  If spacetime is built out of quantum bits, does the shape of space depend on
how the bits are entangled? The ER=EPR conjecture relates the entanglement
entropy of a collection of black holes to the cross sectional area of
Einstein-Rosen (ER) bridges (or wormholes) connecting them. We show that the
geometrical entropy of classical ER bridges satisfies the subadditivity,
triangle, strong subadditivity, and CLW inequalities. These are nontrivial
properties of entanglement entropy, so this is evidence for ER=EPR. We further
show that the entanglement entropy associated to classical ER bridges has
nonpositive interaction information. This is not a property of entanglement
entropy, in general. For example, the entangled four qubit pure state
|GHZ_4>=(|0000>+|1111>)/\sqrt{2} has positive interaction information, so this
state cannot be described by a classical ER bridge. Large black holes with
massive amounts of entanglement between them can fail to have a classical ER
bridge if they are built out of |GHZ_4> states. States with nonpositive
interaction information are called monogamous. We conclude that classical ER
bridges require monogamous EPR correlations.
