  We introduce a way of quantifying how informative a quantum measurement is,
starting from a resource-theoretic perspective. This quantifier, which we call
the robustness of measurement, describes how much `noise' must be added to a
measurement before it becomes completely uninformative. We show that this
geometric quantifier has operational significance in terms of the advantage the
measurement provides over guessing at random in an suitably chosen state
discrimination game. We further show that it is the single-shot generalisation
of the accessible information of a certain quantum-to-classical channel. Using
this insight, we also show that the recently-introduced robustness of coherence
is the single-shot generalisation of the accessible information of an ensemble.
Finally we discuss more generally the connection between robustness-based
measures, discrimination problems and single-shot information theory.
