{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,22]],"date-time":"2025-11-22T17:04:18Z","timestamp":1763831058302,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2019,10,21]],"date-time":"2019-10-21T00:00:00Z","timestamp":1571616000000},"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 address an informational puzzle that appears with a non-Markovian open qubit dynamics: namely the fact that, while, according to the existing witnesses of information flows, a single qubit affected by that dissipative dynamics does not show information returning to it from its environment, instead two qubits do show such information when evolving independently under the same dynamics. We solve the puzzle by adding the so-called quasi-entropies to the possible witnesses of information flows.<\/jats:p>","DOI":"10.3390\/e21101020","type":"journal-article","created":{"date-parts":[[2019,10,21]],"date-time":"2019-10-21T11:37:55Z","timestamp":1571657875000},"page":"1020","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quasi-Entropies and Non-Markovianity"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0712-2057","authenticated-orcid":false,"given":"Fabio","family":"Benatti","sequence":"first","affiliation":[{"name":"Department of Physics, University of Trieste, Strada Costiera 11, I-34151 Trieste, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare (INFN), I-34151 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6000-4308","authenticated-orcid":false,"given":"Luigi","family":"Brancati","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Trieste, Strada Costiera 11, I-34151 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Alicki, R., and Lendi, F. (2007). Quantum Dynamical Semigroups and Applications, Springer.","DOI":"10.1016\/B0-12-512666-2\/00384-9"},{"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","unstructured":"Rivas, A., and Huelga, S.F. (2012). Open Quantum Systems, an Introduction, Springer.","DOI":"10.1007\/978-3-642-23354-8"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"154002","DOI":"10.1088\/0953-4075\/45\/15\/154002","article-title":"Markovianity criteria for quantum evolution","volume":"45","author":"Kossakowski","year":"2012","journal-title":"J. Phys. B-At. Mol. Opt. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"210401","DOI":"10.1103\/PhysRevLett.103.210401","article-title":"Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems","volume":"103","author":"Breuer","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"042108","DOI":"10.1103\/PhysRevA.92.042108","article-title":"Generalised trace-distance measure connecting quantum and classical non-Markovianity","volume":"92","author":"Wimann","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"012112","DOI":"10.1103\/PhysRevA.95.012112","article-title":"Tensor power of dynamical maps and Positive versus Completely Positive divisibility","volume":"95","author":"Benatti","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"032130","DOI":"10.1103\/PhysRevA.97.032130","article-title":"Generalised formalism for information back-flow in assessing Markovianity and its equivalence to divisibility","volume":"97","author":"Chakraborty","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/0034-4877(86)90067-4","article-title":"Quasi entropies for finite quantum systems","volume":"23","author":"Petz","year":"1986","journal-title":"Rep. Math. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Hiai, F., and Petz, D. (2014). Introduction to Matrix Analysis and Applications, Springer.","DOI":"10.1007\/978-3-319-04150-6"},{"key":"ref_11","unstructured":"Petz, D. (2008). Quantum Information Theory and Quantum Statistics, Springer."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1142\/S0129055X11004412","article-title":"Quantum f-divergences and error correction","volume":"23","author":"Hiai","year":"2010","journal-title":"Rev. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Benatti, F., Fannes, M., and Floreanini, R. (2010). Quantum Information, Computation and Cryptography, Springer.","DOI":"10.1007\/978-3-642-11914-9"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3063","DOI":"10.1142\/S0217979205032097","article-title":"Open Quantum Dynamics: Complete Positivity and Entanglement","volume":"19","author":"Benatti","year":"2005","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_15","first-page":"419","article-title":"The proper form of the generator in the weak coupling limit","volume":"34","author":"Spohn","year":"1979","journal-title":"Z. Phys. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1063\/1.522979","article-title":"Completely positive dynamical semigroups of N-level systems","volume":"17","author":"Gorini","year":"1976","journal-title":"J. Math. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1007\/BF01608499","article-title":"On the generators of quantum dynamical semigroups","volume":"48","author":"Lindblad","year":"1976","journal-title":"Comm. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0375-9601(96)00706-2","article-title":"Separability of mixed states: Necessary and sufficient conditions","volume":"223","author":"Horodecki","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"McCracken, J.M. (2014). Negative Quantum Channels, Morgan and Claypool Publishers.","DOI":"10.1007\/978-3-031-02517-4"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"L551","DOI":"10.1088\/0305-4470\/35\/39\/101","article-title":"Complete positivity and Dissipative Factorised Dynamics","volume":"35","author":"Benatti","year":"2002","journal-title":"J. Phys. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"041147","DOI":"10.1103\/PhysRevE.79.041147","article-title":"Structure of completely positive quantum master equations with memory kernel","volume":"79","author":"Breuer","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"070406","DOI":"10.1103\/PhysRevLett.104.070406","article-title":"Non-Markovian quantum dynamics: Local versus non-local","volume":"104","author":"Kossakowski","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"052128","DOI":"10.1103\/PhysRevA.83.052128","article-title":"Measures of non-Markovianity: Divisibility versus backflow of information","volume":"83","author":"Kossakowsky","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"080407","DOI":"10.1103\/PhysRevLett.121.080407","article-title":"Divisibility and information flow notions of quantum Markovianity for non-invertible dynamical maps","volume":"121","author":"Rivas","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"055301","DOI":"10.1088\/1751-8121\/aa5301","article-title":"Positive tensor products of qubit maps and 2-tensor-stable Positive qubit map","volume":"50","author":"Filippov","year":"2017","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s00220-008-0411-y","article-title":"Dividing quantum channels","volume":"279","author":"Wolf","year":"2008","journal-title":"Commun. Math. Phys."},{"key":"ref_27","unstructured":"Szarek, S.J. (2006). On norms of completely positive maps. Topics in Operator Theory, Birkh\u00e4user."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1425","DOI":"10.1016\/j.physleta.2013.04.020","article-title":"Non-Markovian random unitary qubit dynamics","volume":"377","author":"Wudarski","year":"2013","journal-title":"Phys. Lett. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1227","DOI":"10.1063\/1.523789","article-title":"Entropy production for quantum dynamical semigroups","volume":"19","author":"Spohn","year":"1978","journal-title":"J. Math. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1007\/s00023-017-0550-9","article-title":"Monotonicity of the Quantum Relative Entropy Under Positive Map","volume":"Volume 18","author":"Reeb","year":"2017","journal-title":"Annales Henri Poincar\u00e9"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/10\/1020\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:28:08Z","timestamp":1760189288000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/10\/1020"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,21]]},"references-count":30,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2019,10]]}},"alternative-id":["e21101020"],"URL":"https:\/\/doi.org\/10.3390\/e21101020","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,10,21]]}}}