Is there an information loss paradox?

H. D. Zeh (Dec. 2004)

Stephen Hawking's claim of a lost bet recently stirred up a lot of interest and discussion in the media (physical journals included). If correct, it would mean that the "information" falling into a black hole must later come out in some way, for example in terms of correlations existing within the Hawking radiation – even though this can hardly ever be used to recover the original information. The opposite assumption that this information is irretrievably lost (as it should for classical black holes containing a singularity) is generally regarded as a paradox, since it seems to violate unitarity.

Most physicists so far seem to agree that they do not understand Hawking's new arguments against his own bet, which were presented by means of a complicated calculation using specific methods and approximations. However, any calculation must be based on certain concepts and assumptions, in particular about the validity of unitarity on a fundamental level, or about the reality of spacetime. If unitarity were assumed, Hawking's claim would not need any further calculation (as pointed out long ago by Don Page, for example). Can this fundamental question about the validity of unitarity then be answered on the basis of a calculation? I don't see how: all one can learn in this way is what has been tacitly assumed (or what may have been induced by specific approximations).

However, there exist many well known descriptions of natural phenomena, in which information becomes effectively lost. Let me first emphasize that "information" is here always understood in the objective sense of a formal ensemble evolving according to certain dynamical laws. So it is generally accepted to be conserved under deterministic (or unitary in the case of quantum theory) laws. The question of principle therefore regards these hypothetical though empirically successful laws – not what we happen to know or are able to observe or calculate.

In practice, many deviations from information-conserving laws are meaningful and successfully used. For example in classical physics, all master equation (such as Boltzmann's collision equation) are based on the permanent neglect of arising correlations or other kinds of "irrelevant" information when calculating into the future direction of time. The validity of this very restrictive approximation, which requires a special cosmic initial condition, is responsible for the increase of entropy. By definition, ensemble entropy would be conserved under deterministic equations of motion if it were calculated by taking into account all irrelevant information, such as correlations, but this would require a highly non-extensive concept of entropy.

Similarly, in quantum theory, global unitary would warrant the conservation of global entropy or "information" (negentropy), but this global unitarity can hardly ever be probed or used. If one assumes (with Bohr) that quantum theory is not applicable to macroscopic objects, unitarity is not even an issue for them. If one assumes instead (with von Neumann, Pearle or Ghirardi) that the Schrödinger equation has to be modified in order to describe the collapse of the wave function, unitarity is an approximation, valid for microscopic objects only. However, if one assumes that the Schrödinger equation is exact, one has to accept a superposition of myriads of branching Everett "universes". This superposition would not just describe our observed universe, which is quantum mechanically represented by one Everett branch. A single branch by itself describes the same stochastic phenomena as a genuine collapse (that is, measurements in a general sense). Information is here deterministically transformed into unaccessible phase relations between different Everett components. Why then is unitarity such a burning question in quantum gravity or unified quantum field theory – beyond the "normal" and much discussed problem of quantum measurements? It seems that physicists working in special fields are entirely ingnorant about what is known or done in other fields.

Well, unitarity would be a problem (of principle) in the ideal case of completely isolated black holes. However, black holes are macroscopic objects, which are inevitably subject to the permanent action of decoherence in the ordinary quantum mechanical sense of phase relations becoming dislocalized (as described in detail by Claus Kiefer). All quasi-classical concepts (including spacetime) owe their existence to this irreversible process. In this way, the existence of black holes with their future horizons requires the time arrow of radiation and decoherence. Since general relativity is time reversal-invariant, a universe containing advanced radiation only would have to contain time-reversed "white" holes (with "growing hair"). Isolated quantum holes (not affected by decoherence) must be expected to come in time-symmetric superpositions of black and white (that is, in energy eigenstates without any classical interpretation). They can therefore not be responsible for a new (and unnecessary) kind of decoherence that is based on the presumption of (classical) future horizons. A macroscopic "hole" in our time-directed universe must be permanently subject to the information loss (entropy increase) by means of ordinary decoherence, and therefore be "black" (in the sense of apparently possessing a future horizon and obeying the no-hair theorem in the asymptotic future).

Quantum interference (confirming unitarity) has now been observed with rather complex molecules, and it may some day be confirmed for small individual viruses. One may even argue what  an interference experiment with conscious observers – though impossible in practice – would mean (for example, whether they would observe their own passage through one or simultaneously through all slits). However, the assumption of a strictly isolated black hole (required for a corresponding experiment) seems to be self-contradictory. Probing the unitarity in processes containing black holes would require the recoherence (recombination with respect to a local observer) of all thereby arising Everett branches into one branch again! While no black hole can ever evolve unitarily, there is no specific reason on the basis of quantum gravity to have doubts about global unitarity, valid for the superposition of all Everett branches (as far as a concept of dynamical time remains meaningful).

See also Where has all the information gone?
             Sect. 7 of Roots and Fruits of Decoherence
             Sect. 6.2.3 of The Physical Basis of the Direction of Time

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