Suppose we produce two entangled photons with perpendicular polarizations. If we place a polarizer followed by a detector for the direction of the one photon, and that detector some instant of time "beeps", then what kind of polarization will be detected, if any, for the other photon in the pair?
It depends on what is the initial state of the photon pair.
If the initial state is something like $\lvert HV\rangle+\lvert VH\rangle$, then if on one side you detect $H$ on the other side you detect $V$.
If on the other hand the initial state is something like $\lvert HH\rangle+\lvert VV\rangle$, then the same polarization will be detected in both arms.