Understanding Quantum State Preparation: The Significance of |u> and |d> Vectors

[Quarlep]

"Let’s begin by labeling the possible spin states along the three coordinate axes. If A is oriented along the z axis, the two possible states that can be prepared correspond to σ_z= ±1. Let’s call them up and down and denote them by ket-vectors |u> and |d> . Thus, when the apparatus is oriented along the z axis and registers +1, the state |u> has been prepared. " says in The theoretical Minimum

what it means ? In particular "the state |u> has been prepared" Actually I am asking just this part

 

[DrClaude]

It means that the system is in state |u>,
$$
| \psi \rangle = | \mathrm{u} \rangle
$$

 

[jfizzix]

By saying that the state $|u\rangle$ is prepared, one can assume that all subsequent measurements of that same system will be as though the system is in state $|u\rangle$.