  We consider the question of entanglement conservation in the context of the
ER=EPR correspondence equating quantum entanglement with wormholes. In quantum
mechanics, the entanglement between a system and its complement is conserved
under unitary operations that act independently on each; ER=EPR suggests that
an analogous statement should hold for wormholes. We accordingly prove a new
area theorem in general relativity: for a collection of dynamical wormholes and
black holes in a spacetime satisfying the null curvature condition, the maximin
area for a subset of the horizons (giving the largest area attained by the
minimal cross section of the multi-wormhole throat separating the subset from
its complement) is invariant under classical time evolution along the outermost
apparent horizons. The evolution can be completely general, including horizon
mergers and the addition of classical matter satisfying the null energy
condition. This theorem is the gravitational dual of entanglement conservation
and thus constitutes an explicit characterization of the ER=EPR duality in the
classical limit.
