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

$$ I(A; B | C) \ge 0, ........ (1) $$ 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!

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!

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

$$ I(A; B | C) \ge 0, ........ (1) $$ 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, $$ I(A;B|C) := H(A) + H(B) - H(AB). $$ 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!

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

$$ I(A; B | C) \ge 0, ........ (1) $$ 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!