What is reason behind subadditivity property? How does the entropy of a joint system can be less than the sum of entropies of its components?


1 Answer 1


Subadditivity is not necessarily a quantum property. The Shannon entropy is also subadditive. That is,

$$H(AB) \leq H(A) + H(B).$$

The intuition behind it is that if systems $A$ and $B$ have any kind of correlations, then their mutual information is positive. Writing this out, you have

$$I(A:B) = H(A) + H(B) - H(AB) \geq 0,$$

which is a restatement of subadditivity.

Another way to see it is that when $A$ and $B$ are correlated, you learn something about $A$ if you know $B$ and vice versa. This redudant information is double counted when you consider $H(A) + H(B)$ and that's what makes it greater than the information in the joint system $H(AB)$.


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