Significant work has gone into determining the minimal set of entropy
inequalities that determine the holographic entropy cone. Holographic systems
with three or more parties have been shown to obey additional inequalities that
generic quantum systems do not. We consider a two dimensional conformal field
theory that is a single boundary of a holographic system and find four
additional non-linear inequalities which are derived from strong subadditivity
and the formula for the entanglement entropy of a region on the conformal field
theory. We also present an equality obtained by application of a hyperbolic
extension of Ptolemy's theorem to a two dimensional conformal field theory.Comment: 5 pages, 3 figure
