  Observed quantum correlations are known to determine in certain cases the
underlying quantum state and measurements. This phenomenon is known as
(quantum) self-testing.
  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 -- 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.
