C

, with the two input states

ρ_{A BC}, \frac{I _{A}}{d _{A}} \otimes ρ_{BC} .

The channel sends them to

ρ_{A B} \otimes I_{C} / d_{C}

and

I_{A} / d_{A} \otimes ρ_{B} \otimes I_{C} / d_{C}

. Additivity over the common maximally mixed factor reduces data processing to

D (ρ_{A BC} \frac{I _{A}}{d _{A}} \otimes ρ_{BC}) \geq D (ρ_{A B} \frac{I _{A}}{d _{A}} \otimes ρ_{B}) .

- S (ρ_{A BC}) + lo g d_{A} + S (ρ_{BC}) \geq - S (ρ_{A B}) + lo g d_{A} + S (ρ_{B}) .

After cancelling

lo g d_{A}

S (ρ_{B}) + S (ρ_{A BC}) \leq S (ρ_{A B}) + S (ρ_{BC}) .

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