{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,16]],"date-time":"2026-09-16T02:23:45Z","timestamp":1789525425958,"version":"build-2803163510"},"reference-count":51,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,31]],"date-time":"2022-12-31T00:00:00Z","timestamp":1672444800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We propose a model of a quantum N-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of p-adic numbers. As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an N-dimensional p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)\u2014a suitable p-adic counterpart of a POVM in a complex Hilbert space\u2014as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where N=2, thus providing a description of p-adic qubit states and 2-dimensional SOVMs. The analogies\u2014but also the non-trivial differences\u2014with respect to the qubit states of standard quantum mechanics are then analyzed.<\/jats:p>","DOI":"10.3390\/e25010086","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T04:51:22Z","timestamp":1672635082000},"page":"86","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A p-Adic Model of Quantum States and the p-Adic Qubit"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4298-8275","authenticated-orcid":false,"given":"Paolo","family":"Aniello","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica \u201cEttore Pancini\u201d, Universit\u00e0 di Napoli \u201cFederico II\u201d, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3797-3987","authenticated-orcid":false,"given":"Stefano","family":"Mancini","sequence":"additional","affiliation":[{"name":"School of Science and Technology, University of Camerino, Via Madonna delle Carceri, 9, I-62032 Camerino, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Via A. Pascoli, I-06123 Perugia, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9563-6471","authenticated-orcid":false,"given":"Vincenzo","family":"Parisi","sequence":"additional","affiliation":[{"name":"School of Science and Technology, University of Camerino, Via Madonna delle Carceri, 9, I-62032 Camerino, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Via A. Pascoli, I-06123 Perugia, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Vladimirov, V.S., Volovich, I.V., and Zelenov, E.I. (1994). p-Adic Analysis and Mathematical Physics, World Scientific. [1st ed.].","DOI":"10.1142\/1581"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1007\/BF01218590","article-title":"p-adic quantum mechanics","volume":"123","author":"Vladimirov","year":"1989","journal-title":"Commun. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2","DOI":"10.12942\/lrr-2013-2","article-title":"Minimal Length Scale Scenarios for Quantum Gravity","volume":"16","author":"Hossenfelder","year":"2013","journal-title":"Living Rev. Relativ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2854","DOI":"10.1063\/1.528468","article-title":"Quantum mechanics on p-adic fields","volume":"30","author":"Ruelle","year":"1989","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5133","DOI":"10.1142\/S0217751X8900217X","article-title":"Quantum mechanics with p-adic numbers","volume":"4","author":"Meurice","year":"1989","journal-title":"Int. J. Mod. Phys. A"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Khrennikov, A.Y. (1994). p-Adic Valued Distributions in Mathematical Physics, Kluwer Academic Publishers. [1st ed.].","DOI":"10.1007\/978-94-015-8356-5"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Khrennikov, A.Y. (1997). Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers. [1st ed.].","DOI":"10.1007\/978-94-009-1483-4"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1665","DOI":"10.1142\/S021797929600074X","article-title":"p-adic Hilbert space representation of quantum systems with an infinite number of degrees of freedom","volume":"10","author":"Albeverio","year":"1996","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1063\/1.529353","article-title":"p-adic quantum mechanics with p-adic valued functions","volume":"32","author":"Khrennikov","year":"1991","journal-title":"J. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1134\/S2070046610010061","article-title":"Number theory as the ultimate physical theory","volume":"2","author":"Volovich","year":"2010","journal-title":"p-Adic Numbers Ultr. Anal. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/BF01016162","article-title":"Real non-Archimedean structure of spacetime","volume":"86","author":"Khrennikov","year":"1991","journal-title":"Theor. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1070\/RM1990v045n04ABEH002378","article-title":"Mathematical methods of non-Archimedean physics","volume":"45","author":"Khrennikov","year":"1990","journal-title":"Russ. Math. Surv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1142\/S0217732388000763","article-title":"On p-adic functional integrals","volume":"3","author":"Parisi","year":"1988","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_14","first-page":"1380","article-title":"Representation of second quantization over non-Archimedean number fields","volume":"314","author":"Khrennikov","year":"1990","journal-title":"Dokl. Akad. Nauk. SSSR"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1134\/S2070046609020010","article-title":"p-adic valued quantization","volume":"1","author":"Albeverio","year":"2009","journal-title":"p-Adic Numbers Ultr. Anal. Appl."},{"key":"ref_16","first-page":"46","article-title":"Statistical interpretation of p-adic valued quantum field theory","volume":"328","author":"Khrennikov","year":"1993","journal-title":"Dokl. Akad. Nauk. SSSR"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/BF02583040","article-title":"A representation of quantum field Hamiltonian in a p-adic Hilbert space","volume":"112","author":"Albeverio","year":"1997","journal-title":"Teor. Mat. Fiz."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/0370-2693(88)91569-9","article-title":"On the p-adic five-point function","volume":"203","author":"Marinari","year":"1998","journal-title":"Phys. Lett. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1156","DOI":"10.1103\/PhysRevLett.52.1156","article-title":"Nature of the spin-glass phase","volume":"52","author":"Parisi","year":"1984","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1103\/RevModPhys.58.765","article-title":"Ultrametricity for physicists","volume":"58","author":"Rammal","year":"1986","journal-title":"Rev. Mod. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1007\/s100510051063","article-title":"p-adic numbers and replica symmetry breaking","volume":"14","author":"Parisi","year":"2000","journal-title":"Eur. Phys. J. B Condens. Matter Complex Syst."},{"key":"ref_22","first-page":"450","article-title":"Segmentation of images in p-adic and Euclidean metrics","volume":"64","author":"Khrennikov","year":"2001","journal-title":"Dokl. Math."},{"key":"ref_23","first-page":"159","article-title":"Representation and compression of images with the aid of the m-adic coordinate system","volume":"387","author":"Khrennikov","year":"2002","journal-title":"Dokl. Akad. Nauk."},{"key":"ref_24","first-page":"373","article-title":"Compression of images with the aid of representation by p-adic maps and approximation by Mahler\u2019s polynomials","volume":"69","author":"Khrennikov","year":"2004","journal-title":"Dokl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Anashin, V., and Khrennikov, A. (2009). Applied Algebraic Dynamics, Walter de Gruyter & Co.. [1st ed.].","DOI":"10.1515\/9783110203011"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/BF02113290","article-title":"Uniformly distributed sequences of p-adic integers","volume":"55","author":"Anashin","year":"1994","journal-title":"Math. Notes"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.1007\/BF02355442","article-title":"Uniformly distributed sequences in computer algebra, or how to construct program generators of random numbers","volume":"89","author":"Anashin","year":"1998","journal-title":"J. Math. Sci."},{"key":"ref_28","unstructured":"Anashin, V.S. (2004). Pseudorandom number generation by p-adic ergodic transformations. arXiv."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1134\/S1063779620040814","article-title":"Entropy Gain in p-Adic Quantum Channels","volume":"51","author":"Zelenov","year":"2020","journal-title":"Phys. Part. Nuclei"},{"key":"ref_30","unstructured":"Zelenov, E.I. (2022, November 07). Coherent States of the p-adic Heisenberg Group, Heterodyne Measurements and Entropy Uncertainty Relation. [Conference Presentation] New Trends inMathematical Physics 2022, Stekl. Math. Inst. Moscow. Available online: https:\/\/www.mathnet.ru\/php\/presentation.phtml?presentid=36592&option_lang=eng."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Aniello, P., Mancini, S., and Parisi, V. (2022). Trace class operators and states in p-adic quantum mechanics. arXiv.","DOI":"10.1063\/5.0134965"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Folland, G.B. (2015). A Course in Abstract Harmonic Analysis, CRC Press. [2nd ed.].","DOI":"10.1201\/b19172"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Gouv\u00eaa, F.Q. (1997). p-Adic Numbers, an Introduction, Springer. [2nd ed.].","DOI":"10.1007\/978-3-642-59058-0"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Robert, A.M. (2000). A Course in p-Adic Analysis, Springer. [1st ed.].","DOI":"10.1007\/978-1-4757-3254-2"},{"key":"ref_35","unstructured":"Dummit, D.S., and Foote, R.M. (2004). Abstract Algebra, John Wiley & Sons. [3rd ed.]."},{"key":"ref_36","unstructured":"Folland, G.B. (1999). Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons. [2nd ed.]."},{"key":"ref_37","unstructured":"Van Rooij, A.C.M. (1978). Non-Archimedean Functional Analysis, Marcel Dekker. [1st ed.]."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Perez-Garcia, C., and Schikhof, W.H. (2010). Locally Convex Spaces over Non-Archimedean Valued Fields, Cambridge University Press. [1st ed.].","DOI":"10.1017\/CBO9780511729959"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"180","DOI":"10.2307\/1969224","article-title":"On p-adic Hilbert spaces","volume":"48","author":"Kalisch","year":"1947","journal-title":"Ann. Math."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"787","DOI":"10.36045\/bbms\/1197908895","article-title":"Non-Archimedean Hilbert like spaces","volume":"14","author":"Aguayo","year":"2007","journal-title":"Bull. Belg. Math. Soc. Simon Stevin"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1023\/A:1006219101760","article-title":"Non-Archimedean analogues of orthogonal and symmetric operators and p-adic quantization","volume":"57","author":"Albeverio","year":"1999","journal-title":"Acta Appl. Math."},{"key":"ref_42","unstructured":"Narici, L., Beckenstein, E., and Bachman, G. (1971). Functional Analysis and Valuation Theory, Marcel Dekker. [1st ed.]."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Diagana, T., and Ramaroson, F. (2016). Non-Archimedean Operator Theory, Springer.","DOI":"10.1007\/978-3-319-27323-5"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Strocchi, F. (2008). An Introduction to the Mathematical Structure of Quantum Mechanics: A Short Course for Mathematicians, World Scientific. [2nd ed.].","DOI":"10.1142\/7038"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Moretti, V. (2017). Spectral Theory and Quantum Mechanics, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-70706-8"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Bratteli, O., and Robinson, D.W. (1987). Operator Algebras and Quantum Statistical Mechanics 1: C*-and W*-Algebras, Symmetry Groups, Decomposition of States, Springer. [2nd ed.].","DOI":"10.1007\/978-3-662-02520-8"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Holevo, A.S. (2001). Statistical Structure of Quantum Theory, Springer. [1st ed.].","DOI":"10.1007\/3-540-44998-1"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Busch, P., Lahti, P., Pellonp\u00e4\u00e4, J.-P., and Ylinen, K. (2016). Quantum Measurement, Springer. [1st ed.].","DOI":"10.1007\/978-3-319-43389-9"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Heinosaari, T., and Ziman, M. (2012). The Mathematical Language of Quantum Theory, Cambridge University Press. [1st ed.].","DOI":"10.1017\/CBO9781139031103"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.1007\/BF01015763","article-title":"p-adic probability theory and its applications. The principle of statistical stabilization of frequencies","volume":"97","author":"Khrennikov","year":"1993","journal-title":"Theor. Math. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1137\/S0040585X97978920","article-title":"Interpretations of probability and their p-adic extensions","volume":"46","author":"Khrennikov","year":"2002","journal-title":"Theory Probab. Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/86\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:49:15Z","timestamp":1760147355000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/86"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,31]]},"references-count":51,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010086"],"URL":"https:\/\/doi.org\/10.3390\/e25010086","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,31]]}}}