## A Holevo quantity inequality

For some reason, I wanted to know the following fact at some point.

Let $(p_i,\rho_i)$ be an ensemble of states of a bipartite system $AB$. For $\chi$ the Holevo information, we have $\chi_{AB}\leq\chi_A+\chi_B+\bar{I}$ where $\bar{I}=\sum_ip_iI_i$ is the expected value of the quantum mutual information $I_i=S(\rho_i^A)+S(\rho_i^B)-S(\rho_i^{AB})$.

\begin{aligned} \chi_{AB}&=S\left(\sum_{i}p_i\rho_i\right)-\sum_{i}p_iS(\rho_i)\\ &\leq S\left(\sum_{i}p_i\rho^A_i\right)+S\left(\sum_{i}p_i\rho^B_i\right)-\sum_{i}p_iS(\rho_i)\\ &= S\left(\sum_{i}p_i\rho^A_i\right)+\sum_{i}p_iS(\rho^A_i)-\sum_{i}p_iS(\rho^A_i)+S\left(\sum_{i}p_i\rho^B_i\right)-\sum_{i}p_iS(\rho_i)\\ &= \chi_A+\chi_B+\sum_{i}p_iS(\rho^A_i)+\sum_{i}p_iS(\rho^B_i)-\sum_{i}p_iS(\rho_i)\\ &= \chi_A+\chi_B+\sum_{i}p_i\left(S(\rho^A_i)+S(\rho^B_i)-S(\rho_i)\right)\\ &= \chi_A+\chi_B+\sum_{i}p_iI_i \end{aligned}

Since the Holevo information gives an upper bound for the mutual information between the random variable $X\sim (p_i)$ and the outcome of any measurement that can be made on the received state, setting $\chi_A=\chi_B=0$ we see that $\bar{I}$ may be meaningfully taken as an upper bound for the amount of hidden information.

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