  A measurement performed on a quantum system is an act of gaining information
about its state, a view that is widespread in practical and foundational work
in quantum theory. However, the concept of information in quantum theory
reconstructions is multiply-defined, and its conceptual foundations remain
surprisingly under-explored. In this paper, we investigate the gain of
information in quantum measurements from an operational viewpoint. We show that
the continuous extension of the Shannon entropy naturally admits two distinct
measures of information gain, differential information gain and relative
information gain, and that these have radically different characteristics. In
particular, while differential information gain can increase or decrease as
additional data is acquired, relative information gain consistently grows, and
moreover exhibits asymptotic indifference to the data or choice of Bayesian
prior.
  In order to make a principled choice between these measures, we articulate a
Principle of Information Increase, which incorporates Summhammer's proposal
that more data from measurements leads to more knowledge about the system, and
also takes into consideration black swan events. This principle favors
differential information gain as the more relevant metric in two-outcome
quantum systems, and guides the selection of priors for these information
measures. Finally, we show that, of the beta distribution priors, the Jeffreys'
binomial prior is the prior ensures maximal robustness of information gain to
the particular data sequence obtained in a run of experiments.
