{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T19:20:03Z","timestamp":1772738403567,"version":"3.50.1"},"reference-count":59,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"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>Quantum memory effects can be qualitatively understood as a consequence of an environment-to-system backflow of information. Here, we analyze and compare how this concept is interpreted and implemented in different approaches to quantum non-Markovianity. We study a nonoperational approach, defined by the distinguishability between two system states characterized by different initial conditions, and an operational approach, which is defined by the correlation between different outcomes associated to successive measurement processes performed over the system of interest. The differences, limitations, and vantages of each approach are characterized in detail by considering diverse system\u2013environment models and dynamics. As a specific example, we study a non-Markovian depolarizing map induced by the interaction of the system of interest with an environment characterized by incoherent and coherent self-dynamics.<\/jats:p>","DOI":"10.3390\/e24050649","type":"journal-article","created":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T13:10:26Z","timestamp":1651756226000},"page":"649","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Quantum Non-Markovian Environment-to-System Backflows of Information: Nonoperational vs. Operational Approaches"],"prefix":"10.3390","volume":"24","author":[{"given":"Adri\u00e1n A.","family":"Budini","sequence":"first","affiliation":[{"name":"Consejo Nacional de Investigaciones Cient\u00edficas y T\u00e9cnicas (CONICET), Centro At\u00f3mico Bariloche, Avenida E. Bustillo Km 9.5, Bariloche 8400, Argentina"},{"name":"Universidad Tecnol\u00f3gica Nacional (UTN-FRBA), Fanny Newbery 111, Bariloche 8400, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,5]]},"reference":[{"key":"ref_1","unstructured":"van Kampen, N.G. (1992). Stochastic Processes in Physics and Chemistry, North-Holland."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Breuer, H.P., and Petruccione, F. (2002). The Theory of Open Quantum Systems, Oxford University Press.","DOI":"10.1007\/3-540-44874-8_4"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"015001","DOI":"10.1103\/RevModPhys.89.015001","article-title":"Dynamics of non-Markovian open quantum systems","volume":"89","author":"Alonso","year":"2017","journal-title":"Rev. Mod. 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Non-Markovianity of A Central Spin Interacting with a Lipkin\u2013Meshkov\u2013Glick Bath via a Conditional Past\u2013Future Correlation. Entropy, 22.","DOI":"10.3390\/e22080895"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"127246","DOI":"10.1016\/j.physleta.2021.127246","article-title":"Operational non-Markovianity in a statistical mixture of two environments","volume":"397","author":"Ban","year":"2021","journal-title":"Phys. Lett. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"042120","DOI":"10.1103\/PhysRevA.101.042120","article-title":"Detection of quantum non-Markovianity close to the Born-Markov approximation","volume":"101","author":"Walborn","year":"2020","journal-title":"Phys. Rev. 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A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"020101","DOI":"10.1103\/PhysRevA.71.020101","article-title":"Completely positive post-Markovian master equation via a measurement approach","volume":"71","author":"Shabani","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"012147","DOI":"10.1103\/PhysRevE.89.012147","article-title":"Post-Markovian quantum master equations from classical environment fluctuations","volume":"89","author":"Budini","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"030101","DOI":"10.1103\/PhysRevA.87.030101","article-title":"Non-Markovian master equations from piecewise dynamics","volume":"87","author":"Vacchini","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"032115","DOI":"10.1103\/PhysRevA.88.032115","article-title":"Embedding non-Markovian quantum collisional models into bipartite Markovian dynamics","volume":"88","author":"Budini","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"022103","DOI":"10.1103\/PhysRevA.80.022103","article-title":"Non-Markovian nonstationary completely positive open-quantum-system dynamics","volume":"80","author":"Budini","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1103\/RevModPhys.81.865","article-title":"Quantum entanglement","volume":"81","author":"Horodecki","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"032310","DOI":"10.1103\/PhysRevA.92.032310","article-title":"Characterization and measurement of qubit-environment-entanglement generation during pure dephasing","volume":"92","author":"Roszak","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"012306","DOI":"10.1103\/PhysRevA.97.012306","article-title":"Equivalence of qubit-environment entanglement and discord generation via pure dephasing interactions and the resulting consequences","volume":"97","author":"Roszak","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"052344","DOI":"10.1103\/PhysRevA.98.052344","article-title":"Criteria for system\u2013environment entanglement generation for systems of any size in pure-dephasing evolutions","volume":"98","author":"Roszak","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"030403","DOI":"10.1103\/PhysRevLett.120.030403","article-title":"Simulating Open Quantum Systems with Hamiltonian Ensembles and the Nonclassicality of the Dynamics","volume":"120","author":"Chen","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","unstructured":"C\n                                \n                                  p\n                                  f\n                                \n                              \n                              \n                                (\n                                t\n                                ,\n                                \u03c4\n                                )\n                              \n                            \n                            \n                              y\n                              \u02d8\n                            \n                          \n                          \n                            =\n                            d\n                          \n                          \n                            8\n                            \n                              81\n                              \n                                \n                                  (\n                                  \u03b3\n                                  +\n                                  \u03d5\n                                  )\n                                \n                                4\n                              \n                            \n                          \n                          \n                            e\n                            \n                              \u2212\n                              2\n                              t\n                              \u03b3\n                              \u2212\n                              \u03c4\n                              \u03b3\n                              \u2212\n                              3\n                              t\n                              \u03d5\n                              \u2212\n                              2\n                              \u03c4\n                              \u03d5\n                            \n                          \n                          \u03b3\n                          (\n                          (\n                          \u2212\n                          2\n                          \n                            e\n                            \n                              (\n                              t\n                              +\n                              \u03c4\n                              )\n                              (\n                              \u03b3\n                              +\n                              2\n                              \u03d5\n                              )\n                            \n                          \n                          \n                            \n                              (\n                              \u03b3\n                              \u2212\n                              3\n                              \u03d5\n                              )\n                            \n                            2\n                          \n                          \u03d5\n                          \u2212\n                          2\n                          \n                            e\n                            \n                              2\n                              t\n                              \u03b3\n                              +\n                              3\n                              t\n                              \u03d5\n                              +\n                              \u03c4\n                              \u03d5\n                            \n                          \n                          \n                            \n                              (\n                              \u03b3\n                              \u2212\n                              3\n                              \u03d5\n                              )\n                            \n                            2\n                          \n                          \u03d5\n                          \u2212\n                          \n                            e\n                            \n                              2\n                              t\n                              \u03b3\n                              +\n                              \u03c4\n                              \u03b3\n                              +\n                              3\n                              t\n                              \u03d5\n                              +\n                              \u03c4\n                              \u03d5\n                            \n                          \n                          \n                            \n                              (\n                              \u03b3\n                              \u2212\n                              3\n                              \u03d5\n                              )\n                            \n                            2\n                          \n                          \n                            (\n                            \u03b3\n                            +\n                          \n                        \n                      \n                    \n                    \n                      \n                        \n                          \n                            \u03d5\n                            )\n                            \u2212\n                          \n                          \n                            e\n                            \n                              2\n                              t\n                              (\n                              \u03b3\n                              +\n                              \u03d5\n                              )\n                              +\n                              \u03c4\n                              (\n                              \u03b3\n                              +\n                              2\n                              \u03d5\n                              )\n                            \n                          \n                          \n                            \n                              (\n                              \u03b3\n                              \u2212\n                              3\n                              \u03d5\n                              )\n                            \n                            2\n                          \n                          \n                            (\n                            \u03b3\n                            +\n                            \u03d5\n                            )\n                          \n                          \u2212\n                          16\n                          \n                            e\n                            \n                              \u03c4\n                              \u03d5\n                              +\n                              2\n                              t\n                              (\n                              \u03b3\n                              +\n                              \u03d5\n                              )\n                            \n                          \n                          \u03b3\n                          \u03d5\n                          \n                            (\n                            \u03b3\n                            +\n                            \u03d5\n                            )\n                          \n                          \u2212\n                          16\n                          \n                            e\n                            \n                              \u03c4\n                              (\n                              \u03b3\n                              +\n                              \u03d5\n                              )\n                              +\n                              t\n                              (\n                              \u03b3\n                              +\n                              2\n                              \u03d5\n                              )\n                            \n                          \n                          \u03b3\n                          \u03d5\n                          \n                            (\n                            \u03b3\n                            +\n                            \u03d5\n                            )\n                          \n                          +\n                          \n                            e\n                            \n                              2\n                              t\n                              \u03b3\n                              +\n                              \u03c4\n                              \u03b3\n                              +\n                              3\n                              t\n                              \u03d5\n                              +\n                              2\n                              \u03c4\n                              \u03d5\n                            \n                          \n                          \n                            \n                              (\n                              \u03b3\n                              \u2212\n                              3\n                              \u03d5\n                              )\n                            \n                            2\n                          \n                          \n                            (\n                            \u03b3\n                            +\n                          \n                        \n                      \n                    \n                    \n                      \n                        \n                          \n                            3\n                            \u03d5\n                            )\n                            +\n                          \n                          \n                            e\n                            \n                              (\n                              2\n                              t\n                              +\n                              \u03c4\n                              )\n                              (\n                              \u03b3\n                              +\n                              \u03d5\n                              )\n                            \n                          \n                          \n                            \n                              (\n                              \u03b3\n                              +\n                              \u03d5\n                              )\n                            \n                            2\n                          \n                          \n                            (\n                            \u03b3\n                            +\n                            9\n                            \u03d5\n                            )\n                          \n                          +\n                          2\n                          \n                            e\n                            \n                              t\n                              \u03b3\n                              +\n                              2\n                              t\n                              \u03d5\n                              +\n                              \u03c4\n                              \u03d5\n                            \n                          \n                          \u03d5\n                          (\n                          9\n                          \n                            \u03b3\n                            2\n                          \n                          +\n                          2\n                          \u03b3\n                          \u03d5\n                          +\n                          9\n                          \n                            \u03d5\n                            2\n                          \n                          )\n                          )\n                          ."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"032206","DOI":"10.1103\/PhysRevA.104.032206","article-title":"Solvable class of non-Markovian quantum multipartite dynamics","volume":"104","author":"Budini","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"180602","DOI":"10.1103\/PhysRevLett.94.180602","article-title":"Experimental Test of the Fluctuation Theorem for a Driven Two-Level System with Time-Dependent Rates","volume":"94","author":"Schuler","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_59","unstructured":"In the Laplace domain, f(u)=\u222b0\u221edte\u2212utf(t), it reads 4\u03c1ue4=A(u)\/B(u), where A(u)=\u03b3(u+\u03d5)(2u+\u03b3+\u03d5)+(3(u+\u03d5)+2\u03b3)\u03a92 and \n\t\tB(u)=3u(u+\u03d5)(u+\u03b3+\u03d5)(2u+\u03b3+\u03d5)+6u(3(u+\u03d5)+\u03b3)\u03a92."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/5\/649\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:06:26Z","timestamp":1760137586000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/5\/649"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,5]]},"references-count":59,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["e24050649"],"URL":"https:\/\/doi.org\/10.3390\/e24050649","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,5]]}}}