Why can data be compressed only once?

Question

So the compression process takes a chunk of binary data A and outputs a smaller chunk of binary data B. What characteristics of B make it unable to go through this process again?

Answer 1

Data has something called entropy: the amount of new information each new bit gives. For example, 10101010101010101010 has low entropy because you don't need the next bit to know what comes next. A perfect compression algorithm would compress to maximum entropy, so every bit gives information and so cannot be removed, making the size a minimum.

Answer 2

It is not true that data that is already compressed cannot be compressed again. If you take a file consisting of 1 million zeros and compress it using gzip, the resulting compressed file is 1010 bytes. If you compress the compressed file again it is further reduced to just 75 bytes.

$ python
>>> f = open('0.txt', 'w')
>>> f.write('0'*1000000)
>>> f.close()
>>>
$ wc -c 0.txt
1000000 0.txt

$ gzip 0.txt
$ wc -c 0.txt.gz
1010 0.txt.gz

$ mv 0.txt.gz 0.txt
$ gzip 0.txt
$ wc -c 0.txt.gz
75 0.txt.gz

The reason why it is unlikely that compression works twice is because the compression process removes redundancy. When you have less redundancy it is harder to compress the file further.