= Solution
A <spin measurement> along unit vector $\mathbf a$ has outcome projector $\Pi_s(\mathbf a)=(I+s\mathbf a\cdot\boldsymbol\sigma)/2$, with $s=\pm1$. The <spin singlet state> obeys $\langle\sigma_i\otimes I\rangle=\langle I\otimes\sigma_i\rangle=0$ and $\langle\sigma_i\otimes\sigma_j\rangle=-\delta_{ij}$. Applying the <Born rule>,
$$
\boxed{P_{st}(\mathbf a,\mathbf b)
=\operatorname{Tr}\!\left[\rho_-\Pi_s(\mathbf a)\otimes\Pi_t(\mathbf b)\right]
=\frac{1-st\mathbf a\cdot\mathbf b}{4}.}
$$
Thus $P_{++}=P_{--}=(1-\mathbf a\cdot\mathbf b)/4$ and $P_{+-}=P_{-+}=(1+\mathbf a\cdot\mathbf b)/4$. Substitution into part (a) gives
$$
\boxed{C(\mathbf a,\mathbf b)=-\mathbf a\cdot\mathbf b.}
$$
Equal axes give perfect anticorrelation, while orthogonal axes give zero correlation. Each local marginal remains uniform.
Back to article page
