  The classical information metric provides a unique notion of distance on the
space of probability distributions with a well-defined operational
interpretation: two distributions are far apart if they are readily
distinguishable from one another. The quantum information metric generalizes
this to the space of quantum states, and thus defines a notion of distance on
an arbitrary continuous family of quantum field theories via their vacua that
is proportional to the metric on moduli space when restricted appropriately. In
this paper, we study this metric and its operational interpretation in a
variety of examples. We specifically focus on why and how infinite distance
singularities appear. We argue that two theories are infinitely far apart if
they are hyper-distinguishable: that is, if they can be distinguished from one
another, with certainty, using only a few measurements. We explain why such
singularities appear for the simple harmonic oscillator yet are absent for
quantum field theories near a typical quantum critical point, and show how an
infinite distance point can emerge when a tower of fields degenerates in mass.
Finally, we use this perspective to provide a potential bottom-up motivation
for the Swampland Distance Conjecture and indicate how we might extend it
beyond current lampposts.
