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

Are entangled particles connected by wormholes? Support for the ER=EPR conjecture from entropy inequalities