  We study the question of how reliably one can distinguish two quantum field
theories (QFTs). Each QFT defines a probability distribution on the space of
fields. The relative entropy provides a notion of proximity between these
distributions and quantifies the number of measurements required to distinguish
between them. In the case of nearby conformal field theories, this reduces to
the Zamolodchikov metric on the space of couplings. Our formulation quantifies
the information lost under renormalization group flow from the UV to the IR and
leads us to a quantification of fine-tuning. This formalism also leads us to a
criterion for distinguishability of low energy effective field theories
generated by the string theory landscape.
