[66,112],"conventional":[67],"theories":[68],"equilibration.":[73],"We":[74,102],"review":[75],"unifying":[77],"framework,":[78],"rooted":[79],"information":[82,142],"theory":[83],"principles":[85],"maximum":[87],"entropy,":[88],"explains":[90],"different":[92],"ergodicity":[95],"which":[96],"arise":[97],"under":[98],"various":[99],"physical":[100],"constraints.":[101],"open":[104],"questions":[105],"active":[107],"research":[108],"directions,":[109],"including":[110],"generalizations":[111],"ensembles":[113],"pure":[115],"states,":[117],"phase":[118],"transitions":[119],"tied":[123],"to":[124,130],"deeper":[125],"ergodicity-breaking,":[128],"connections":[129],"broader":[131],"topics":[132],"ergodicity,":[137],"applications":[139],"science,":[143],"such":[144],"as":[145],"for":[146],"benchmarking":[147],"or":[148],"tomography.":[149]},"counts_by_year":[{"year":2026,"cited_by_count":3}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-26T00:00:00"}
