I Curved spacetime and measurement direction

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How are parallel directions for measurements of entangled particles defined in locations which are far from each other?
To be specific, let's say that two photons in EPR entangled state were sent to Alice and Bob separated by billions of light years. We know that if they measure their photon polarizations in the same direction, they certainly get the same result. My question is, what is the same direction in their case?
I assume that this is determined by the parallel transport in curved spacetime and depends on trajectories of the photons between their origin and Alice and Bob respectively. Is this assumption correct?
 
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