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In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascu\u00e9s-Pironio-Ac\u00edn hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.<\/jats:p>","DOI":"10.22331\/q-2024-08-27-1445","type":"journal-article","created":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T15:02:36Z","timestamp":1724770956000},"page":"1445","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":39,"title":["Device-independent lower bounds on the conditional von Neumann entropy"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9593-0136","authenticated-orcid":false,"given":"Peter","family":"Brown","sequence":"first","affiliation":[{"name":"Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France"},{"name":"T\u00e9l\u00e9com Paris, LTCI, Institut Polytechnique de Paris, 91120 Palaiseau, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hamza","family":"Fawzi","sequence":"additional","affiliation":[{"name":"DAMTP, University of Cambridge, United Kingdom"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Omar","family":"Fawzi","sequence":"additional","affiliation":[{"name":"Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2024,8,27]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Charles H. 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