Electron Spin — Question 7

Question 7

Three spin projections reconstruct a physical Bloch vector

A repeated preparation has measured positive-outcome probabilities $P_x=.8$, $P_y=.5$, and $P_z=.9$. A spin density matrix is $\rho=(I+\boldsymbol r\cdot\boldsymbol\sigma)/2$, with $r_j=2P_j-1$ and physical condition $|\boldsymbol r|\le1$.

1. Reconstruct the vector and determine whether the state is pure.

2. Predict the positive probability along $\hat n=(3,0,4)/5$ using $P_n=(1+\boldsymbol r\cdot\hat n)/2$.

Question 7 - Solution

Three spin projections reconstruct a physical Bloch vector

(a) Method and calculation

$$\boxed{\boldsymbol r=(.6,0,.8)},\quad |\boldsymbol r|^2=.36+.64=1.$$$$\mathrm{Tr}(\rho^2)=(1+|\boldsymbol r|^2)/2=\boxed{1\ \text{(pure)}}.$$

(b) Interpretation and further calculation

$$\boldsymbol r\cdot\hat n=.6(.6)+.8(.8)=1.$$$$\boxed{P_n=1},\qquad\hat n=\boldsymbol r.$$

Physical check

These probabilities describe a fully polarized state along the reconstructed direction. A vector longer than one would signal inconsistent ideal-state data.