{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:39:46Z","timestamp":1760402386728,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2020,4,28]],"date-time":"2020-04-28T00:00:00Z","timestamp":1588032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11461061; 11861057"],"award-info":[{"award-number":["11461061; 11861057"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>As a discrete-time quantum walk model on the one-dimensional integer lattice    Z   , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897\u20131908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer     d \u2265 2    , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice     Z d    , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.<\/jats:p>","DOI":"10.3390\/e22050504","type":"journal-article","created":{"date-parts":[[2020,4,28]],"date-time":"2020-04-28T05:05:32Z","timestamp":1588050332000},"page":"504","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2916-8478","authenticated-orcid":false,"given":"Ce","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1173-8118","authenticated-orcid":false,"given":"Caishi","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1103\/PhysRevA.48.1687","article-title":"Quantum random walks","volume":"48","author":"Aharonov","year":"1993","journal-title":"Phys. Rev. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1080\/00107151031000110776","article-title":"Quantum random walks: An introductory overview","volume":"44","author":"Kempe","year":"2003","journal-title":"Contemp. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1007\/s11128-012-0432-5","article-title":"Quantum walks: A comprehensive review","volume":"11","year":"2012","journal-title":"Quantum Inf. Process."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s11128-017-1737-1","article-title":"Stationary amplitudes of quantum walks on the higher-dimensional integer lattice","volume":"16","author":"Komatsu","year":"2017","journal-title":"Quantum Inf. Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1179","DOI":"10.2969\/jmsj\/1150287309","article-title":"A new type of limit theorems for one-dimensional quantum random walks","volume":"57","author":"Konno","year":"2005","journal-title":"J. Math. Soc. Jpn."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/978-3-540-69365-9_7","article-title":"Quantum walks","volume":"Volume 1954","author":"Franz","year":"2008","journal-title":"Quantum Potential Theory, Lecture Notes in Mathematics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"053528","DOI":"10.1063\/1.3431028","article-title":"Discrete-time quantum Bernoulli noises","volume":"51","author":"Wang","year":"2010","journal-title":"J. Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"023502","DOI":"10.1063\/1.4939920","article-title":"Quantum Markov semigroups constructed from quantum Bernoulli noises","volume":"57","author":"Wang","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1897","DOI":"10.1007\/s11128-016-1259-2","article-title":"Quantum walk in terms of quantum Bernoulli noises","volume":"15","author":"Wang","year":"2016","journal-title":"Quantum Inf. Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1169","DOI":"10.1017\/S0960129507006354","article-title":"Decoherence in quantum walks\u2014A review","volume":"17","author":"Kendon","year":"2007","journal-title":"Math. Struct. Comput. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2745","DOI":"10.1088\/0305-4470\/35\/12\/304","article-title":"Quantum walks in higher dimensions","volume":"35","author":"Mackay","year":"2002","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1142\/S0219749908004456","article-title":"Limit theorems for quantum walks driven by many coins","volume":"6","author":"Segawa","year":"2008","journal-title":"Int. J. Quantum Inf."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"042337","DOI":"10.1103\/PhysRevA.84.042337","article-title":"Alternate two-dimensional quantum walk with a single-qubit coin","volume":"84","author":"Gettrick","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s11128-017-1653-4","article-title":"Localization of a multi-dimensional quantum walk with one defect","volume":"16","author":"Fuda","year":"2017","journal-title":"Quantum Inf. Process."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"052323","DOI":"10.1103\/PhysRevA.69.052323","article-title":"Localization of two-dimensional quantum walks","volume":"69","author":"Inui","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"042315","DOI":"10.1103\/PhysRevA.67.042315","article-title":"Decoherence can be useful in quantum walks","volume":"67","author":"Kendon","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"032304","DOI":"10.1103\/PhysRevA.67.032304","article-title":"Quantum random walks with decoherent coins","volume":"67","author":"Brun","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_18","first-page":"741","article-title":"Crossovers induced by discrete-time quantum walks","volume":"11","author":"Chisaki","year":"2011","journal-title":"Quantum Inf. Comput."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/504\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:32:09Z","timestamp":1760362329000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/5\/504"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,28]]},"references-count":18,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2020,5]]}},"alternative-id":["e22050504"],"URL":"https:\/\/doi.org\/10.3390\/e22050504","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,4,28]]}}}