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This is the dephased Bell-state mixture with coherence parameter $2x$. The Pauli correlation tensor isso spin measurements along unit vectors have $E(\mathbf{a},\mathbf{b})={\mathbf{a}}^{T}T\mathbf{b}$. ChooseThen $E(a,b)-E(a,b^{\prime })=2/\sqrt{1+4x^2}$ and $E(a^{\prime },b)+E(a^{\prime },b^{\prime })=8x^2/\sqrt{1+4x^2}$. The CHSH inequality expression is thereforeThese axes also attain the CHSH optimum from the correlation tensor, since the two largest eigenvalues of $T^{T}T$ are $1$ and $4x^2$. At $x=1/2$ the value is $2\sqrt{2}$; at $x=0$ the explicit LHV model in part (c) applies. Thus exactly $0<x\le 1/2$ has correlations incompatible with a local hidden-variable description.

Arbitrarily small positive coherence suffices in this particular one-parameter family, with optimally chosen axes; the violation above two tends to zero quadratically as $x\to 0$. This does not imply that every entangled mixed state violates a CHSH inequality.

In the computational basis ordered as $00,01,10,11$, the Hermitian matrix isIts trace is one, and its eigenvalues are $(1+\alpha )/2$, $(1-\alpha )/2$, $0$, and $0$. Thus the remaining density operator condition, positivity, holds exactly when both nonzero candidates are nonnegative:This is the admissible parameter interval for the dephased Bell-state mixture.