Entropy Preserving Functions Let F be a deterministic compression function that takes fixed length 128 bit inputs and outputs 8 bits. Each input to the function has 128 bits of entropy, does that mean that the output byte has 128 bits of entropy ?
Of course not; 8 bits have a maximum of 8 bits of entropy.  There does not exist a probability distribution over the 256 possible values of those 8 bits that gives an entropy value of more than 8.

What this means is that your function can have more entropy on the input than the output.  This is not uncommon, and typically happens if you have a function with fewer possible outputs than inputs (the only exception is if the probability distribution that the input occurs has some inputs with zero probability).