Paradox No More: How Stimulated Emission of Radiation Preserves Information Absorbed by Black Holes

TL;DR

The paper argues that information is not lost in black holes because stimulated emission accompanies Hawking radiation, enabling a positive Holevo capacity for classical information to be transmitted through the black-hole channel. By modeling spontaneous and stimulated emission via a Bogoliubov-like unitary and analyzing both early-time and late-time channels, it shows that information can be reconstructed from exterior radiation, resolving the apparent information destruction and restoring time-reversal symmetry. The work links black-hole radiation to quantum-optical amplification and backreaction physics, suggesting a unitary, information-preserving process that aligns with concepts such as the fully quantum Slepian-Wolf protocol. These insights recast black holes as information-preserving quantum amplifiers, with implications for CPT symmetry and the foundational understanding of black-hole thermodynamics.

Abstract

Black holes have been implicated in two paradoxes that involve apparently non-unitary dynamics. According to Hawking's theory, information that is absorbed by a black hole is destroyed, and the originally pure state of a black hole is converted to a mixed state upon complete evaporation. Here we address one of the two, namely the apparent loss of (classical) information when it crosses the event horizon. We show that this paradox is due to a mistake in Hawking's original derivation: he ignored the contribution of the stimulated emission of radiation that according to Einstein's theory of blackbody radiance must accompany the spontaneous emission (the Hawking radiation). Resurrecting the contribution of stimulated emission makes it possible to calculate the (positive) classical information transmission capacity of black holes, which implies that information is fully recoverable from the radiation outside the black hole horizon.

Figures (7)

• Figure 1: Penrose diagram showing the early-time modes $a$ and $b$, propagated backwards in time from future infinity ${\mathscr I}^+$ towards past infinity ${\mathscr I}^-$ (just as the black hole formed) and traveling just outside and just inside of the event horizon. The mode at future infinity is annihilated by $A$. In this depiction, modes $a$ and $b$ are shown separated for clarity, but they are just inside and just outside the horizon and coincide at ${\mathscr I}^-$.
• Figure 2: Outgoing particles (arrows pointing down) for $m$ incident particles on the horizon. In this sketch, $\langle n\rangle$ ($\langle \bar{n}\rangle$) refer to the mean number of particles (anti-particles) given by Eq. (\ref{['hawking']}) that are generated in front and behind the horizon (to conserve particle number).
• Figure 3: Early-time capacity $\chi$ for a binary non-reflecting ($\Gamma=1$) black hole channel as a function of the parameter $z=e^{-\omega/T_{{\rm BH}}}$. The limit $z=0$ corresponds to a black hole with infinite mass (vanishing surface gravity), while $z\to1$ as the mass of the black hole tends to zero.
• Figure 4: Penrose diagram showing early-time modes ($a$ and $b$) and late-time modes $c$. Late-time modes are scattered at the horizon with probability ${\pazocal R}=1-\alpha=1-\Gamma_0$ (the black hole reflectivity). A perfectly absorbing black hole has ${\pazocal R}=0$.
• Figure 5: Capacity $\chi$ for a binary non-reflecting black ($\Gamma_0=1$) hole channel as a function of the parameter $z=e^{-\omega/T_{bh}}$. The solid line represents the late-time capacity, while the dashed line is the capacity for early-time modes. Note that since $g\approx z$, we can see this plot also as depicting the dependence of the information transmission capacity on the mode coupling strength in the black-hole Hamiltonian.