General relativity contains solutions in which two distant black holes are connected through the interior via a wormhole, or Einstein-Rosen bridge. These solutions can be interpreted as maximally entangled states of two black holes that form a complex EPR pair. We suggest that similar bridges might be present for more general entangled states. In the case of entangled black holes one can formulate versions of the AMPS(S) paradoxes and resolve them. This suggests possible resolutions of the firewall paradoxes for more general situations.
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One of the most important papers in this field is [0]. (As is common in science, it built on a long history of nearby ideas, e.g. [1].) An important calculation done around this idea is presented in [2], in which remarkable progress is made on Hawking's black hole information paradox. All of these papers have some amount of mathematical backing, but you'll find plenty of formulas and calculations in [2]!
[0] https://arxiv.org/abs/1306.0533
[1] https://arxiv.org/abs/1005.3035
[2] https://arxiv.org/abs/1908.10996
I been juggling the following three papers for a bit and trying to brush my maths up. I don't know why I feel a hunch between them, but I do know that the hunch is to do with this problem.
https://arxiv.org/pdf/1803.06824.pdf
https://arxiv.org/pdf/1306.0533.pdf
https://arxiv.org/pdf/1204.2779.pdf
I will almost certainly be wrong about whatever it is. However I may have more updates on how wrong and what it is I am wrong about, in maybe a decade or so ;)
The links to the original papers:
older one: http://arxiv.org/abs/1005.3035v1
recent one: http://arxiv.org/abs/1306.0533
Very cool. The paper suggests a connection between the non-local properties of EPR pairs and the non-local properties of wormholes that connect distant regions of space time.
EPR pair = Einstein-Podolsky-Rosen pair, e.g.
|Phi〉 = |0〉⊗|0〉+ |1〉⊗ |1〉. (1)
The vectors to the left of ⊗ represent the state of Alice, the vectors to the right of the ⊗ represent Bob's state. The + in the equation refers to the notion of quantum superposition.
There is nothing mysterious about quantum superposition. You know how y=sin(x) and y=cos(x) are both solutions to…
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y''+y=0? Well then any linear combination of sin and cos is also a solution:
y = a*sin(x) + b*cos(x).
Similarly, in QM, a quantum states |v〉is described by linear combinations of some reference vectors |0〉and |1〉
|v〉 = a|0〉+ b|1〉.
For example, the vector |+〉=|0〉+ |1〉 points half-way between the direction |0〉and |1〉.
However, we cannot describe |Phi〉as the tensor product of two local descriptions:
|0〉⊗|0〉+ |1〉⊗ |1〉 ≠ (a|0〉+ b|1〉) ⊗ (c|0〉+ d|1〉) for any a,b,c,d \in C.
Because |Phi〉, the quantum state of an EPR pair cannot be written in terms of local descriptions. We say EPR pairs are non-local. The equation of the state |Phi〉(see equation (1) above) is a description of the combined state of Alice and Bob.
Their states are entangled. If upon measuring Alice's state we find it to be |0>, then we know immediately that Bob's state is also |0>. Because of the entanglement, the collapse in the superposition on Alice's side immediately causes a collapse of the superposition on Bob's side. Note that Bob's collapse will occur IMMEDIATELY, and no matter how far Bob's system is from Alice's. This is what the authors Einstein, Podolsky and Rosen called "spooky action at a distance."
Now imagine you have have a whole bunch of physical systems prepared so their state is an EPR pair. Alice has the left halfs of the EPR pairs, Bob has all the right halfs. Now Alice's systems collapse into a black hole and Bob's systems collapse into their own black hole. The connection that will exist between these black holes appears to be similar to the wormholes connecting black holes ("Einstein-Rosen bridge"s), which have been studied in general relativity.
In honor of Alan Turing's hundredth birthday, I unwisely set out some thoughts about one of Turing's obsessions throughout his life, the question of physics and free will.