In Quantum Algorithm Implementations for Beginners, page 7 it is stated,
Suppose we have a three qubit state, $\vert\psi\rangle$, but we only measure the first qubit and leave the other two qubits undisturbed. What is the probability of observing a $\vert 0\rangle$ in the first qubit? This probability will be given by,
The state of the system after this measurement will be obtained by normalizing the state
Applying this paradigm to the state in Eq. (5) we see that the probability of getting $\vert0\rangle$ in the first qubit will be 0.5, and if this result is obtained, the final state of the system would change to $\vert 000\rangle$. On the other hand, if we were to measure $\vert 1\rangle$ in the first qubit we would end up with a state $\vert 111\rangle$.
This feels wrong to me. Why would measuring the first qubit determine the states of the other two qubits? Is this an error in the paper?