A firewall confusion related to disentangling the vacuum

[S.Daedalus]

So while reading some old threads/blog posts on the black hole firewall paradox, it occurred to me that I had some residual confusion regarding why firewalls (supposedly) form at all. IIUC, the argument is that usually, the vacuum is a highly entangled state, and that disentangling it (in order to be able to save entanglement monotony and preserve unitarity of Hawking radiation) produces a maximally mixed state, which is infinitely hot. Basically, we trace out the vacuum on the other side of some boundary (e.g. the horizon), and what's left will be maximally mixed, i.e. a convex combination of states containing any possible number of modes.

But of course, the resulting reduced density matrix would have been the state according to an infalling observer, anyway---even if the entanglement is not broken, if an observer has access to only one part of a system in a maximally entangled state, his local physics are described by a system in a maximally mixed state. In the simplest case, if I have two entangled qubits, and one is behind the horizon, if I jump in, and have no access to the second qubit anymore, the qubit I have with me will be maximally mixed. No?

I think the resolution of this must somehow be that I'm mischaracterizing what an observer has access to. I'm imagining something like a 'bubble of vacuum', which is entangled with its exterior, and which, if we don't have access to the exterior, ought to be in a maximally mixed state. But actually, the 'state I have access to' is always given by (in the extreme) my cosmological horizon, or my causal past: whatever's influenced my state here and now. So I don't only have access to my bubble of vacuum, but also to its purification---and thus, observe it in a pure state. It's only when I'd do something to break the entanglement of the bubble with the rest of the vacuum that I would create a ball of fire, with the energy for this presumably coming from whatever disentanglement process I have performed.

Does this make sense to anybody?

 

[Haelfix]

Somewhat. I'm a little confused by a few things you are asking. Locally if you take a Hawking pair (call them b and b'), and you have Bob in the zone and Alice that falls into the black hole as usual. Is your question whether from the viewpoint of Bob that b and b' are NOT maximally entangled? For concreteness, a measurement might be putting a box around empty space, and then measuring the z component of the spin of a particle in the box.

Of course from Alice's point of view, she can still receive a signal from Bob, so she can actually verify the entanglement.

In the early days of the Firewall business, a lot of people were trying to play games with each observers causal patch in the spirit of complementarity to see if they could evade the argument. My feeling is that those attempts were broadly unsuccessful and extra arguments were devised that defeated most of those attempts. Currently only the Papodimas and Raju point of view still stands (and that is wonderfully subtle business).

see
arXiv:1207.5192 for the argument focusing on what observers have access too or some of Susskinds early papers on the subject for a review (also see arXiv:1409.1231 for lecture notes)

 

[S.Daedalus]

Thanks for your answer, Haelfix. I think this part:

Of course from Alice's point of view, she can still receive a signal from Bob, so she can actually verify the entanglement.

is actually the origin of my confusion. I thought that, quite apart from firewalls, etc., when Alice jumped into a black hole, the infalling Hawking radiation ought to be, to her, in a mixed state no matter what, but that seemed to lead to absurd conclusions. But of course, the part behind the horizon is not all she has access to: as you say, she can still receive signals from the outside. That's what I was groping towards with the whole 'what I have access to'-thing.

Also, I'm happy to see my outside-observer impression that the PR 'state-dependence' proposal may be the best contender in the firewall race at least somewhat substantiated (not that I can claim to understand it in its full subtlety).