# Strong sub-additivity

**Strong sub-additivity** is a relationship between various entropic quantities. In quantum information theory, it gives <amsmath>S(AB)+S(BC)\geq S(ABC)+S(B)</amsmath> One can write it suggestively in terms of the [[conditional entropy as S(A|BC)\leq S(A|B) where it follows from the fact that the rate of quantum state merging can only be better if the receiver has access to additional information.

Strong sub-additivity implies sub-additivity, namely that *S*(*A*) + *S*(*B*) ≥ *S*(*A**B*), which is another way of saying that the quantum mutual information is always positive. To see this result, just make the state on system "B" pure.

Currently, it is open problem to find non-constrained inequalities among entropies which are not implied by strong subadditivity.

It is of interest which states saturate the strong sub-additivty bound.

Category:Quantum Information Theory Category:Handbook of Quantum Information