The structure of a point in space and the meaning of distance

The classical concepts of space and time, including the very
notion of a point in space, are likely to break down below the Planck
scale. I use the holographic duality (AdS/CFT) -- an equivalence
between quantum gravity in anti-de Sitter space and a
lower-dimensional field theory without gravity -- to illustrate how
spacetime emerges from the underlying fundamental theory. A useful
tool for this goal is the Ryu-Takayanagi proposal -- an equality
between a classical object in gravity (area of a minimal surface) and
an essentially quantum quantity in field theory (entanglement
entropy). I explain how to use this proposal to give a
non-perturbative definition of a point in space in the language of the
dual quantum field theory. From this point of view, the geodesic
distance between two points is related to the entanglement cost of a
"restricted merging protocol" in information theory. These results
show that holographic duality relies on an intimate relation between
the geometry of anti-de Sitter space and information theory: for
example, the triangle inequality is equivalent to the strong
subadditivity of entropy.