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For the Schmidt-basis Pauli correlation tensor, choose Alice's axes ${\mathbf{e}}_z,{\mathbf{e}}_x$ and Bob's $({\mathbf{e}}_z\pm s{\mathbf{e}}_x)/\sqrt{1+s^2}$. The CHSH inequality expression is $2\sqrt{1+s^2}$. It exceeds two exactly when both Schmidt coefficients are nonzero, reaching $2\sqrt{2}$ for maximal quantum entanglement. A product state gives $s=0$ and saturates rather than violates the inequality.

Take ${\alpha }^2+{\beta }^2=1$ and label the first vector of each measurement basis by outcome $0$. Assign the Pauli measurement value $+1$ to outcome $0$ and $-1$ to outcome $1$. The four observables areHere $X$ and $Z$ are the Pauli X gate and Pauli Z gate matrices. These follow by subtracting the two rank-one basis projectors; a real basis rotated through $\gamma$ has observable $\sin (2\gamma )X+\cos (2\gamma )Z$.

The Schmidt-basis Pauli correlation tensor of the state givesConsequently $E_{11}=1/2$, $E_{21}=E_{22}=\sqrt{3}\alpha \beta$ and $E_{12}=-1/2$, where $E_{jk}=⟨A_j\otimes B_k⟩$. For binary outcomes, $\mathbb{E}([A-B{]}_2)=P(A\ne B)=(1-E_{AB})/2$, whereas the final offset term has $\mathbb{E}([B-A-1{]}_2)=P(A=B)=(1+E_{AB})/2$. The chained modular Bell inequality left side is thereforeThe local bound is $I\ge 1$, so the exact violation condition isEquality saturates the bound. The maximal violation for these fixed measurements occurs at $\alpha =\beta =1/\sqrt{2}$ or their common negative, giving $I=(3-\sqrt{3})/2$. Opposite signs do not violate this particular inequality with these fixed bases, although other measurement choices can reveal the state's entanglement. If unnormalized real amplitudes are used, replace $\alpha \beta$ throughout by $\alpha \beta /({\alpha }^2+{\beta }^2)$.

The Schmidt decomposition theorem states that a normalized vector in a finite-dimensional tensor product ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$ has the form ${\sum }_{j=1}^r{s}_j|u_j⟩|v_j⟩$, where the two families are orthonormal, $s_j>0$, ${\sum }_j{s}_j^2=1$, and $r\le \min (\dim {\mathcal{H}}_A,\dim {\mathcal{H}}_B)$. Extending each family to an orthonormal basis for two qubits givesThe nonzero Schmidt coefficients can be made positive by absorbing phases into the basis vectors. Both are positive exactly when the pure state is entangled. A product state has Schmidt rank one, so one coefficient is zero; the source's assertion of two positive coefficients for every pure state needs this exception.

Choose each local orthonormal basis independently as the computational basis. In that basis ${\sigma }_z=|0⟩⟨0|-|1⟩⟨1|$. This is a local change of coordinates, implemented by separate unitary matrices, rather than a physical restriction on the original quantum state. The associated Pauli operators supply the other two local axes. Up to an irrelevant common phase, each unitary change of qubit basis corresponds to a rotation of its Bloch sphere.

Put $s=2{\lambda }_0{\lambda }_1$. The Schmidt-basis Pauli correlation tensor is diagonal. The Pauli operators ${\sigma }_x\otimes {\sigma }_x$ exchange $|00⟩$ and $|11⟩$, while ${\sigma }_y\otimes {\sigma }_y$ do so with minus signs, and ${\sigma }_z\otimes {\sigma }_z$ leaves both fixed. HenceMixed components vanish: those containing one $z$ and one transverse Pauli operator map the occupied basis vectors outside their span, while the $xy$ and $yx$ matrix elements are purely imaginary and cancel for real Schmidt coefficients. By bilinearity, for arbitrary real vectors,For a CHSH inequality test take unit quantum measurement axesThese CHSH axes for an entangled pure two-qubit state giveThe local bound is two: for each hidden state, $A(B+B^{\prime })+A^{\prime }(B-B^{\prime })$ is $\pm 2$ when all four outcomes are $\pm 1$, and averaging cannot increase its absolute value. Thus every entangled pure two-qubit state violates a CHSH inequality, the content of Gisin's theorem. A product state has $s=0$ and does not violate it, so the unqualified final claim in the question is false for that case. The maximum $2\sqrt{2}$ occurs for equal Schmidt coefficients. This excludes local hidden-variable theories satisfying measurement independence, but does not permit faster-than-light signalling: the local reduced density matrix and its quantum measurement probabilities are unchanged by the remote choice of axis.