Probability of spin in the +x direction

[srallaba]

Consider a qubit.
Given a state equation, how to calculate the probability of finding spin in +x direction?

 

[vanhees71]

If your system is prepared in the state $\hat{\rho}$ then the probability for the $x$-component of the spin being $\sigma_x$ is
$$P(\sigma_x)=\langle \sigma_x|\hat{\rho}|\sigma_x \rangle.$$

 

[olgerm]

How to do it if I know spinor instead of $\hat{\rho}$?

 

[vanhees71]

If your spin is prepared in a pure state, represented by the ket $|\psi \rangle$, then the statistical operator is the projection operator $\hat{\rho}=|\psi \rangle \langle \psi|$, and the general formula gives you the Born rule for pure states, as it should be:

$$P(\sigma_x)=\langle \sigma_x \psi \rangle \langle \psi|\sigma_x \rangle=\psi(\sigma_x) \psi^*(\sigma_x)=|\psi(\sigma_x)|^2.$$