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For dichotomic observables with $A_i^2=B_j^2=I$, let $\mathcal{B}=A_0\otimes (B_0+B_1)+A_1\otimes (B_0-B_1)$. Expanding gives ${\mathcal{B}}^2=4I-[A_0,A_1]\otimes [B_0,B_1]$. Each commutator has operator norm at most two, so $\Vert \mathcal{B}{\Vert }^2\le 8$ and $|⟨\mathcal{B}⟩|\le 2\sqrt{2}$. Appropriate spin measurements of a Bell state attain equality. The local hidden-variable bound of the corresponding CHSH inequality is two.

An electron has spin one-half. Its intrinsic spin angular momentum has two possible components, $\pm \hslash /2$, along a specified unit vector $\mathbf{n}$. These are the two eigenvalues of the corresponding spin measurement; the spin is distinct from the electron's orbital angular momentum.

A spin measurement along unit vector $\mathbf{a}$ has outcome projector ${\Pi }_s(\mathbf{a})=(I+s\mathbf{a}\cdot \mathbf{\sigma })/2$, with $s=\pm 1$. The spin singlet state obeys $⟨{\sigma }_i\otimes I⟩=⟨I\otimes {\sigma }_i⟩=0$ and $⟨{\sigma }_i\otimes {\sigma }_j⟩=-{\delta }_{ij}$. Applying the Born rule,Thus $P_{++}=P_{--}=(1-\mathbf{a}\cdot \mathbf{b})/4$ and $P_{+-}=P_{-+}=(1+\mathbf{a}\cdot \mathbf{b})/4$. Substitution into part (a) givesEqual axes give perfect anticorrelation, while orthogonal axes give zero correlation. Each local marginal remains uniform.

The projective measurement probabilities are $P({\sigma }_j=\pm 1)=\mathrm{Tr}[\rho (I\pm {\sigma }_j)/2]$. Both density operators have zero $z$-component of their Bloch vector, soTheir full ${\sigma }_z$ outcome distributions, not merely their mean values, coincide. Along $x$, however,Thus repeated spin measurement along $x$ distinguishes the preparations statistically. A single negative outcome excludes A, while a single positive outcome is compatible with either; under B the probability of $N$ independent positive outcomes is $2^{-N}$.

The opposite $x$-axis pure states have projectors $(I\pm {\sigma }_x)/2$. Averaging them givesTherefore C and B cannot be distinguished by any measurement on the atoms alone. Every POVM outcome has the same probability $\mathrm{Tr}(E\rho )$ for the two preparations; this is the operational equivalence of different quantum state ensembles. Accessible preparation labels or additional correlations would be extra information. C can be distinguished from A by the $x$-axis spin measurement in part (c).

The Pauli matrices are the complex two-by-two matricesEach is a traceless matrix, a Hermitian matrix and a unitary matrix. Their eigenvalues are $+1,-1$, and direct matrix multiplication gives ${\sigma }_x{\sigma }_y=i{\sigma }_z$, with cyclic permutations and a minus sign for reversed order. This is the Pauli matrix multiplication law.

Together with the identity matrix, they form the Pauli operator basis of all complex two-by-two matrices. For spin one-half, the angular momentum operators are $S_j=\hslash {\sigma }_j/2$, so a spin measurement along a unit vector $n$ has outcomes $\pm \hslash /2$. For example, ${\sigma }_z$ has the standard basis as its eigenbasis, whereas ${\sigma }_x$ has normalized eigenvectors $(1,\pm 1{)}^T/\sqrt{2}$. This makes changing the measurement axis an explicit change of basis.^[ref]

For a spin one-half system, an ideal projective measurement along a unit vector $\mathbf{n}$ measures ${\sigma }_{\mathbf{n}}=\mathbf{n}\cdot \mathbf{\sigma }$. Its eigenvalues are $\pm 1$, corresponding to physical spin angular momentum components $\pm \hslash /2$, and its orthogonal projections are ${\Pi }_{\pm }=(I\pm {\sigma }_{\mathbf{n}})/2$. For density matrix $\rho =(I+\mathbf{r}\cdot \mathbf{\sigma })/2$, the Born rule and the Pauli matrix multiplication law give the displayed probabilities. Thus a spin measurement along one direction tests only that component of the Bloch vector; measurements along three independent directions determine the entire density matrix.