Noncommutative Geometry Seminar

The Ryu-Takayanagi formula relates the entanglement entropy
in a conformal field theory to the area of a minimal surface in its
holographic dual. We show that this relation can be inverted for any
state in the conformal field theory to compute the bulk stress-energy
tensor near the boundary of the bulk spacetime, reconstructing the local
data in the bulk from the entanglement on the boundary. We also show
that positivity, monotonicity, and convexity of the relative entropy for
small spherical domains between the reduced density matrices of any
state and of the ground state of the conformal field theory are
guaranteed by positivity conditions on the bulk matter energy density.
As positivity and monotonicity of the relative entropy are general
properties of quantum systems, this can be interpreted as a derivation
of bulk energy conditions in any holographic system with the
Ryu-Takayanagi prescription applies. We discuss an information
theoretical interpretation of the convexity in terms of the Fisher
metric. This is joint work with Jennifer Lin, Matilde Marcolli
and Hirosi Ooguri, arXiv:1412.1879.