I'm working on a problem involving the calculation of probabilities for outcomes of a measurement on a quantum state perturbed by an error. The state in question is a GHZ state $|\text{GHZ}\rangle = \alpha|000\rangle + \beta|111\rangle$, with the normalization condition $|\alpha|^2 + |\beta|^2 = 1$. An error occurs which is modeled by applying the $\sigma_x$ operator to the first qubit, resulting in the state $\sigma_x^A|\text{GHZ}\rangle = \alpha|100\rangle + \beta|011\rangle$.
I am familiar with the concept of the expectation value $ \langle \Psi | \hat{O} | \Psi \rangle $ and how to calculate it for an observable $ \hat{O} $ in the state $ | \Psi \rangle $. However, I am now interested in the measurement of the observable $ \sigma_z^A \sigma_z^B $ on the new state $ \sigma_x^A|\text{GHZ}\rangle $ and explicitly calculating the probabilities for each outcome. While I understand the theoretical framework that the probability $ P_i $ of measuring an eigenvalue $ o_i $ is given by Born's rule $ P_i = |\langle \phi_i | \Psi \rangle|^2 $, where $ | \phi_i \rangle $ are the eigenstates of $ \hat{O} $, I am unsure about applying it in this particular situation.
I would be very grateful if someone could provide a few hints on how to apply Born’s rule to find the probabilities for each possible outcome in this scenario.
