Type checks: rays, density operators, mixtures, local quantities and outcome laws

Let $\psi :=\frac{1}{\sqrt{2}}(e_{+}+e_{-})$, so $\mathrm{\parallel }\psi \mathrm{\parallel }=1$ by orthonormality. The pure preparation is the unit ray $[\psi ]=\{z\psi :z\in \mathbb{C}\setminus \{0\}\}$: the vector $e^{i\theta }\psi$ has the same norm and defines the same ray, and the associated density operator $T_{\psi ,\psi }$ satisfies $\mathrm{tr}{T}_{\psi ,\psi }=\mathrm{\parallel }\psi {\mathrm{\parallel }}^2=1$ by [L2]; hence it is a density operator in the sense of [L1].

Let $0<w<1$ and let ${\psi }_1,{\psi }_2$ be orthonormal vectors of $H$, and put $\rho :=w{T}_{{\psi }_1,{\psi }_1}+(1-w)T_{{\psi }_2,{\psi }_2}$. By [L1] $\rho$ is a density operator with $\mathrm{tr}\rho =w+(1-w)=1$; orthogonality gives $\rho {\psi }_1=w{\psi }_1$ and $\rho {\psi }_2=(1-w){\psi }_2$. Both eigenvalues are positive, so the range of $\rho$ contains the two linearly independent vectors ${\psi }_1,{\psi }_2$; hence $\rho$ has rank at least two and is not a rank-one operator $T_{\phi ,\phi }$ for any $\phi$. The mixture of two orthogonal pure preparations is therefore a preparation of a different type from either component.

On the composite $H\widehat{\otimes }H$ consider the product state $\mathrm{\Psi }:=e_{+}\otimes e_{+}$ and the local observable $S_z\otimes I$. By [L3] and the self-adjointness of $S_z$ in [L4], this tensor operator is self-adjoint, and $(S_z\otimes I)\mathrm{\Psi }=(S_z{e}_{+})\otimes e_{+}=\frac{\mathrm{\hslash }}{2}\mathrm{\Psi }$. On the product basis of $H\widehat{\otimes }H$, the spectral projection for $\mathrm{\hslash }\mathrm{/}2$ is $P_{+}\otimes I$, so the local outcome probability for $\{\mathrm{\hslash }\mathrm{/}2\}$ is $\mathrm{\parallel }(P_{+}\otimes I)\mathrm{\Psi }{\mathrm{\parallel }}^2=1$.

For the state $\psi =\frac{1}{\sqrt{2}}(e_{+}+e_{-})$ the outcome law of $S_z$ is $B\mapsto {\mu }_{\psi }^{S_z}(B)=\mathrm{tr}(T_{\psi ,\psi }{E}_{S_z}(B))=\mathrm{\parallel }{E}_{S_z}(B)\psi {\mathrm{\parallel }}^2$. By [L4] the projections are $P_{\pm }$ onto the lines $\mathbb{C}{e}_{\pm }$, so ${\mu }_{\psi }^{S_z}(\{\mathrm{\hslash }\mathrm{/}2\})=\mathrm{\parallel }{P}_{+}\psi {\mathrm{\parallel }}^2=\frac{1}{2}$ and ${\mu }_{\psi }^{S_z}(\{-\mathrm{\hslash }\mathrm{/}2\})=\mathrm{\parallel }{P}_{-}\psi {\mathrm{\parallel }}^2=\frac{1}{2}$. Thus the law is $\frac{1}{2}{\delta }_{\mathrm{\hslash }\mathrm{/}2}+\frac{1}{2}{\delta }_{-\mathrm{\hslash }\mathrm{/}2}$ on the two-point set $\sigma (S_z)$ and is not concentrated at a single outcome; its finite mean is $\frac{1}{2}(\mathrm{\hslash }\mathrm{/}2)+\frac{1}{2}(-\mathrm{\hslash }\mathrm{/}2)=0$.

Combining steps 1.1-1.4: rays, density operators, mixtures, local tensor actions and outcome laws of the adopted postulates all have the mathematical types claimed for them, and the displayed computations are the ones the definitions and lemmas give.