If I understood properly, you wanna compress one Byte of data. i.e. 8 bits.

No, if you can represent 8 bits into lesser bits let's say 7 bits, then atleast half of the 256 bit strings will map to some compressed bit pattern that already been mapped from some other 8 bits pattern. Thats pigeonhole principle.

The pigeonhole principle is a mathematical concept that states that if there are more items than containers, then at least one container will have more than one item.

Lets assume 8 bits can be represented into lesser bits, then that lesser bits can further be compressed to much lesser bits using the same technique, till left with 0 or 1. So, TeraBytes and GigaBytes of data could have stored in a single bit. AND IT HAPPENS. Thats called Hash like SHA256, MD5, SHA1 etc. can represent any data into fixed length hash e.g. SHA256 generates 256bits hash of any size of data.

But the hard part is when decompressing the data. from 1 bit it is almost impossible to generate 8 bits or GigaBytes of data.

So, YES YOU CAN compress but can't decompress.