Electron Spin — Question 6

Question 6

A mixed spin ensemble can share z statistics with a pure state

Compare the pure state $|+x\rangle=(|+z\rangle+|-z\rangle)/\sqrt2$ with an incoherent ensemble containing equal numbers of $|+z\rangle$ and $|-z\rangle$. Use the z basis.

1. Write both density matrices and their purities $\mathrm{Tr}(\rho^2)$.

2. Find the positive-x probability for each preparation.

Question 6 - Solution

A mixed spin ensemble can share z statistics with a pure state

(a) Method and calculation

$$\rho_p=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix},\quad\rho_m=\tfrac12\begin{pmatrix}1&0\\0&1\end{pmatrix}.$$$$\boxed{\mathrm{Tr}(\rho_p^2)=1,\quad\mathrm{Tr}(\rho_m^2)=1/2}.$$

(b) Interpretation and further calculation

$$P_p(+x)=\langle+x|\rho_p|+x\rangle=\boxed{1}.$$$$P_m(+x)=\langle+x|I/2|+x\rangle=\boxed{1/2}.$$

Physical check

Both preparations give equal z probabilities. Their off-diagonal coherences differ, and an x measurement distinguishes them statistically.