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Corresponding author.

Corresponding author.

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We use Holevo information in the two-dimensional conformal field theory (CFT) with a large central charge

The black hole information paradox lies in the fact that a pure state seems to evolve into a thermal state through Hawking radiation, and thus it violates the unitarity of quantum mechanics. This paradox can be partially resolved if there exists black hole microstates, which are pure states that cannot be distinguished from the underlying thermal state. This resolution however calls for a complete theory of quantum gravity, which is beyond the reach at this moment. However, with the help of the anti–de Sitter/conformal field theory (

Recently, it was proposed in

By construction

To investigate the information loss paradox of black hole in Einstein gravity in the three-dimensional anti–de Sitter (

We consider a 2D large

By using the holographic entanglement entropy (HEE)

We find that Holevo information is not vanishing as long as the length of the interval is nonvanishing, and this indicates that the black hole microstates are distinguishable from thermal state as long as the measuring region is nonvanishing. We also find the Holevo information is smaller than the thermal entropy as long as the interval is shorter than the whole system.

For calculation convenience we choose that the interval

For a canonical ensemble thermal state we have

The holographic Holevo information

We consider only contributions from the vacuum conformal family, and will briefly discuss the contributions from nonvacuum conformal families in the end of the Letter. For the short interval

For the long interval

We denote the results of Eqs.

The leading

In low temperature limit, we have

In CFT, the above total indistinguishability can be lifted by taking into account the finite-size effect exponentially suppressed by the factor

We now consider the microcanonical ensemble thermal state with fixed high energy

The Cardy formula can be generalized to the cases of various multipoint correlation functions on a torus

For the long interval case, we use the OPE of twist operators in Refs.

Lastly, we consider the leading contribution to Holevo information from a nonidentity primary operator

For the conclusion of this Letter, we would like to mention the implication of the almost vanishing short interval Holevo information to our recent finding of nongeometric states in Ref.

We would like thank Alice Bernamonti, Pasquale Calabrese, Federico Galli, Manuela Kulaxizi, Hong Liu, Andrei Parnachev, Tadashi Takayanagi, and Erik Tonni for helpful discussions. J. Z. would like to thank the Galileo Galilei Institute for Theoretical Physics and the organizers of the workshop “Entanglement in Quantum Systems” for hospitality and for being given the opportunity to present part of the result of work, and to thank participants of the workshop for helpful discussions. W. Z. G. is supported in part by the National Center of Theoretical Science (NCTS). F. L. L. is supported by Taiwan Ministry of Science and Technology through Grant No. 103-2112-M-003-001-MY3. J. Z. is supported in part by Italian Ministero dell’Istruzione, Università e Ricerca (MIUR), and Istituto Nazionale di Fisica Nucleare (INFN) through the “Gauge Theories, Strings, Supergravity” (GSS) research, and by Fondazione Cariplo and Regione Lombardia, Grant No. 2015-1253.