# How can I find the probabilities corresponding to measurement results of an observable of the GHZ states?

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.

• it sounds like you're asking how to compute the squared overlaps of $\sigma_x^A|\operatorname{GHZ}\rangle$ on the eigenvectors of $\sigma_z^A\sigma_z^B$. What are you uncertain about in this calculation?
– glS
Commented Mar 9 at 13:20
• My issue is, that for: Computing the expectation value of an Operator O and now I want to get the corresponding probability to this value. So for this I need to take the inner product of all eigenvalues of the operator O with the GHZ state and then square it? (So using Born’s rule)? Commented Mar 9 at 18:43