Fluctuation in the fidelity of information recovery from Hawking radiation

The interior of a pure-state black hole is known to be reconstructed from the Petz map by collecting a sufficiently large amount of the emitted Hawking radiation. This was established based on the Euclidean replica wormhole, which comes from an ensemble averaging over gravitational theories. On the other hand, this means that the Page curve and the interior reconstruction are both ensemble averages; thus, there is a possibility of large errors. In the previous study [R. Bousso and M. Miyaji, Fluctuations in the entropy of Hawking radiation], it was shown that the entropy of the Hawking radiation has fluctuation of order <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>e</a:mi><a:mrow><a:mo>−</a:mo><a:msub><a:mi>S</a:mi><a:mrow><a:mi mathvariant="bold">BH</a:mi></a:mrow></a:msub></a:mrow></a:msup></a:math>, thus is typical in the ensemble. In the present article, we show that the fluctuations in the information recovery are similarly suppressed by the factor <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msup><d:mi>e</d:mi><d:mrow><d:mo>−</d:mo><d:msub><d:mi>S</d:mi><d:mrow><d:mi mathvariant="bold">BH</d:mi></d:mrow></d:msub></d:mrow></d:msup></d:math> by studying the fluctuations of the relative entropy difference in the encoding map and the entanglement fidelity of the Petz map. This result establishes the typicality of the information recovery in the ensemble. In addition, we also compute the entanglement loss of the encoding map.