Let’s see if we can make some more sense of the statement of subadditivity, and in particular, demonstrate that some particular Rényi entropies violate it.

In this example, we will explicitly construct a counterexample for Rényi entropy with parameter α → ∞, and in Exercise 12.1, you will work to generalize this result. Let’s consider a system of two spin-1/2 particles, call them 1 and 2, and their density matrix is diagonal and can be expressed as

ρ12 =12|↑1↑2⟩⟨↑1↑2 |+14|↑1↓2⟩⟨↑1↓2 |+14|↓1↑2⟩⟨↓1↑2 | , (12.37)

where the subscripts denote the specific particle’s spin. What does subadditivity look like for this density matrix?