Random pure Gaussian states and Hawking radiation

Abstract

A black hole evaporates by Hawking radiation. Each mode of that radiation is thermal. If the total state is nevertheless to be pure, modes must be entangled. Estimating the minimum size of this entanglement has been an important outstanding issue. We develop a new theory of constrained random symplectic transformations, based on that the total state is pure and Gaussian with given marginals. In the random constrained symplectic model we then compute the distribution of mode-mode correlations, from which we bound mode-mode entanglement. Modes of frequency much larger than $\frac{k_B{T}_H(t)}{\hslash }$ are not populated at time $t$ and drop out of the analysis. Among the other modes we find that correlations and hence entanglement between relatively thinly populated modes (early-time high-frequency modes and/or late modes of any frequency) to be strongly suppressed. Relatively highly populated modes (early-time low-frequency modes) can on the other hand be strongly correlated, but a detailed analysis reveals that they are nevertheless very unlikely to be entangled. Our analysis hence establishes that restoring unitarity after a complete evaporation of a black hole does not require any significant quantum entanglement between any pair of Hawking modes. Our analysis further gives exact general expressions for the distribution of mode-mode correlations in random, pure, Gaussian states with given marginals, which may have applications beyond black hole physics.

Published version, with supplementary material. Main paper 7 pages, 3 figures. Supplementary material 30 pages, 1 figure

References in corpus (29)

• Quantum entanglement
• Quantum information with continuous variables
• Gaussian Quantum Information
• Introducing Quantum Discord
• A bound on chaos
• Average Entropy of a Subsystem
• Black holes as mirrors: quantum information in random subsystems
• Information in Black Hole Radiation
• Fast Scramblers
• The foundations of statistical mechanics from entanglement: Individual states vs. averages
• Replica Wormholes and the Entropy of Hawking Radiation
• The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole
• The entropy of Hawking radiation
• Quantum versus classical correlations in Gaussian states
• Jerusalem Lectures on Black Holes and Quantum Information
• Better Late than Never: Information Retrieval from Black Holes
• Bound entangled Gaussian states
• Hawking Radiation and Black Hole Thermodynamics
• An Infalling Observer in AdS/CFT
• Information Loss
• Quantum marginal problem and N-representability
• The Black Hole information problem: past, present, and future
• The Black Hole Interior in AdS/CFT and the Information Paradox
• Gaussian quantum marginal problem
• Minimal energy cost of entanglement extraction
• Spatially Overlapped Partners in Quantum Field Theory
• Pure State Entanglement Harvesting in Quantum Field Theory
• Ergodic observables in non-ergodic systems: the example of the harmonic chain
• Quantum information in Hawking radiation

Cited by in corpus (3)

• Quantum entanglement of Hawking-Partner modes in expanding cavities
• Average entanglement entropy of a small subsystem in a constrained pure Gaussian state ensemble
• Average mutual information for random fermionic Gaussian quantum states