# Exercise 6

## from Example Sheet 3

### Exercise 6.

Let $$\rho=\sum_ip_i\rho_i$$ where $$\rho_i$$ are density matrices which have support on orthogonal subspaces. Then prove that $S\bigg(\sum_ip_i\rho_i\bigg)=H(p)+\sum_ip_iS(\rho_i)$ where $$p=\{p_i\}$$ is a probability distribution, and $$H(p)$$ is the corresponding Shannon entropy.

Let us write the total number of states as $$n$$. Recall $$S(\rho) := - \operatorname{tr}[\rho \log \rho]$$. In fact, we don’t need to restrict $$S$$ to act on states; for any $$X\geq 0$$, we can define $S(X) = - \operatorname{tr}[X\log X].$ We notice for $$p> 0$$ and a state $$\rho$$, we have \begin{aligned} S(p\rho) &= - \operatorname{tr}[ p \rho \log (p \rho)] = -p \operatorname{tr}[ \rho (\log p I + \log \rho)] = - p \operatorname{tr}(\rho) \log p - p \operatorname{tr}[ \rho \log \rho]\\ &= -p \log p + p S(\rho). \end{aligned}(1) Now, if the supports of the $$\rho_i$$ are all orthogonal, then in particular their eigenvectors are too. We can thus make an orthonormal basis of the Hilbert space by the eigenvectors of all the $$\rho_i$$, so that each $$\rho_i$$ is diagonal in this basis. We see that $\sum_i p_i \rho_i = \begin{pmatrix} p_1 \rho_1 \\ &&p_2 \rho_2 \\ &&& \ddots \\ &&&& p_n \rho_n \end{pmatrix}$ where here we write $$\rho_1 \equiv \rho_1|_{\operatorname{supp}\rho_1}$$ to mean the block matrix of $$\rho_1$$ restricted to its support. Then $-\big(\sum_i p_i \rho_i \big) \log \big(\sum_i p_i \rho_i \big) = \begin{pmatrix} -p_1 \rho_1 \log (p_1 \rho_1) \\ && -p_2 \rho_2\log(p_2 \rho_2) \\ &&& \ddots \\ &&&& -p_n \rho_n \log(p_n \rho_n) \end{pmatrix},$ and taking the trace, we see $S\left(\sum_i p_i \rho_i\right) = \sum_i S(p_i \rho_i) = -\sum_i p_i \log p_i + p_i S(\rho_i)$ by (1).