I've been trying to perform Grover's algorithm on entangled states, e.g. $|00\rangle + |11\rangle$. However, the algorithm apparently doesn't seem to amplify the amplitude of the state $|11\rangle$ which I have marked. Instead it only applies a net effect of a phase flip, despite applying both oracle and diffusion operator. Is Grover search only applicable to uniform, "pure" states?
The problem is that you have only two states in your database. So when you mark state $|00\rangle$ its amplitude is $-1$ while the other state has amplitued $1$ (note that I ignore normalization constants). Hence an average of amplitudes is zero. When you flip the amplitudes around the zero average, the amplitudes still have absoute value $1$. As a result, you cannot increase a probability of the marked state.