I Curved spacetime and measurement direction

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The discussion centers on the measurement of photon polarizations in an EPR entangled state between two observers, Alice and Bob, separated by vast distances. It raises the question of how to define "the same direction" for their measurements in curved spacetime. The assumption is that this direction is determined by parallel transport, which is influenced by the trajectories of the photons as they travel to Alice and Bob. The inquiry suggests that the curvature of spacetime plays a crucial role in defining measurement directions for entangled particles. Understanding this relationship is essential for deeper insights into quantum entanglement in the context of general relativity.
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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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