Mathematical proof or any reference of "Deterministic random bit generator cannot produce more randomness than the randomness of seed"

Question

The heading tells everything. Any proof or any kind of reference is welcome regarding this:

Can we get more randomness from a deterministic random bit generator than the entropy that we feed to the generator?

Answer 1

Define the Mutual Information of a pair of random variables. $$I(X; Y) = H(X) - H(X\mid Y)$$ For discrete random variables we hae that $H(X\mid X) = 0$, so: $$I(X; X) = H(X)$$

The Data-Procesing Inequality states that for any function $f$, we have that: $$I(f(X); f(Y)) \leq I(X; Y)$$ While we won't need it here, this includes randomized functions, provided they use randomness independent of either $X$ or $Y$. Combining these two, we get that:

$$H(f(X)) - H(f(X)\mid f(X)) \leq H(X) - H(X\mid X)$$ If $f$ is deterministic, I believe the conditional entropies become 0, giving us: $$H(f(X)) \leq H(X)$$ So the output distribution of a DRBG is always lower entropy than the distribution that its seed is drawn from.