<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.3 20210610//EN" "JATS-journalpublishing1-3.dtd">
<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="en"><front><journal-meta><journal-id journal-id-type="publisher-id">najo</journal-id><journal-title-group><journal-title xml:lang="en">Nanosystems: Physics, Chemistry, Mathematics</journal-title><trans-title-group xml:lang="ru"><trans-title>Наносистемы: физика, химия, математика</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">2220-8054</issn><issn pub-type="epub">2305-7971</issn><publisher><publisher-name>Университет ИТМО</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.17586/2220-8054-2026-17-4-491-494</article-id><article-id custom-type="elpub" pub-id-type="custom">najo-1911</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="en"><subject>PHYSICS</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>ФИЗИКА</subject></subj-group></article-categories><title-group><article-title>Entropy for two weakly coupled quantum waveguides: asymptotic approach</article-title><trans-title-group xml:lang="ru"><trans-title>Энтропия для двух слабо связанных квантовых волноводов: асимптотический подход</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-4320-6917</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Трифанова</surname><given-names>Е. С.</given-names></name><name name-style="western" xml:lang="en"><surname>Trifanova</surname><given-names>E. S.</given-names></name></name-alternatives><bio xml:lang="en"><p>Ekaterina S. Trifanova – Information Technologies and Programming Faculty </p><p>Kronverkskiy, 49, 197101, St. Petersburg</p></bio><email xlink:type="simple">etrifanova@gmail.com</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Юрова</surname><given-names>Т. С.</given-names></name><name name-style="western" xml:lang="en"><surname>Yurova</surname><given-names>T. S.</given-names></name></name-alternatives><bio xml:lang="en"><p>Tatiana S. Yurova – Institute of Mathematics </p><p>Kronverkskiy, 49, 197101, St. Petersburg</p></bio><email xlink:type="simple">tsyurova@itmo.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-2115-2479</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Блинова</surname><given-names>И. B.</given-names></name><name name-style="western" xml:lang="en"><surname>Blinova</surname><given-names>I. V.</given-names></name></name-alternatives><bio xml:lang="en"><p>Irina V. Blinova – Institute of Mathematics </p><p>Kronverkskiy, 49, 197101, St. Petersburg</p></bio><email xlink:type="simple">rin-a@yandex.ru</email><xref ref-type="aff" rid="aff-1"/></contrib><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-5251-5327</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Попов</surname><given-names>И. Ю.</given-names></name><name name-style="western" xml:lang="en"><surname>Popov</surname><given-names>I. Y.</given-names></name></name-alternatives><bio xml:lang="en"><p>Igor Y. Popov – Institute of Mathematics </p><p>Kronverkskiy, 49, 197101, St. Petersburg</p></bio><email xlink:type="simple">popov1955@gmail.com</email><xref ref-type="aff" rid="aff-1"/></contrib></contrib-group><aff xml:lang="en" id="aff-1"><institution>ITMO University</institution><country>Russian Federation</country></aff><pub-date pub-type="collection"><year>2026</year></pub-date><pub-date pub-type="epub"><day>31</day><month>08</month><year>2026</year></pub-date><volume>17</volume><issue>4</issue><fpage>491</fpage><lpage>494</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Trifanova E.S., Yurova T.S., Blinova I.V., Popov I.Y., 2026</copyright-statement><copyright-year>2026</copyright-year><copyright-holder xml:lang="ru">Трифанова Е.С., Юрова Т.С., Блинова И.B., Попов И.Ю.</copyright-holder><copyright-holder xml:lang="en">Trifanova E.S., Yurova T.S., Blinova I.V., Popov I.Y.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://nanojournal.ifmo.ru/jour/article/view/1911">https://nanojournal.ifmo.ru/jour/article/view/1911</self-uri><abstract><p>Electron in a system of two plane quantum waveguides coupled through small window is considered. The method of matching the asymptotic expansions of the boundary problem solution is used. The dependence of entropy on the window width is described. for the case of small window in the framework of the asymptotic approach.</p></abstract><trans-abstract xml:lang="ru"><p>Рассматривается электрон в системе из двух плоских квантовых волноводов, связанных через малое окно. Используется метод согласования асимптотических разложений решения краевой задачи. Описывается зависимость энтропии от ширины окна для случая малого окна в рамках асимптотического подхода.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>квантовые волноводы</kwd><kwd>энтропия</kwd><kwd>асимптотика</kwd></kwd-group><kwd-group xml:lang="en"><kwd>quantum waveguides</kwd><kwd>entropy</kwd><kwd>asymptotics</kwd></kwd-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Fu J., Tang S.-F. Quantum computations with transverse modes of an optical field propagating in waveguides. Chin. Phys. Lett., 2003, 20, 1426.</mixed-citation><mixed-citation xml:lang="en">Fu J., Tang S.-F. Quantum computations with transverse modes of an optical field propagating in waveguides. Chin. Phys. Lett., 2003, 20, 1426.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Gavrilov M.I., Gortinskaya L.V., Pestov A.A., Popov I.Yu., Tesovskaya E.S. Quantum computer elements based on coupled quantum waveguides. Phys. Part. Nucl. Lett., 2007, 4 (2), P. 237–243.</mixed-citation><mixed-citation xml:lang="en">Gavrilov M.I., Gortinskaya L.V., Pestov A.A., Popov I.Yu., Tesovskaya E.S. Quantum computer elements based on coupled quantum waveguides. Phys. Part. Nucl. Lett., 2007, 4 (2), P. 237–243.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Kok P., Williams C.P., Dowling J.P. Construction of a quantum repeater with linear optics. Phys. Rev. A, 2003, 68, 022301.</mixed-citation><mixed-citation xml:lang="en">Kok P., Williams C.P., Dowling J.P. Construction of a quantum repeater with linear optics. Phys. Rev. A, 2003, 68, 022301.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Knill E., Laflamme R., Milburn G.J. A scheme for efficient quantum computation with linear optics. Nature, 2001, 409, P. 46–52.</mixed-citation><mixed-citation xml:lang="en">Knill E., Laflamme R., Milburn G.J. A scheme for efficient quantum computation with linear optics. Nature, 2001, 409, P. 46–52.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Guselnikov M.S., Gaidash A.A., Miroshnichenko G.P., Kozubov A.V. Properties of multi-moded phase-randomized coherent states. Nanosystems: Physics, Chemistry, Mathematics, 2025, 16 (3), P. 311–316.</mixed-citation><mixed-citation xml:lang="en">Guselnikov M.S., Gaidash A.A., Miroshnichenko G.P., Kozubov A.V. Properties of multi-moded phase-randomized coherent states. Nanosystems: Physics, Chemistry, Mathematics, 2025, 16 (3), P. 311–316.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Stepanov I., Goncharov R., Kiselev A.D. Sub-Poissonian light in fluctuating thermal-loss bosonic channels. Nanosystems: Physics, Chemistry, Mathematics, 2025, 16 (3), P. 333–342.</mixed-citation><mixed-citation xml:lang="en">Stepanov I., Goncharov R., Kiselev A.D. Sub-Poissonian light in fluctuating thermal-loss bosonic channels. Nanosystems: Physics, Chemistry, Mathematics, 2025, 16 (3), P. 333–342.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Latypov I.Z., Chistyakov V.V., Fadeev M.A., et al. Hybrid quantum communication protocol for fiber and atmosphere channel. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15, P. 654–657.</mixed-citation><mixed-citation xml:lang="en">Latypov I.Z., Chistyakov V.V., Fadeev M.A., et al. Hybrid quantum communication protocol for fiber and atmosphere channel. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15, P. 654–657.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Pachos J.K. Introduction to Topological Quantum Computation. Cambridge, Cambridge University Press, 2012.</mixed-citation><mixed-citation xml:lang="en">Pachos J.K. Introduction to Topological Quantum Computation. Cambridge, Cambridge University Press, 2012.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Popov I.Y., Trifanova E.S., Bagmutov A.S., Lytaev A.A. Boundary composed of small Helmholtz resonators: asymptotic approach. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15 (6), P. 736–741.</mixed-citation><mixed-citation xml:lang="en">Popov I.Y., Trifanova E.S., Bagmutov A.S., Lytaev A.A. Boundary composed of small Helmholtz resonators: asymptotic approach. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15 (6), P. 736–741.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Melikhova A.S., Popov A.I., Blinova I.V., Popov I.Y. Mathematical model of weakly coupled spherical resonator chains under the influence of external magnetic field. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15 (2), P. 155–159.</mixed-citation><mixed-citation xml:lang="en">Melikhova A.S., Popov A.I., Blinova I.V., Popov I.Y. Mathematical model of weakly coupled spherical resonator chains under the influence of external magnetic field. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15 (2), P. 155–159.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Exner P., Vugalter S. Bound state asymptotic estimates for window-coupled Dirichlet strips and layers. J. Phys. A: Math. Gen., 1997, 30, P. 7863– 7878.</mixed-citation><mixed-citation xml:lang="en">Exner P., Vugalter S. Bound state asymptotic estimates for window-coupled Dirichlet strips and layers. J. Phys. A: Math. Gen., 1997, 30, P. 7863– 7878.</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Borisov D. Discrete spectrum of a pair of non-symmetric waveguides coupled by a window. Sbornik Mathematics, 2006, 197 (4), P. 475–504.</mixed-citation><mixed-citation xml:lang="en">Borisov D. Discrete spectrum of a pair of non-symmetric waveguides coupled by a window. Sbornik Mathematics, 2006, 197 (4), P. 475–504.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Dittrich J., Kriz J. Bound states in straight quantum waveguides with combined boundary conditions. J. Math. Phys., 2002, 43 (8), P. 3892–3915.</mixed-citation><mixed-citation xml:lang="en">Dittrich J., Kriz J. Bound states in straight quantum waveguides with combined boundary conditions. J. Math. Phys., 2002, 43 (8), P. 3892–3915.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Figotin A. Analytic theory of coupled-cavity traveling wave tubes. J. Math. Phys., 2023, 64, 042705.</mixed-citation><mixed-citation xml:lang="en">Figotin A. Analytic theory of coupled-cavity traveling wave tubes. J. Math. Phys., 2023, 64, 042705.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Popov I.Yu. Asymptotics of bound state for laterally coupled waveguides. Rep. on Math. Phys., 1999, 43 (3), P. 427–437.</mixed-citation><mixed-citation xml:lang="en">Popov I.Yu. Asymptotics of bound state for laterally coupled waveguides. Rep. on Math. Phys., 1999, 43 (3), P. 427–437.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Popov I.Yu. Asymptotics of Bound States and Bands for Waveguides Coupled Through Small Windows. Appl. Math. Lett., 2001, 14, P. 109–113.</mixed-citation><mixed-citation xml:lang="en">Popov I.Yu. Asymptotics of Bound States and Bands for Waveguides Coupled Through Small Windows. Appl. Math. Lett., 2001, 14, P. 109–113.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Chogle F., Teklu B. Quantum Information Measures of a Dirichlet Waveguide with Neumann Window(s). Adv. Quant. Technol., 2026, 9, e70326.</mixed-citation><mixed-citation xml:lang="en">Chogle F., Teklu B. Quantum Information Measures of a Dirichlet Waveguide with Neumann Window(s). Adv. Quant. Technol., 2026, 9, e70326.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Chogle F., Olendski O. Bound-state evolution of Dirichlet waveguide with Neumann window(s) in transverse electric fields. Phys. Scr., 2024, 99, 095509.</mixed-citation><mixed-citation xml:lang="en">Chogle F., Olendski O. Bound-state evolution of Dirichlet waveguide with Neumann window(s) in transverse electric fields. Phys. Scr., 2024, 99, 095509.</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Olendski O. Comparative analysis of electric field influence on the quantum wells with different boundary conditions. I. Energy spectrum, quantum information entropy and polarization. Ann. Phys. (Berlin), 2015, 527 (3–4), P. 278–295.</mixed-citation><mixed-citation xml:lang="en">Olendski O. Comparative analysis of electric field influence on the quantum wells with different boundary conditions. I. Energy spectrum, quantum information entropy and polarization. Ann. Phys. (Berlin), 2015, 527 (3–4), P. 278–295.</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
