{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,23]],"date-time":"2026-05-23T06:07:24Z","timestamp":1779516444485,"version":"3.53.1"},"reference-count":55,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"6","license":[{"start":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T00:00:00Z","timestamp":1780272000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Trans. Inform. Theory"],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1109\/tit.2026.3678348","type":"journal-article","created":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T19:49:36Z","timestamp":1774640976000},"page":"4041-4055","source":"Crossref","is-referenced-by-count":0,"title":["Geometrical Constructions of Purity Testing Protocols and Their Applications to Quantum Communication"],"prefix":"10.1109","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2839-0472","authenticated-orcid":false,"given":"R\u00f3bert","family":"Tr\u00e9nyi","sequence":"first","affiliation":[{"name":"Department of Theoretical Physics, University of the Basque Country (UPV\/EHU), Bilbao, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4845-2084","authenticated-orcid":false,"given":"Simeon","family":"Ball","sequence":"additional","affiliation":[{"name":"Departament de Matem&#x00E0;tiques, Universitat Polit&#x00E8;cnica de Catalunya, Barcelona, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6399-7771","authenticated-orcid":false,"given":"David G.","family":"Glynn","sequence":"additional","affiliation":[{"name":"College of Science and Engineering, Flinders University, Adelaide, SA, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0330-6828","authenticated-orcid":false,"given":"Marcos","family":"Curty","sequence":"additional","affiliation":[{"name":"Department of Signal Theory and Communications, and the atlanTTic Research Center, Vigo Quantum Communication Center, the Escuela de Ingenier&#x00ED;a de Telecomunicac&#x00ED;on, University of Vigo, Vigo, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"263","reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.81.865"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physrep.2009.02.004"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1038\/s42254-018-0003-5"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.92.217903"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.94.020501"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.70.1895"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.95.045006"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1126\/science.aam9288"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.53.2046"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.76.722"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.54.3824"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.77.2818"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.78.574"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.75.010303"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1038\/nphoton.2015.195"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1126\/science.aan0070"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1109\/SFCS.2002.1181969"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.72.012338"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.131.190601"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1109\/CCC.2002.1004345"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1126\/science.283.5410.2050"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.85.441"},{"key":"ref23","volume-title":"Projective Geometry: From Foundations to Applications","author":"Beutelspacher","year":"1998"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198502951.001.0001"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780199298853.001.0001"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1016\/s0924-6509(08)x7030-8"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2014.2317312"},{"key":"ref28","article-title":"The universal composable security of quantum message authentication with key recyling","author":"Hayden","year":"2016","journal-title":"arXiv:1610.09434"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.72.042309"},{"key":"ref30","article-title":"Quantum ciphertext authentication and key recycling with the trap code","author":"Dulek","year":"2018","journal-title":"arXiv:1804.02237"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-63715-0_12"},{"key":"ref32","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-56617-7_11","article-title":"Quantum authentication and encryption with key recycling","author":"Fehr","year":"2017"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1109\/FOCS.2006.68"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1007\/11426639_17"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.22331\/q-2020-09-30-337"},{"key":"ref36","article-title":"Stabilizer codes and quantum error correction","author":"Gottesman","year":"1997"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/70\/8\/R03"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1038\/nature03074"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1038\/nature14270"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.129.030501"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1201\/b15006"},{"key":"ref42","volume-title":"Finite Fields","author":"Lidl","year":"2009"},{"key":"ref43","volume-title":"Introduction to Algebraic Theories","author":"Albert","year":"1937"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.4171\/aihpd\/160"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1090\/psapm\/068\/2762145"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1017\/cbo9780511976667"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316257449"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1137\/0209059"},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-62012-6"},{"key":"ref50","article-title":"Interactive proofs for quantum computations","author":"Aharonov","year":"2008","journal-title":"arXiv:0810.5375"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-49175-2_4"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-40084-1_20"},{"key":"ref53","article-title":"Private quantum channels and the cost of randomizing quantum information","author":"Mosca","year":"2000","journal-title":"arXiv:quant-ph\/0003101"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.26421\/QIC3.6-2"},{"key":"ref55","volume-title":"Bounds on the Minimum Distance of Linear Codes and Quantum Codes","author":"Grassl","year":"2003"}],"container-title":["IEEE Transactions on Information Theory"],"original-title":[],"link":[{"URL":"http:\/\/xplorestaging.ieee.org\/ielx8\/18\/11534061\/11457055.pdf?arnumber=11457055","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,23]],"date-time":"2026-05-23T05:46:05Z","timestamp":1779515165000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/11457055\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6]]},"references-count":55,"journal-issue":{"issue":"6"},"URL":"https:\/\/doi.org\/10.1109\/tit.2026.3678348","relation":{},"ISSN":["0018-9448","1557-9654"],"issn-type":[{"value":"0018-9448","type":"print"},{"value":"1557-9654","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6]]}}}