How can ER = EPR ?
Consider two sets, Alice’s set A and Bob’s set B, of huge number of particles such that each individual particle in A is entangled with one particle in B (forming Bell pairs). These two sets of particles were taken away by Alice and Bob to a far away distance, space-like separated, from each other. Now Alice and Bob squeeze and collapse their respective set of particles into black holes. And the assumption made here is that, during the collapse of the particles to form black holes, the entanglement between Bell pairs survive unaffected! With this, The ER bridge between the black holes is argued to be quantum entanglement between the same black holes.
My question is that, how come the entanglement between Bell pairs survive during their collapse to form black holes? It’s impossible. This impossibility is assumed to be `somehow’ possible and then ER is argued to be equal to EPR. It seems to me that, it’s a desperate attempt to make somehow ER = EPR, which are two seemingly unrelated physical phenomena. Please educate me in this context if I am wrong.
As an extension to the above question, suppose that only Alice collapses her set of particles to form black hole but not Bob. He retains his set of particles as they are. Then, how this situation can be described by entanglement?
Why it is impossible for the entanglement between Bell pairs to survive during the collapse to form black holes? Here we just, by certain (hyphothesis) way, squeeze two sets so that they form 2 spheres (for example) whose radii are smaller than the Schwarzschild radius. During this process, the entanglement can survive or not, and here the entanglement are assumed to survive.
The idea here, i think, is as follows. First we have the ER = EPR for entangled particles. Remember that this is only a conjecture, which supposes that two entangles particles are connected by a wormhole so that their states should be described together. Now, we have two sets of entangled particles which are squeezed to form two black-holes. In General Relativity, two black-holes should be connected by a Schwarzschild wormhole, or a ER bridge, which is a solution of the Einstein field equation. The question now is: what could be the origin of the ER bridge for those two black-hole? The argument, then, is that: since two black-holes are formed from two entangled sets of particles, and it is assumed that ER = EPR for each entanged pair, so the ER bridge may be the quantum entanglement between two blackhole (we have ER = EPR for particles, then it's true for the black-holes).
It's well-known how delicate the entanglement between two quantum particles is. Consider two maximally entangled particles `a' and `b' separated by a large distance. If one makes another quantum particle `c' to interact with `a', then `a' and `c' becomes entangled such that the particle `b' is thrown into a definite quantum state and the entanglement between `a' and `b' becomes broken. (Further, if one makes a joint measurement of `a' and `c' and communicates this result classically to the `b' place, then it's possible to project `b’ into a quantum state exactly as that of `c' and is known as quantum teleportation which is an established experimental fact). The whole point is that, the moment `c' interacts with `a', the earlier entanglement between `a' and `b' are broken instantaneously. But, in the ER=EPR case, even though the particles in the set `A' are interacting among them self to form back-holes, it is assumed to maintain entanglement with the particles in set `B'. Once this kind of impossibilities are assumed to be `some how' possible, then anything can be conjectured be equal to anything else.
For simplicity, let us consider N number of spin 1/2 particles in set A and also in set B (of course one can consider any other property instead of spin). If all the particles in A and B collapse to form black holes, then at the best one can have a black hole A with spin having a value between N/2 and -N/2 (for z-component). Then the disentangled black hole B’s spin will be thrown into a direction opposite to the previous one because the particles are assumed to form spin singlets. The quantum entanglement between A and B has gone even though they gain opposite spin along z-axis. Whether they are connected by ER bridge or not is another question. It depends on whether Einstein's equations permit such a solution or not.
If only Alice collapses her set of particles to form black hole but not Bob. Then how can you describe this situation? But it’s not difficult to understand this situation by quantum mechanics. Whatever the total spin state of black hole A, that is the spin state of all the particles in set B, which didn’t collapse to form black hole. There is no more entanglement between black hole A and the particles in set B. But, all the particles in set B become entangled such that their total spin along z-axis is exactly opposite to that of black hole A. It’s impossible to claim from Einstein’s equations any ER bridges between black hole A and the particles in set B.