Understanding the definition of entropy in the joint entropy theorem derivation

Question

From section 11.3.2 of Nielsen & Chuang:

(4) let $\lambda_i^j$ and $\left|e_i^j\right>$ be the eigenvalues and corresponding eigenvectors of $\rho_i$. Observe that $p_i\lambda_i^j$ and $\left|e_i^j\right>$ are the eigenvalues and eigenvectors of $\sum_ip_i\rho_i$ and thus \begin{align}S\left(\sum_ip_i\rho_i\right) &= -\sum_{ij}p_i\lambda_i^j\log p_i\lambda_i^j \\ &= - \sum_ip_i\log p_i - \sum_ip_i\sum_j\lambda_i^j\log\lambda_i^j \\ &= H\left(p_i\right) + \sum_ip_iS\left(\rho_i\right)\end{align}

I do not understand the last step $- \sum_ip_i\log p_i = H\left(p_i\right)$. I thought the definition of binary entropy is $H(p_{i}) = -p_{i}\log p_{i} - (1-p_{i})\log (1-p_{i})$. This does not give the desired equality.

Moreover, it is rather confusing that $i$ is used as a running index, and suddently it becomes a fixed index in the last equality.

Thanks for any helps.

Answer 1

It is not just the binary entropy that is denoted $H(p_i)$. The quantity that is relevant here is the Shannon entropy of the distribution $\{p_i\}$ which is defined as $$ H(p_i) = - \sum_i p_i \log p_i. $$ Note that when the distribution has only two elements we recover the binary entropy.