Lecture 6: How close are pure and mixed states?
In this lecture, we explored a simple question in quantum statistical mechanics. From the point of view of physical observations, how closely do typical pure states resemble mixed states. We found that typical superpositions of states from a given energy band are exponentially close to the microcanonical density matrix.
This simple result, when applied to black-hole evaporation, tells us that Hawking's original argument is not precise enough to lead to a paradox. Hawking argued, using an approximate computation of expectation values of simple observables, that black-hole evaporation would produce a mixed state. But since pure states can resemble mixed states to exponentially accuracy, and since Hawking's computation did not keep track of such corrections, this argument is invalid.
Our result teaches us that, since black-holes are statistical systems, it is not surprising that simple correlators at null infinity are approximately thermal. In fact, it would have been surprising had this not been the case.
