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The algorithm is accurate if a rank condition for marginal states within the smoothing environment is satisfied and provides an upper bound otherwise. Central to our method is a novel SDP, for which we establish primal and dual formulations and prove strong duality. With the direct application of bounding the one-shot distillable key of a quantum state, this contribution extends SDP-based techniques in quantum information theory. Thereby it improves the capabilities to compute or estimate information measures with application to various quantum information processing tasks.\n                  <\/jats:p>","DOI":"10.1007\/s11128-026-05101-8","type":"journal-article","created":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T05:44:19Z","timestamp":1771998259000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Computation of the smooth max-mutual information via semidefinite programming"],"prefix":"10.1007","volume":"25","author":[{"given":"Christopher","family":"Popp","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tobias C.","family":"Sutter","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Beatrix C.","family":"Hiesmayr","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,25]]},"reference":[{"key":"5101_CR1","unstructured":"Wilde, M.M.: Quantum Information Theory Cambridge University Press, Cambridge, (2013).https:\/\/www.cambridge.org\/core\/books\/quantum-information-theory\/9DC2CA59F45636D4F0F30D971B677623"},{"key":"5101_CR2","unstructured":"Khatri, S., Wilde, M.M.: Principles of Quantum Communication Theory: A Modern Approach (2024). 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