Lecture 21: The Strong Subadditivity Paradox

In this lecture, we first reviewed the strong subadditivity of the von Neumann entropy. This is the statement that, given three systems, A, B, C, we have:S_{AB} + S_{BC} \geq S_{A} + S_{C}.

We can apply this to black hole evaporation to obtain a paradox as follows. Consider an old black hole, past its Page time. Divide the geometry into three regions: a near-horizon region outside the black hole, B, an analogous region inside the black hole, C, and the rest of the exterior geometry, A. Then the Page curve tells us that: S_{AB} < S_{A}.

The fact that the horizon is smooth, and that the Hawking radiation is thermal tells us that: S_{BC} < S_{C}.

This leads to a seeming contradiction with the strong subadditivity of entropy. This paradox was first outlined by Mathur in arXiv:0909.1038 and later explored further by AMPS in arXiv:1207.3123

Lecture 21 notes