A simple naïve question on the information paradox

Question

The 'information paradox', as I understand it naïvely, points out that there is a conflict between i) the 'classical' picture of a black hole, in which information passed through the event horizon will be lost after the phenomenon of Hawking evaporation; and ii) the 'quantum' or maybe more accurately put the 'semi-classical' picture in which a black whole seen as a system in quantum-field theory should obey a unitary evolution in its Hilbert space of states, which precludes loss of information.

I have difficulties to see the paradox here for two reasons, one probably not very serious, the other more serious. First reason: the classical (and even the semi-classical) pictures of a black hole are non-definitive as long as we don't know how to combine gravitation and quantum effects into a sound theory. Of course one might reply that this is precisely the point made by the paradox.

More serious reason: the unitary evolution in quantum mechanics holds only as long as no measurement is made on the system. Wave function collapse phenomena (which are non-unitary and even nonlinear) appear to be put under the rug in the statement of the paradox. What happens to a black hole when it is observed (e.g. how decoherence might play a role) ? Is there a discussion of this question and how it affects the information paradox to be found in the literature?

Comments

are we even completely sure that Hawking does not know the solution to this paradox? if a pair is created near the horizon, then there seems to be a copy of the information both inside and outside it? i don't expect him to know the complicated answer, but maybe in terms of pair creation he might know it?

Answer 1

The reason the black hole information loss problem is a problem, is that one expects semi-classical gravity *should* be a valid approximation to quantum gravity, for large black holes, because the invariant quantities measuring the geometry are small in units of the Planck scale. However, taking this "effective field theory" picture seriously leads to an apparent contradiction, since on the one hand any quantum theory must have unitary evolution, and on the other hand the state of the system moves from a pure state (as the black hole is formed) to a mixed state (the thermal radiation after the black hole has evaporated).

Appealing to measurement doesn't fix the problem. First you would need to identify the point at which the measurement is supposed to happen. But more to the point, a measurement of a pure state would project the state into another pure state. If anything the issue is the reverse in that the problem is that we are losing access to measurable information. Once the information falls behind the event horizon, an outside observer can no longer access it, and so to describe the quantum state must "trace over" the state inside the horizon; this is what leads to the external state being a mixed state. After the black hole evaporates, the black hole interior is gone, and the mixed state is all that is left. But evolving from a pure to a mixed state can never happen in unitary evolution.

Comments

"The reason the black hole information loss problem is a problem, is that [semi-classical gravity should be a valid approximation to quantum gravity]

This is exactly correct, but if you add the (for large black holes) part, your statement is wrong.

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Considering "information" as a physical concept I find problematic. The paradox can certainly also be seen as considering what happens if part of a quantum system in a pure state "falls behind the event horizon". 

The problem here is that how, from the point of view of an external observer, can such a part (a subsystem) fall behind the event horizon within a finite interval of the observer's time? 

If it cannot, the paradox is irrelevant.

If it can, then the following should be considered: In decoherence, the states of the environment are traced over in the density matrix of the total system, i.e., the observed system plus the environment, leading (usually) to a density matrix of the observed system that looks like the density matrix of a mixed state. However, the state of the total system remains a pure state. 

If one uses the same approach w.r.t. the states of the black hole (with the subsystem fallen in), tracing them out, then similarly the density matrix for the external system may only look like that of a mixed state, while the total state (including the state of the subsystem in the black hole) is still a pure state. A more exact treatment of the evaporation might perhaps recover the pure state.

As long as the internal quantum states of a black hole are not really known, it may be questioned whether such a tracing out of its states is justified. In the case of a system in a laboratory, the states of the environment, e.g., the states of the air molecules in a vicinity of the system within which they have any significant chance of interacting with the system, may change sufficiently between repeated measurements so that they can at each repetition be considered unknown. Is the same guaranteed for the states inside a black hole?

As a further remark: In view of

Unruh, 1976, Phys. Rev. D, 14, number 4, p. 870, "Notes on black-hole evaporation"

it appears questionable how well established theoretically black-hole evaporation is.

Answer 2

Information is a very subjective notion. For somebody something new is an information since it brings a new knowledge; for another person the same is not new and does not increase his knowledge (noise rather than information).

Informamiton is barely conserved (preserved) while transferring from one person to another. This is our human feature.

Sending some information to a black hole makes its retreival much more difficult, so the general statement is that the information is lost or distorted while exchange.