Entropy inequalities, recovery boundaries and thermal entanglement — Examples

The vector $\mathrm{\mid }\mathrm{\Phi }⟩$ has norm one because $\mathrm{\mid }00⟩$ and $\mathrm{\mid }11⟩$ are orthonormal, so ${\rho }_{\mathrm{\Phi }}$ is a rank-one projection with trace one. The two orthogonal basis projections defining ${\rho }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$ are positive and each has trace one, so their half-weighted sum is also positive with trace one. Expanding ${\rho }_{\mathrm{\Phi }}$ gives the four terms $\frac{1}{2}\mathrm{\mid }00⟩⟨00\mathrm{\mid }$, $\frac{1}{2}\mathrm{\mid }00⟩⟨11\mathrm{\mid }$, $\frac{1}{2}\mathrm{\mid }11⟩⟨00\mathrm{\mid }$, and $\frac{1}{2}\mathrm{\mid }11⟩⟨11\mathrm{\mid }$. In the partial trace over the second factor, the two off-diagonal terms vanish because the traced basis indices differ, by [F2]; in the trace over the first factor, the same is true by [F4]. In both cases the diagonal terms each contribute one half of the corresponding basis projection, so both marginals are $I_2\mathrm{/}2$. Applying the same two formulas to ${\rho }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$ gives these marginals directly; they are density operators by [F3].

The eigenvalues of ${\rho }_{\mathrm{\Phi }}$ are $1,0,0,0$, those of ${\rho }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}}$ are $1\mathrm{/}2,1\mathrm{/}2,0,0$, and those of each marginal are $1\mathrm{/}2,1\mathrm{/}2$. Substitution into [F1] yields $S({\rho }_{\mathrm{\Phi }})=0$, $S({\rho }_{\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}})=\log 2$, and $S(I_2\mathrm{/}2)=\log 2$. Therefore the mutual informations are $2\log 2$ and $\log 2$, respectively, as stated.