Distinguishability of black hole microstates

We use the Holevo information to estimate distinguishability of microstates of a black hole in anti-de Sitter space by measurements one can perform on a subregion of a Cauchy surface of the dual conformal field theory. We find that microstates are not distinguishable at all until the subregion reaches a certain size and that perfect distinguishability can be achieved before the subregion covers the entire Cauchy surface. We will compare our results with expectations from the entanglement wedge reconstruction, tensor network models, and the bit threads interpretation of the Ryu-Takayanagi formula.

Figure 1

The Holevo information $\chi (A)$ in ${\mathrm{AdS}}_3/{\mathrm{CFT}}_2$, as a function of the radius ${\ell }_A$ of the subregion $A$. Note that $\chi (A)$ is identically equal to zero until $A$ covers one half of the Cauchy surface of CFT. Above this point, $\chi (A)$ increases monotonically until it achieves the maximum value, which is the Bekenstein-Hawking entropy $S_{\mathrm{BH}}$ of the black hole, at ${\ell }_{\mathrm{crit}}$ defined in Sec. 2.

Figure 2

The Ryu-Takayanagi surfaces for region $A$ for the thermal ensemble $\rho =\sum _i{p}_i{\rho }_i$.

Figure 3

Two possible Ryu-Takayanagi configurations for $A_1\cup A_2$. The total area of the surfaces on the left is $S({\rho }_{A_1\cup A_2})=S({\rho }_{{\overline{A}}_2})+S({\rho }_{A_1\cup {\overline{A}}_2\cup A_2})$, while that on the right is $S({\rho }_{A_1\cup A_2})=S({\rho }_{A_1})+S({\rho }_{A_2})$.