{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T02:53:48Z","timestamp":1771210428304,"version":"3.50.1"},"reference-count":58,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,11,20]],"date-time":"2024-11-20T00:00:00Z","timestamp":1732060800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation","award":["1908730"],"award-info":[{"award-number":["1908730"]}]},{"name":"National Science Foundation","award":["1910571"],"award-info":[{"award-number":["1910571"]}]},{"name":"National Science Foundation","award":["1907615"],"award-info":[{"award-number":["1907615"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"A colloquial interpretation of entropy is that it is the knowledge gained upon learning the outcome of a random experiment. Conditional entropy is then interpreted as the knowledge gained upon learning the outcome of one random experiment after learning the outcome of another, possibly statistically dependent, random experiment. In the classical world, entropy and conditional entropy take only non-negative values, consistent with the intuition that one has regarding the aforementioned interpretations. However, for certain entangled states, one obtains negative values when evaluating commonly accepted and information-theoretically justified formulas for the quantum conditional entropy, leading to the confounding conclusion that one can know less than nothing in the quantum world. Here, we introduce a physically motivated framework for defining quantum conditional entropy, based on two simple postulates inspired by the second law of thermodynamics (non-decrease of entropy) and extensivity of entropy, and we argue that all plausible definitions of quantum conditional entropy should respect these two postulates. We then prove that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world. All of our arguments are based on constructions of physical processes that respect the first postulate, the one inspired by the second law of thermodynamics.<\/jats:p>","DOI":"10.22331\/q-2024-11-20-1529","type":"journal-article","created":{"date-parts":[[2024,11,20]],"date-time":"2024-11-20T14:18:16Z","timestamp":1732112296000},"page":"1529","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Inevitability of knowing less than nothing"],"prefix":"10.22331","volume":"8","author":[{"given":"Gilad","family":"Gour","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, Technion - Israel Institute of Technology, Haifa 3200003, Israel"}]},{"given":"Mark M.","family":"Wilde","sequence":"additional","affiliation":[{"name":"School of Electrical and Computer Engineering, Cornell University, Ithaca, New York 14850, USA"}]},{"given":"S.","family":"Brandsen","sequence":"additional","affiliation":[{"name":"Department of Physics, Duke University, Durham, North Carolina 27708, USA"}]},{"given":"Isabelle Jianing","family":"Geng","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Institute for Quantum Science and Technology, University of Calgary, Alberta, Canada T2N 1N4"}]}],"member":"9598","published-online":{"date-parts":[[2024,11,20]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Albert Einstein, Boris Podolsky, and Nathan Rosen. ``Can quantum-mechanical description of physical reality be considered complete?''. 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