 Maximal CHSH witness for a fixed Bell state

The Hermitian operator $\mathcal{B}=\sqrt{2}(X\otimes X-Z\otimes Z)$ has eigenvalues $2\sqrt{2},0,0,-2\sqrt{2}$ on ${\Psi }^{+},{\Phi }^{+},{\Psi }^{-},{\Phi }^{-}$, respectively. Thus any density matrix satisfying $\mathrm{Tr}(\rho \mathcal{B})=2\sqrt{2}$ must be $|{\Psi }^{+}⟩⟨{\Psi }^{+}|$: 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.