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?

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.

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.