Bits, Bits, Wherefore Art Thou Bits?

Black holes evaporate via the process of Hawking radiation. If we take the initial pure state describing a body which will collapse and form a black hole which can evaporate, then it would appear that this pure state will evolve, after the complete evaporation, to a state which is mixed. Since it is doing this without leaving any remnant for quantum information to hide in, this would seem to violate the unitarity of quantum theory, which does not allow a closed system pure state to evolve to a closed system mixed state. Similarly, if we consider the an evaporating black hole, there are spacelike surfaces which contain the collapsing body and the outgoing Hawking radiation: therefore if we somehow believe that the outgoing Hawking radiation contains quantum information which we have thrown into the black hole, this process must have cloned this information and thus violated the linearity of quantum theory. This reasoning has been hounding theoretical physicists for ages and is known as the black hole information paradox.
A few years ago, Horowitz and Maldacena proposed a solution to this problem (see hep-th/0310281, “The black hole final state”.) The basic idea of their solution is to use boudary conditions at the black hole singularity to fix this problem. Actually the basic idea of their solution is to use post-selected quantum teleportation. How does this work? Well consider the initial pure state describing the collapsing matter. In addition to this collapsing matter, there will be the Unruh state of the infalling and outgoing Hawking radiation. Now it works out that this Unruh state is basically a maximally entangled quantum state. So what Horowitz and Maldacena propose is that, in order to get the pure state describing the collapsing matter “out” of the black hole, one need only “teleport” this information by using the Unruh radiation. Indeed, we can consider such teleportation: perform a measurement of the pure state describing the collapsing matter and the state of the infalling Hawking radiation in the appropriate (generalized) Bell basis. Now, of course, in order to complete the teleportation procedure we need to send the result of this measurement to the outgoing radiation and apply the appropriate unitary rotation to get the pure state of the collapsing matter outside of the black hole. But, of course, we can’t do this: we can’t send the classical information from inside the black hole to outside. So what Horowitz and Maldacena propose is that instead of performing the full teleportation, a particular result is post-selected for this teleportation. And this result will be the one where you do nothing in the teleportatin protocol. In other words, they postulate that the black-hole singularity acts like a measurement in which one always gets a particular outcome. This is the “final state” projection postulate.
This is a very nice way to try to avoid the black-hole information paradox. One reason is that it seems to put all the craziness at the black-hole singularity, and not of the craziness elsewhere, where we would expect our current ideas about physics to be rather robustly true. But does this really solve the problem? Gottesman and Preskill, in hep-th/0311269 argue that it does not. What Gottesman and Preskill point out is that the collapsing matter and the infalling Unruh radiation will, in general, interact with each other. In this case, the problem is that this interaction can cause the information to no longer be faithfully reproduced outside of the black hole. The problem is that this interaction causes post-selection onto a state which is no longer maximally entangled: the state will then fail at producing the appropriate teleported copy of the state outside of the black hole. Sometimes, in fact, this interaction can completely destroy the effect Horrowitz and Maldacena are attempting to achieve (if we disentangle the post-selected state.) This does not bode well for this post-selected answer to the black-hole information paradox.
So is this the end? Well it is never the end! One might try to show that the amount of information destroyed in the Gottesman-Preskill scheme is small. Now this, in some sense, would be comforting: who would notice a single qubit missing in a world so large? On the other hand, while this might be comforting, it would certainly cause certain consternation among those of us who would like an explanation which is satisfying without giving up even an ounce of information. This question, of the amount of informatin lost, is addressed, in part, in the recent paper “Almost Certain Escape from Black Holes in Final State Projection Models” by Seth Lloyd, Physical Review Letters, 061302 (2006)
Consider the Horowitz and Maldacena scheme as modified by Gottesman and Preskill with an interaction between the infalling and collapsing matter state described by U. Lloyd calculates the fidelity of this scheme, when averaged over all unitaries U according to the Harr measure (this fidelity is the overlap between the initial pure collapsing matter state and the outgoing state after the Horowitz-Maldacena post-selection.) Lloyd finds that this fidelity is 0.85…. In the paper, Seth argues that this is indeed large enough as to indicate that most of the information survives. So, I ask, is 0.85… fidelity really satisfying? I would argue that it is not, even if I accept that averaging over the unitaries makes any sense at all. Why? Well suppose that you daisy chain this procedure. Then it seems that your average fidelity could be as small as desired. Thus while you might argue that there is only a small loss of unitarity for random unitaries, there are physical processes for which this loss of unitarity is huge. This seems to be one of the lessons of quantum theory: destroying only a little bit of information is hard to do without bringing down the whole wagon. Daniel Gottesman says, I think, exactly this in this New Scientist article. Even Lloyd, at the end of his article, hedges his bets a bit:

Final-state projection will have to await experimental and theoretical confirmation before black holes can
be used as quantum computers. It would be premature to jump into a black hole just now.

What about you? Would you jump into a black hole with the fidelity that we could put you back together of 0.85… on average?