= Maximal CHSH witness for a fixed Bell state
{title2=$\mathcal B=\sqrt2(X\otimes X-Z\otimes Z)$}
The <Hermitian operator> $\mathcal B=\sqrt2(X\otimes X-Z\otimes Z)$ has <eigenvalues> $2\sqrt2,0,0,-2\sqrt2$ on $\Psi^+,\Phi^+,\Psi^-,\Phi^-$, respectively. Thus any <density matrix> satisfying $\operatorname{Tr}(\rho\mathcal B)=2\sqrt2$ must be $|\Psi^+\rangle\langle\Psi^+|$: an average can equal the largest eigenvalue only when all weight lies in its one-dimensional <eigenspace>. The <purity decouples a subsystem from its purification> lemma then excludes correlations with an external environment. This argument assumes the stated measurement operators; <CHSH self-testing> handles unknown representations with additional qualifications.
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