The von Neumann entropy satisfies a further inequality among three systems called strong subadditivity. \({ }^{20}\) As with subadditivity, strong subadditivity is only a property of the von Neumann entropy, and not of the general Rényi entropies. Consider three systems \(A, B\), and \(C\) and let \(A B, B C\), and \(A B C\) be their combined systems, appropriately. The statement of strong subadditivity is that

\[\begin{equation*}S_{\mathrm{vN}}^{A B C}+S_{\mathrm{vN}}^{B} \leq S_{\mathrm{vN}}^{A B}+S_{\mathrm{vN}}^{B C} \tag{12.149}\end{equation*}\]

(a) Show that strong subadditivity holds if the systems \(A, B\), and \(C\) are all independent.

(b) Show that strong subadditivity holds if the combined state \(A B C\) is pure.

If \(A B C\) is pure, then the von Neumann entropies of systems \(A\) and \(B C\), say, are equal.

(c) Consider the pure state \(|\psiangle\) of three spins, where one spin is up and two are down:

\[\begin{equation*}|\psiangle=\frac{1}{\sqrt{3}}\left(\left|\uparrow_{1}\rightangle\left|\downarrow_{2}\rightangle\left|\downarrow_{3}\rightangle+\left|\downarrow_{1}\rightangle\left|\uparrow_{2}\rightangle\left|\downarrow_{3}\rightangle+\left|\downarrow_{1}\rightangle\left|\downarrow_{2}\rightangle\left|\uparrow_{3}\rightangle\right) . \tag{12.150}\end{equation*}\]

Show that this state is normalized and construct the density matrices of the systems of spins 12, 23, and 2 alone.

(d) Now show that this state \(|\psiangle\) satisfies strong subadditivity. What is the inequality that you find?