How to prove the positivity of the conditional quantum mutual information, $I(A;B|C)\ge0$?

Question

I was reading Wilde's 'Quantum Information Theory' and saw the following theorem at chapter 11 $(11.7.2)$:

$$ I(A; B | C) \ge 0, $$ where, $$ I(A;B|C) := H(A|C) + H(B | C) - H(AB|C). $$

I know that the mutual information is non-negative, i.e. $$ I(A;B) \ge 0, $$ where, $$ I(A;B) := H(A) + H(B) - H(AB). $$ Now if we have access to an additional subsystem $C$, this can't decrease the mutual information of $A$ and $B$. But I was looking for sort of formal proof of this. I was trying to apply the non-negativity of mutual info. into this, but not sure how to proceed. Thanks in advance!

Answer 1

Here's a relatively simple proof just based on the data processing inequality (DPI) for the relative entropy $D(\rho\|\sigma) = \mathrm{tr}[\rho (\log \rho - \log \sigma)]$ -- if you're willing to accept the DPI as a basis for a formal proof. Recall that the DPI says that for any channel $\Phi$ we have $$ D(\rho \|\sigma) \geq D(\Phi(\rho)\|\Phi(\sigma)). $$

Now $$ \begin{aligned} I(A:B|C) &= H(A|C) + H(B|C) - H(AB|C) \\ &= H(AC) - H(C) + H(BC) - H(C) - H(ABC) + H(C) \\ &= H(AC) + H(BC) - H(ABC) - H(C) \\ &= H(A|C) - H(A|BC) \\ &= -D(\rho_{AC} \| I_A \otimes \rho_C) + D(\rho_{ABC} \| I_A \otimes \rho_{BC}). \end{aligned} $$ Thus $I(A:B|C) \geq 0$ is equivalent to $$ D(\rho_{ABC} \|I_A \otimes \rho_{BC}) \geq D(\rho_{AC} \|I_A \otimes \rho_C), $$ but this follows immediately from the DPI by taking the channel $\Phi$ to be the partial trace over the $B$ system.

Note also that the fourth line of the derivation shows this result is equivalent to strong subadditivity of the von Neumann entropy, as mentioned in the comments by @Purva Thakre.

Answer 2

To expand on @Purva Tharke's comment, the strong subadditivity inequality states: $$H(ABC)+H(C) \le H(AC) + H(BC)$$ $$=H(ABC)+H(C) +H(C) -H(C) \le H(AC) + H(BC)$$ $$=H(AB|C) \le H(A|C) + H(B|C)$$ $$=0\le H(A|C) + H(B|C) - H(AB|C)=H(A;B|C)$$

Edit: A good proof of SS for entropies can be found in Nielsen and Chuang, in case you wanted to take a look.