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Addendum to Computational Complexity and Black Hole Horizons
Leonard Susskind
TL;DR
The addendum addresses limited evidence for identifying ERB length with computational complexity and a paradox involving Alice’s measurement of the left black hole. It analyzes shockwave geometries and GHZ tripartite entanglement, finding additional support for the length-complexity relation and reconciling Alice’s and Charlie’s descriptions of signaling.
Problem
Evidence for the ERB-length–complexity relation had mainly been established for the thermofield-double state, while Alice’s complete measurement raises questions about firewalls and signaling to Bob.
Method
The addendum studies Shenker-Stanford shockwave geometries using minimal-surface calculations and analyzes Alice’s measurement as a unitary operation on the left black hole and memory, interpreted through GHZ entanglement.
Results
The shockwave calculations provide additional support for identifying computational complexity with ERB length, while Charlie can signal only by applying a sufficiently complex precursor that erases Alice’s memory.
Takeaways & Limitations
ERB length can continue to track complexity beyond the thermofield-double setting, but vertical entanglement ceases to measure increasing complexity after its minimal surface disconnects.
Takeaways & Limitations
The shockwave calculations are not conclusive, and complexity cannot grow beyond the maximal value Cmax = eS, where the classical ERB geometry is expected to break down.
Abstract
from arXiv · showhide
In this addendum to [arXiv:1402.5674] two points are discussed. In the first additional evidence is provided for a dual connection between the geometric length of an Einstein-Rosen bridge and the computational complexity of the quantum state of the dual CFT's. The relation between growth of complexity and Page's ``Extreme Cosmic Censorship" principle is also remarked on. The second point involves a gedanken experiment in which Alice measures a complete set of commuting observables at her end of an Einstein-Rosen bridge is discussed. An apparent paradox is resolved by appealing to the properties of GHZ tripartite entanglement.
1 Complexity and Wormhole Length
The addendum develops evidence that Einstein-Rosen bridge length tracks the dual state's computational complexity, extending beyond the thermofield-double case to shockwave geometries. It also relates complexity growth to black-hole interior stretching and discusses its behavior near maximal complexity.
- The proposed duality identifies bridge length with computational complexity divided by entanglement entropy.
- 1. Complexity and Wormhole Length: For the TFD state, both bridge length and complexity grow linearly with time until the classical recurrence time.
- 1.1 Evidence from Shockwave Geometries: Connected minimal surfaces remain available as a proxy for complexity and ERB distance after the absolute extremal surface disconnects.
- 1.1 Evidence from Shockwave Geometries: In the BTZ analysis, shockwave geometries provide additional evidence that precursor complexity and later complexity growth match the corresponding bridge-length dependence.
- 1.1 Evidence from Shockwave Geometries: The precursor contribution includes a factor of 2 because chaotic evolution prevents cancellation between U and U†.
- 1.2 Stretching and Extreme Cosmic Censorship: The stretching hypothesis is restated as black holes forming with states whose complexity increases, excluding half the Hilbert-space states by time reversal.
2 Measurements and GHZ States
Alice’s complete measurement produces GHZ tripartite entanglement among the right black hole, left black hole, and memory, resolving the apparent signaling paradox. The measurement does not create a firewall at Bob’s end, while Charlie can signal only by applying a sufficiently complex precursor that erases the memory.
- Measurement setup: Alice’s measurement couples S memory qubits to the S qubits of the left black hole, with each pair interacting through a single gate.The memory records the corresponding left-black-hole qubit state.
- Firewall question: The measurement introduces complexity no larger than S, below the scrambling complexity S log S required to send a shockwave to Bob.Therefore it does not reach deeply enough into the near-horizon region to create a firewall at Bob’s end.
- GHZ state: The post-measurement state is a tensor product of S GHZ triplets involving the right black hole, left black hole, and memory.GHZ entanglement connects all three parties in a form not described by conventional classical geometry.
- GHZ properties: In a GHZ state, each party is maximally entangled with the union of the other two, while tracing out any party leaves a separable bipartite state.Thus Bob’s black hole is maximally entangled with the combined left-black-hole and memory system, despite no pairwise entanglement between individual parties.
- Alice’s operations: Alice cannot signal to Bob by manipulating her memory because the memory is completely unentangled with Bob’s black hole.Carrying the full S-bit record into Bob’s black hole would also massively disturb it because the record cannot fit in the original black hole.
- Resolving the paradox: Charlie can signal to Bob only by applying a very complex precursor on the combined left-and-memory system that undoes the measurement and erases Alice’s memory.This reconciles Charlie’s description with Alice’s conclusion that she cannot signal from her subsystem after measurement.