"noise" can still be compressed to reduce the size...

Good encryption should be indistinguishable from random strings. What is the chance that it could be compressed by at least one byte? There are 256^n strings with a length of n bytes. Since we want to be able to recover the original string, each compressed string must decompress to one decompressed string. There are 256^(n-1) strings of length n-1 bytes. This means that only (256^(n-1))/(256^n) = 1/256 strings of length n can be compressed by one byte. For more highly compressed strings, the chances get worse. This is why trying to compress the data would be useless.