KITP: Hawking radiation and proposed decoding

Research date: 2026-10-05. The speculative trampoline model is paused. This note concerns published models. Lecture pages and slides were inspected; recordings were located, not watched.

Lecture route

  1. Alexei Kitaev — Efficient decoding for the Hayden-Preskill protocol, KITP, October 12, 2017. The closest match to the requested decoding formula. Lecture and recording · Original slides. Local copy: Kitaev_QInfo17Conf_KITP.pdf.

  2. John Preskill — The ghost in the radiation: Robust encodings of the black hole interior, KITP, January 17, 2020. Explores computational protection of interior information under a pseudorandomness assumption. Lecture · Paper with Isaac Kim and Eugene Tang. Local slides: Preskill_QGravity20Conf_KITP.pdf.

  3. Ahmed Almheiri — Is Information Thrown into a Black Hole Lost Forever? Resolving Stephen Hawking’s Black Hole Information Paradox, KITP public chalk talk, January 22, 2020. An accessible introduction to information recovery and the developments behind the Page curve. Talk page.

Hawking’s emission formula

For an uncharged, nonrotating black hole of mass M:

[ T_H=\frac{\hbar c^3}{8\pi G M k_B}. ]

The temperature decreases as mass increases. Here hbar is reduced Planck’s constant, c the speed of light, G Newton’s constant, and k_B Boltzmann’s constant. The thermal spectrum predicts emission statistics; it does not specify how to decode an individual message. Hawking, Particle creation by black holes (1975).

Kitaev’s equations and slide locations

Page numbers below are PDF pages, counted from 1; pages 4–17 were visually inspected.

  • Page 4: A is the message, B the black hole, B′ its accessible entangled partner, C the remaining hole, D new radiation. U maps AB to CD; a decoder acts on DB′.

  • Page 6: Recovery is possible when the reference R decouples from C: rho_RC approximately equals rho_R tensor rho_C. The model uses delta = d_R d_C Tr(rho_RC squared) − 1, with delta much less than 1.

  • Pages 11–12: Apply U* on auxiliary systems and project DD′ onto an EPR state. Success probability Delta = (1 + delta)/(d_A d_R); conditional fidelity is at least 1/(1 + delta).

  • Pages 13–16: Grover amplification uses order sqrt(d_A d_R) applications of U* and U^T.

  • Page 17: Generalizing to realistic thermal states remains an explicit limitation.

U* denotes entrywise complex conjugation in the specified basis; U^T is the transpose. Neither notation means that entanglement automatically supplies an inverse message. These equations describe a controlled quantum circuit.

Cost and assumptions

For k message qubits with d_A = d_R = 2^k, successful postselection scales as 4^(-k). The deterministic construction costs O(2^k C_U), where C_U is the circuit cost of U. It assumes access to the requisite entangled partner and implementation of the evolution operations. Preparing that resource is a separate difficulty. Yoshida–Kitaev paper, sections 1–2 and 4–5.

Preskill’s later work explains why recoverable information can nevertheless resist efficient outside access, conditional on pseudorandomness. This is not an unconditional result about every physical black hole.

Research implication

The useful next object to understand is the Hayden–Preskill channel and its decoder. Emission temperature, recovery fidelity, and decoding cost are different quantities. These sources do not establish a RedTail-X hash, a SHA-256 replacement, or a white-hole mechanism. No new simulation was run for this literature review.