Entropy/information flux in Hawking radiation

TL;DR

The paper analyzes whether Hawking evaporation preserves information by contrasting classical thermodynamic entropy flow with quantum entanglement entropy, arguing that unitarity requires hidden information to be carried by correlations. It extends Page's average-subsystem framework from a bipartite system to a tripartite one that includes the environment, showing that entropy budgets balance continuously as the black hole evaporates and that the mutual information between the hole and radiation remains small (bounded by ~1/2 nat) and vanishes in the infinite-environment limit. Key results include a per-photon entanglement-information content of about $3.9$ bits and a precise thermodynamic equivalence: $S_{\rm Bekenstein}(t) + \langle S_{\rm Hawking}(t)\rangle \approx S_{\rm Bekenstein,0}$, with the environment ensuring consistent purification. The findings argue that there is no information paradox in a realistic setting when the environment is accounted for, and they suggest that the Page curve’s interpretation should be revised in favor of a continuous, tripartite balance that recovers standard thermodynamics in the infinite-environment limit.

Abstract

Blackbody radiation contains (on average) an entropy of 3.9\pm2.5 bits per photon. If the emission process is unitary, then this entropy is exactly compensated by "hidden information" in the correlations. We extend this argument to the Hawking radiation from GR black holes, demonstrating that the assumption of unitarity leads to a perfectly reasonable entropy/information budget. The key technical aspect of our calculation is a variant of the "average subsystem" approach developed by Page, which we extend beyond bipartite pure systems, to a tripartite pure system that considers the influence of the environment.

Figures (5)

• Figure 1: Clausius (thermodynamic) entropy balance: As the black hole Bekenstein entropy (defined in terms of the area of the horizon) decreases the Clausius entropy of the radiation increases to keep total entropy constant and equal to the initial Bekenstein entropy.
• Figure 2: Page curve, bipartite entanglement entropy: Under the "average subsystem" assumption applied to a pure-state bipartite system consisting of (black hole) plus (Hawking radiation) the entanglement entropy rises from zero to one half the initial Bekenstein entropy before dropping back to zero.
• Figure 3: Page curves for entanglement entropy and (asymmetric) subsystem information: Note the "kinked" behaviour of the (asymmetric) subsystem information and that the "not-quite sum rule" $\langle\tilde{I}_{{\mathrm{H}},{\mathrm{R}}}\rangle + \langle\tilde{I}_{{\mathrm{R}},{\mathrm{H}}}\rangle+ 2 \langle\hat{S}_{\mathrm{H}} \rangle = \hat{S}_\mathrm{Bekenstein,0}$ is satisfied.
• Figure 4: Modified Page curves, bipartite mutual information and (asymmetric) subsystem information: Note that the "sum rule" $\langle\tilde{I}_{{\mathrm{H}},{\mathrm{R}}}\rangle + \langle\tilde{I}_{{\mathrm{R}},{\mathrm{H}}}\rangle + \langle\hat{I}_{{\mathrm{H}}:{\mathrm{R}}}\rangle = \hat{S}_\mathrm{Bekenstein,0}$ is satisfied.
• Figure 5: Tripartite quantum (von Neumann) entropy flux: The quantum (von Neumann) analysis now reproduces the Clausius (thermodynamic) analysis. As black hole Bekenstein entropy (entanglement entropy) decreases, the entanglement entropy of the radiation increases, to keep total entropy approximately constant, at least to within 1 nat. In the limit where the environment (rest of universe) becomes arbitrarily large the correspondence is exact.