Not a new paper but, in this version, I’ve added some comments on the difference between gauge theories and gravity and the relevance to the AMPS paradox, about which I got a number a questions.
It seems to have been clear for a while that the resolution to the Mathur/AMPS paradox is that operators in the black-hole interior can be expressed as complicated combinations of operators outside. This was also a feature of the construction of the interior that Kyriakos Papadodimas and I set up. However, I’ve long felt that it would be nice to have a toy model where this could be demonstrated as cleanly as possible with no conjectures or hand waving. This paper sets out such a model.
The technical tool is to use Bell correlators to express the monogamy of entanglement. (Bell correlators are far better controlled than the von Neumann entropy). Then one can show that that, even in empty AdS, if one is allowed to use arbitrarily complicated operators from distinct regions, it is possible to violate the monogamy inequalities precisely like the Mathur/AMPS paradox. Moreover, everything can be calculated cleanly, with explicit numbers. In this model, I violate the monogamy inequalities by 42.5%!
What this shows is that, in quantum gravity, operators which appear to be in different regions may secretly be acting on the same degrees of freedom and so can access the same quantum information.
It is important that this phenomenon appears only in gravity, and not in gauge theories. In gauge theories, because there exist exactly local gauge-invariant operators, one cannot read off information about interior excitations from a far-away region; someone could act with a little localized Wilson loop in the interior, while leaving every observable for the far-away observer invariant. So, while in gauge theories, operators from different regions may not commute because their Wilson lines intersect, “complementarity” (in the sense described above) seems to be a feature unique to quantum gravity.
Comments (especially critical ones!) are welcome. https://arxiv.org/abs/1809.10154