Bell State, if Bob applies a Pauli Gate? After Alice and Bob share a Bell state, Bob applies a Pauli gate to his qubit. What will be the situation of the Bell state? What happens?
Then Alice applies the same gate to her qubit – again, what happens?
I would be thankful if you could explain it slowly, as I am a beginner in this field.
 A: Pauli gates applied to one qubit will just transform one Bell state into another one. Take, for example, the antisymmetric singlet state:
$$|\psi_{AB}^-\rangle = \frac{1}{\sqrt{2}}(|\uparrow_A\rangle\otimes|\downarrow_B\rangle - |\downarrow_A\rangle\otimes|\uparrow_B\rangle). $$
Alice applies the Pauli X gate, defined as $\sigma^x|\uparrow\rangle = |\downarrow\rangle$, $\sigma^x|\downarrow\rangle = |\uparrow\rangle$. The gate is local, so it acts as the identity on Bob's qubit (it does nothing to Bob's qubit). This is written as
\begin{eqnarray}(\sigma^x_A\otimes 1_B) |\psi_{AB}^-\rangle &=& \frac{1}{\sqrt{2}}(\sigma^x_A|\uparrow_A\rangle\otimes 1_B|\downarrow_B\rangle - \sigma^x_A|\downarrow_A\rangle\otimes 1_B|\uparrow_B\rangle) \\ &=&\frac{1}{\sqrt{2}}(|\downarrow_A\rangle\otimes|\downarrow_B\rangle - |\uparrow_A\rangle\otimes|\uparrow_B\rangle) \\ &=& |\phi_{AB}^-\rangle,\end{eqnarray}
which is just a different Bell state. This demonstrates the local unitary equivalence of Bell states.
A: In your question, you were also wondering what would happen if both Alice and Bob apply the same Pauli gate on their respective qubit. To complement Mark's answer, I would like to expand on this point. Interestingly, the answer is that nothing happens, the action of their individual gates is equivalent to the identity map! This is a consequence of the more general result that for any maximally entangled pure state $|\psi\rangle$ and any unitary $U$, it holds that
$$U\otimes U^T|\psi\rangle=|\psi\rangle$$
where $U^T$ is just the transpose of $U$. All the Bell states are maximally entangled and the Pauli operators satisfy $\sigma_x^T=\sigma_x$, $\sigma_y^T=-\sigma_y$, $\sigma_z^T=\sigma_z$ so that if both Alice and Bob apply the same Pauli gate, from the above result the state is unchanged (up to a global phase).
