[Paper Review] ER = EPR revisited: On the Entropy of an Einstein-Rosen Bridge

Herman Verlinde|arXiv (Cornell University)|Mar 29, 2020

Black Holes and Theoretical Physics34 references42 citations

TL;DR

Verlinde argues that an ER bridge can carry macroscopic quantum information with entropy SBH = A/4GN, describing a thermo-mixed double state with only classical correlations, supported by Island/replica wormhole calculations in JT and AdS3 gravity.

ABSTRACT

We propose a new link between entropy and area: an eternal black hole with an ER bridge with cross-section $A$ can carry a macroscopic amount of quantum information, or be in a mixed state, with entropy bounded by $S \leq A/4G_N$. We substantiate our proposal in the context of AdS3 and JT gravity, by using the Island prescription and replica wormhole method for computing the black hole entropy. We argue that the typical mixed state of a two sided black hole takes the form of an entangled `thermo-mixed double' state with only classical correlations between the two sides. Our result for the von Neumann entropy of a post-Page time two-sided black hole is smaller by a factor of two from previous answers. Our reasoning implies that black hole quantum information is topologically protected, similar to the information stored inside a topological quantum memory.

Motivation & Objective

• Reframe the Bekenstein-Hawking entropy as the maximal information content of an ER bridge in a two-sided black hole.
• Show that a macroscopic entropy SBH = A/4GN can be stored in the ER bridge as a mixed state rather than a pure TFD state.
• Demonstrate, via Island prescription and replica wormholes, that the two-sided black hole tends to a thermo-mixed double form with classical correlations between sides.
• Connect holographic entropy with quantum error correction and topological protection concepts in AdS/CFT.

Proposed method

• Compute entanglement entropy using the Island prescription and replica wormholes in JT gravity and pure AdS3 gravity to derive the TMD form (SBH) of the two-sided black hole.
• Propose a geometric ‘janus pacman’ representation for the TMD density matrix with Island regions mediating entanglement with the environment.
• Show that the replica wormhole saddle points yield tr(ρ^k_TMD) = Z(kβ)/Z(β)^k, matching the SBH entropy.
• Relate the TMD state to generalized TFD states with non-local phases and discuss decoherence to a balanced, topologically protected information structure.
• Discuss implications for bulk reconstruction, QEC, RT, and Poincaré recurrence within the code subspace of low-energy QFT.

Experimental results

Research questions

• RQ1 What is the maximal entropy or quantum information that can be contained in a two-sided black hole with an ER bridge?
• RQ2 Does the entanglement structure across the ER bridge necessarily require a pure TFD-like state, or can a thermo-mixed double with classical correlations suffice?
• RQ3 How do Island prescriptions and replica wormholes constrain the form of the two-sided black hole density matrix and its entropy?
• RQ4 How does holographic quantum error correction relate to the topological protection of information stored in the ER bridge?

Key findings

• A two-sided black hole with an ER bridge of cross-section A can carry entropy SBH = A/4GN, corresponding to macroscopic quantum information.
• The typical mixed state of a two-sided black hole can take the form of a thermo-mixed double with only classical correlations between the sides (ρTMD).
• Replica wormhole calculations in JT gravity and AdS3 gravity support the TMD form and yield tr(ρ^k_TMD) = Z(kβ)/Z(β)^k.
• The von Neumann entropy after Page time for the two-sided black hole is SBH, not twice SBH as in a fully decohered product state, reflecting balanced decoherence.
• The state of the black hole plus environment can be understood via a tripartite purifications where bulk information is topologically protected and accessible only through combined information from multiple regions (L, R, E).
• The framework connects ER = EPR with quantum error correction and RT, showing that entanglement structure across the bridge is compatible with holographic code subspaces.