In quantum mechanics, entanglement is a phenomenon where two or more particles become correlated in such a way that the state of one particle cannot be described independently of the others. When an entangled system undergoes a measurement and the wave function collapses, the state of the particles becomes determined, but the collapse of one particle's wave function does not instantaneously affect the other particle's wave function.
If one of the entangled particles travels near the speed of light and experiences time dilation, it means that from the perspective of an observer at rest, time appears to pass more slowly for that particle. However, the concept of wave function collapse and the resulting communication of properties is independent of the observer's reference frame. From any observer's perspective, whether at rest or in motion, the wave function collapse occurs simultaneously for both entangled particles.
This is because wave function collapse is a non-local phenomenon. When a measurement is performed on one of the particles, the collapse of the wave function instantaneously determines the properties of both particles, regardless of the spatial or temporal separation between them. This phenomenon has been experimentally confirmed in various experiments, such as Bell's theorem tests, which have shown that entangled particles can indeed communicate their properties instantaneously.
Therefore, despite the time dilation experienced by one of the particles, the collapse of the wave function and the communication of properties between the entangled particles would still occur simultaneously, regardless of the observer's frame of reference.