# nLab ER = EPR

Contents

### Context

#### Gravity

gravity, supergravity

# Contents

## Idea

The symbols ER = EPR (pronounced: “ee arr is ee pee arr”) are the name of a hypothesis about quantum gravity, claiming, somewhat vaguely, that entangled particles (Einstein-Podolsky-Rosen phenomenon) are connected by a wormhole (Einstein-Rosen bridge).

This idea was proposed by Maldacena-Susskind 13 as a solution for the black hole firewall problem.

## As a slogan for the Ryu-Takayanagi formula

The idea may usefully be compared – and maybe is in parts a way of turning into prose – the Ryu-Takayanagi formula (which has a more well-defined content as a theorem in a class of discrete tensor network-models), which equates

with

Concretely, in tensor network-models of AdS2/CFT1 duality, the entanglement entropy $S_A$ of any Majorana dimer code-tensor network state turns out to count the number of dimers that cross between a (connected) subregion $A$ and its complement:

In the case that the dimers correspond to the hyperbolic tesselation $\{5,4\}$ from the HaPPY code, this entropy formula

$S_A \;=\; \tfrac{1}{2} \ln \big( 2^{ \left\vert Chords(A, \bar A) \right\vert } \big)$

recovers the Ryu-Takayanagi formula (JGPE 19 (78)), as here the number of chords crossing any hyperbolic geodesic grows linearly with the length of this geodesic.

A precursor to this picture is the “bit-thread”-interpretation of entanglement entropy due to Freedman & Headrick 16 (notice the use of “EPR pair” in their Figure 4.)

Following Sati-Schreiber 19c we recognize the above Majorana dimer/bit-thread networks from JGPE 19, Yan 19 as chord diagram-encodings of holographic bulks (p. 38).

Notice that the formula for the entanglement entropy in the Majorana dimer code holds generally, for any configuration and hence any underlying chord diagram.

In conclusion, note how each chord here reflects at the same time:

1. (ER) one entangled pair of qbits in the boundary quantum system;

2. (EPR) a geodesic through the hyperbolic plane bulk spacetime,

which is a (rigorous) state of affairs clearly reminiscent of the “ER = EPR” slogan (except that wormholes are replaced by minimal area hypersurfaces, here: geodesics).

For more on this see at holographic entanglement entropy.

## References

The original article: