Parallel transport and entanglement

[Heidi]

Hi Pfs,
When Bob and Alice receive maximally entangled particles, Bob can choose a direction and measure the spin along it.
If Alice does the same thing in the same direction she will get the same result. But what is "same direction" when space time is curved between them? Have we to use parallel transport of Bob's direction toward Alice?
Il looks like when you make a measurement on a particle, if nothing acts on the particle. repeating the same measurement
will give you the same result but if there is the action of an hamiltonian, doing the same measurement to get the same result is also something that evolves (with tim here)

 

[gentzen]

Have we to use parallel transport of Bob's direction toward Alice?

No. In case the two particles were initially created in the singlet state (i.e. with total spin 0), then you could transport the direction along the trajectories of the particles. (But note that in this case, "If Alice does the same thing in the opposite direction she will get the same result".)

In general, you just have to "know" how the directions map to each other, because being maximally entangled does not yet fully specify the state. For example, if you have two particles in the singlet state, you can apply a unitary (2x2) matrix to one of the particles, and the state will remain maximally entangled. This unitary matrix gives you some mapping of directions.

 

[PeroK]

Hi Pfs,
When Bob and Alice receive maximally entangled particles, Bob can choose a direction and measure the spin along it.
If Alice does the same thing in the same direction she will get the same result. But what is "same direction" when space time is curved between them? Have we to use parallel transport of Bob's direction toward Alice?

I would assume the entangled state would act effectively as a gyroscope through curved spacetime. I wonder to what extent this has been tested?

 

[vanhees71]

You'd have to use quantum field theory in the given background spacetime to see, what comes out. For sure it's highly non-trivial.

 

[Heidi]

What do you think of the repeated measurement seen as entanglement (with timelike interval) ?

 

[PeterDonis]

Have we to use parallel transport of Bob's direction toward Alice?

No, because any such transport in curved spacetime will not be unique.

The best approach I can see, theoretically, would be to Fermi-Walker transport (the more robust version of parallel transport that can handle non-geodesic curves) each of the measurement directions backwards along the worldlines of the two particles, to the event of their common creation, and then compare the directions. I think that is what a calculation using QFT in curved spacetime would end up amounting to. But that's just my intuitive guess; I have not actually tried to do the calculation (which, as @vanhees71 points out, would be highly non-trivial).