Suppose we have a pair of entangled photons, one photon sent to Alice and the other to Bob. A and B are at rest in the same reference frame. When A receives and measures the polarization of her photon, the photon sent to B is still traveling and he measures its polarization later. Does B's photon acquire its correlated polarization instantaneously when A measures her photon, or when B makes his measurement later on?
As tparker points out in the comment I just upvoted, there is more than one observationally equivalent way to model this. But according to standard interpretations, your measurement collapses the pair into a new state, the new state is unentangled, and therefore you can say that from that moment on the second particle has a state of its own. Of course you can't expect that state to necessarily be an eigenstate of some measurement that nobody's yet made.