Do black holes comply with the principle of unitary evolution? Claus Kiefer, "Quantum gravity", 3rd ed., page 220/221, says in chapter 7 "Quantization of black holes":

"A theory of quantum gravity should give a definite answer to the
  question of whether unitarity (with respect to an outside observer) is
  preserved or not."

I am not able to see the problem here. According to the basic characteristics of a black hole, an outside observer can never observe what is happening inside the event horizon, and for everything happening outside the event horizon there is no unitarity issue.
So is there an error in the cited phrase or am I missing something? What exactly should a theory of quantum gravity provide?
 A: The quote is referring to the information loss paradox. 
The paradox arises when considering quantum fields in a curved black hole background. Hawking has shown long ago that black holes (where black stems for zero emitting objects) actually becomes radiating objects with a black (!) body spectrum. Moreover this radiation make them slowly evaporate, therefore unitarity of quantum mechanics and evaporating black hole solutions seems to be mutually incompatible, even though they are solid predictions of the respective theories.
A quantum theory of gravity should:


*

*Provide a mechanism to describe unitary black hole evaporation (or the contrary, even though few people (no one?) are willing to give up unitarity) 

*Get rid of the central singularity

*Give a microscopic interpretation of the entropy formula for black holes

*Give a gravitational descriptions of these microstates
String theory has achieved 1) and 3) 20 years ago. Some approaches (the fuzzball proposal for instance) are working on solving even 2) and 4). In some particular (but physically unrealistic) systems the problem has been completely solved (D1D5 system).
A: Semi-classical gravity does infer a situation which complicates the subject of unitarity within black hole physics. The evolution of two states after forming the black holes are identical, leading to a mixed state obtained through integrating the thermal Hawking radiation states. It leads to the information paradox.
The problem with this is that the final states are identical - we cannot recover the initial state of the evolution just by knowing the final state, even in principle. This contradicts unitarity evolution in quantum mechanics.
Reference: "Black Holes, Information, and Hilbert Space for Quantum Gravity" (2012).
In principle unitarity preserves the ability to recover the initial state if we know the final state by applying the inverse of the time evolution $e^{+iHt}$.
