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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.

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