Hayden-Preskill protocol and decoding Hawking radiation at finite temperature

TL;DR

This paper analyzes the Hayden-Preskill protocol at finite temperature, deriving conditions for information recovery from Hawking radiation and showing that decoding fidelity decreases due to noise, decoherence, and finite temperature effects.

Contribution

It extends the Hayden-Preskill protocol to finite temperature scenarios, including noise and decoherence, and calculates the associated decoding fidelities.

Findings

Decoding fidelities are generally less than one at finite temperature.

Finite temperature, noise, and decoherence make decoding more challenging.

Decoding strategies are less effective compared to the infinite temperature case.

Abstract

We study the Hayden-Preskill thought experiment at finite temperature and obtain the decoupling condition that the information thrown into an old black hole can be extracted by decoding the Hawking radiation. We then consider the decoding Hayden-Preskill protocol at finite temperature assuming the observer outside the black hole who has the access to the full radiation and the unitary dynamics of the black hole. We also consider the cases when the Hawking radiation has noise and decoherence in the storage. The decoding probabilities and the corresponding fidelities are calculated. It is shown that for all the three cases we have considered, the decoding fidelities are less than unity in general. This result indicates that at finite temperature, the decoding strategy and the recovery algorithm is harder to realize than that at infinite temperature.