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This phenomenon is known as <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">(quantum) self-testing<\/mml:mtext><\/mml:mrow><\/mml:math>.Self-testing constitutes a significant research area with practical and theoretical ramifications for quantum information theory. But since its conception two decades ago by Mayers and Yao, the common way to rigorously formulate self-testing has been in terms of operator-algebraic identities, and this formulation lacks an operational interpretation. In particular, it is unclear how to formulate self-testing in other physical theories, in formulations of quantum theory not referring to operator-algebra, or in scenarios causally different from the standard one.In this paper, we explain how to understand quantum self-testing operationally, in terms of causally structured dilations of the input-output channel encoding the correlations. These dilations model side-information which leaks to an environment according to a specific schedule, and we show how self-testing concerns the relative strength between such scheduled leaks of information. As such, the title of our paper has double meaning: we recast conventional quantum self-testing in terms of information-leaks to an environment \u2013 and this realises quantum self-testing as a special case within the surroundings of a general operational framework.Our new approach to quantum self-testing not only supplies an operational understanding apt for various generalisations, but also resolves some unexplained aspects of the existing definition, naturally suggests a distance measure suitable for robust self-testing, and points towards self-testing as a modular concept in a larger, cryptographic perspective.<\/jats:p>","DOI":"10.22331\/q-2022-04-27-699","type":"journal-article","created":{"date-parts":[[2022,4,27]],"date-time":"2022-04-27T15:27:10Z","timestamp":1651073230000},"page":"699","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["An Operational Environment for Quantum Self-Testing"],"prefix":"10.22331","volume":"6","author":[{"given":"Matthias","family":"Christandl","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2200 Copenhagen, Denmark"}]},{"given":"Nicholas Gauguin","family":"Houghton-Larsen","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2200 Copenhagen, Denmark"}]},{"given":"Laura","family":"Mancinska","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2200 Copenhagen, Denmark"}]}],"member":"9598","published-online":{"date-parts":[[2022,4,27]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Samson Abramsky and Bob Coecke, A Categorical Semantics of Quantum Protocols, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004, IEEE, 2004, https:\/\/doi.org\/10.1109\/LICS.2004.1319636.","DOI":"10.1109\/LICS.2004.1319636"},{"key":"1","doi-asserted-by":"publisher","unstructured":"John C. 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