JI/poly = MIP* = RE?

Recently, the string theorist Leonard Susskind proposed a new computational complexity class, "JI/poly", which includes all computations that can be performed in polynomial time by "jumping into" (j.i.) a black hole. We should also bear in mind his "ER=EPR" work with Maldacena, in which a wormhole is actually two entangled black holes. There is interest in the scenario in which an observer enters each black hole, and can never make it out the other side (this is for the case where the wormhole is non-traversable), but the observers can briefly meet inside the wormhole. I think JI/poly is meant to apply in scenarios like this. 

(Susskind also seems to say that the length of a wormhole has something to do with quantum computational complexity. This may be a holographic statement - that the length of the wormhole has something to do with the computational complexity of processes in its nongravitational dual. I don't have any opinion or use for this idea yet, but I mention it here for future reference.) 

Also recently, and closer to the mainstream of quantum gravity research, Witten et al proposed "an algebra of observables for de Sitter space" (finding observables for de Sitter space has been on Witten's mind for at least twenty years). They mention that this is a Type II1 von Neumann algebra (the algebra for an ordinary, nongravitational quantum field theory in flat space is Type III). 

This rang a bell for me, and indeed, there is an old conjecture in the theory of von Neumann algebras, due to Connes. Apparently the conjecture boils down to asking whether entanglement created by tensor products, is capable of approximating, arbitrarily closely, all forms of entanglement that can be created by commuting operators. This conjecture was falsified in 2020, as a corollary of the result in complexity theory that "MIP* = RE". 

MIP* is the set of computations that can be performed by two "provers" that cannot communicate, but which share an "infinite" amount of entanglement. RE is the set of "recursively enumerable" decision problems. This means there's a program which eventually lists every entity with a given property. So if you want to know whether entity X has property Y, and if there is a program that lists everything with property Y, then you can run the program and wait to see if X ever turns up in the list. Unfortunately, it could take a really long time for X to show up, and in general, you have no way of knowing if X ever will show up. 

But what MIP* = RE seems to mean, is that if X is in the list of entities with property Y, two infinitely entangled provers can prove it to you, in polynomial time. This is surprising because RE includes decision problems where it can take arbitrarily supra-exponential time for X to show up in the list. The provers, however, have infinite entanglement to work with - the equivalent of infinitely many entangled qubits - and that allows them to overcome the classical time complexity. 

I was just re-reading the thread on Scott Aaronson's complexity blog announcing the result, reliving the struggle of the commenters to understand what it means to have two provers that can't communicate but which have infinite entanglement - and suddenly I thought: that sounds like the two entangled black holes, in the Maldacena-Susskind model of a non-traversable wormhole! The observers can't pass through the wormhole - this is the counterpart of being unable to communicate - but if they both jump in, they can meet in the middle - something made possible by the entanglement of the black holes, since that is what creates the wormhole bridge. 

Does it mean that JI/poly = MIP* = RE? At least for the specific case of "two observers meeting in a non-traversable wormhole"... And what about the work by Witten et al? Well, they consider the case of two "static patches" in de Sitter space. Perhaps the two patches are analogous to the two event horizons of the non-traversable wormhole... 

I still have a lot to learn about all this, but I was sufficiently excited by the connection, that I decided to blog about it right away. :-)