Are entangled particles connected by wormholes? Evidence for theER=EPRconjecture from entropy inequalities

Abstract

If spacetime is built out of quantum bits, does the shape of space depend on how the bits are entangled? The $\mathrm{ER}=\mathrm{EPR}$ 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 Cadney-Linden-Winter inequalities. These are nontrivial properties of entanglement entropy, so this is evidence for $\mathrm{ER}=\mathrm{EPR}$. We further show that the entanglement entropy associated with classical ER bridges has nonpositive tripartite information. This is not a property of entanglement entropy, in general. For example, the entangled four qubit pure state $|GH{Z}_{4}⟩=(|0000⟩+|1111⟩)/\sqrt{2}$ has positive tripartite 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 $|GH{Z}_{4}⟩$ states. States with nonpositive tripartite information are called monogamous. We conclude that classical ER bridges require monogamous EPR correlations.