{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T11:19:10Z","timestamp":1773659950810,"version":"3.50.1"},"reference-count":30,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T00:00:00Z","timestamp":1773619200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Research Council","award":["No. 884762"],"award-info":[{"award-number":["No. 884762"]}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"crossref","award":["ANR-22-PETQ-0006"],"award-info":[{"award-number":["ANR-22-PETQ-0006"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution.\nIn this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.<\/jats:p>","DOI":"10.22331\/q-2026-03-16-2031","type":"journal-article","created":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:31:06Z","timestamp":1773657066000},"page":"2031","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["A posteriori error estimates for the Lindblad master equation"],"prefix":"10.22331","volume":"10","author":[{"given":"Paul-Louis","family":"Etienney","sequence":"first","affiliation":[{"name":"Laboratoire de Physique de l'\u00c9cole Normale Sup\u00e9rieure, Mines Paris, Inria, CNRS, ENS-PSL, Sorbonne Universit\u00e9, PSL Research University, Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R\u00e9mi","family":"Robin","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique de l'\u00c9cole Normale Sup\u00e9rieure, Mines Paris, Inria, CNRS, ENS-PSL, Sorbonne Universit\u00e9, PSL Research University, Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pierre","family":"Rouchon","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique de l'\u00c9cole Normale Sup\u00e9rieure, Mines Paris, Inria, CNRS, ENS-PSL, Sorbonne Universit\u00e9, PSL Research University, Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2026,3,16]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Daniel Appel\u00f6and Yingda Cheng ``Kraus is king: High-order completely positive and trace preserving (CPTP) low rank method for the Lindblad master equation'' Journal of Computational Physics 534, 114036 (2025).","DOI":"10.1016\/j.jcp.2025.114036"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Sahel Ashhab, Felix Fischer, Davide Lonigro, Daniel Braak, and Daniel Burgarth, ``Finite-dimensional approximations of generalized squeezing'' Phys. 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