Entanglement is a phenomenon in quantum mechanics where two or more particles become correlated in such a way that the state of one particle is instantaneously related to the state of the other particle, regardless of the distance between them. This instantaneous correlation is often referred to as "spooky action at a distance."
When two particles are entangled, their wave functions are intertwined, and measuring the state of one particle can instantaneously determine the state of the other, regardless of the spatial separation between them. This means that if one particle collapses its wave function, the state of the other particle is immediately known, regardless of the distance between them.
However, when one of the entangled particles is moving near the speed of light and experiences time dilation, it is important to consider the effects of relativity. According to the theory of relativity, an observer moving at a high velocity relative to another observer will experience time dilation, which means that time appears to pass more slowly for the moving observer.
In the scenario you described, if one entangled particle is moving near the speed of light, its time dilation would cause time to pass more slowly for that particle compared to a stationary observer. However, the entanglement between the particles is not affected by the relative velocities of the particles.
From the perspective of the moving particle, it would still observe the instantaneous collapse of the wave function of its entangled partner, regardless of the time dilation it experiences. Similarly, from the perspective of a stationary observer, the collapse of the wave function of the moving particle would appear instantaneous.
In summary, even though the entangled particles may experience time dilation due to relativistic effects, the collapse of their wave functions would still be observed as instantaneous by each respective observer, ensuring that their properties are communicated at the same time, regardless of the relative velocities.