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ER = EPR revisited: On the Entropy of an Einstein-Rosen Bridge

Herman Verlinde

TL;DR

It is argued 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, which implies that black hole quantum information is topologically protected, similar to the information stored inside a topological quantum memory.

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.

ER = EPR revisited: On the Entropy of an Einstein-Rosen Bridge

TL;DR

It is argued 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, which implies that black hole quantum information is topologically protected, similar to the information stored inside a topological quantum memory.

Abstract

We propose a new link between entropy and area: an eternal black hole with an ER bridge with cross-section can carry a macroscopic amount of quantum information, or be in a mixed state, with entropy bounded by . 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.

Paper Structure

This paper contains 11 sections, 58 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Three thermal states
  3. Global geometry of an ER bridge
  4. Non-local phases and generalized TFD states
  5. Dephasing and the thermo-mixed double
  6. Replica wormholes and Renyí entropy
  7. QEC, RT and Poincaré recurrence
  8. Decoherence of a two-sided black hole
  9. Conclusion
  10. Acknowledgements

Figures (5)

  • Figure 1: Spatial slice of an ER bridge. The two asymptotic regions ${\cal E}_L$ and ${\cal E}_R$ are connected via the interface $I$.
  • Figure 2: Penrose diagram of an ER bridge. The time variables defined on the asymptotic regions ${\cal E}_L$ and ${\cal E}_R$ are connected via an identification $I$.
  • Figure 3: The Island is contained inside the entanglement wedge of any pair of the three regions $L$, $R$ and ${\cal E}$. The two regions $L$ and $R$ combined are in the thermal mixed state, with entropy given by the area of the red RT surface.
  • Figure 4: Penrose diagram of an ER bridge with an Island region, bounded by two quantum extremal surfaces.
  • Figure 5: The two entanglement cuts surrounding the near horizon region of the two-sided black hole introduce edge states on each side of the cut. The combined state of edges across each cut are described by a boundary state of the holographic CFT.