One bit has two states and could represent either of two four-bit sequences for an algorithm set to produce both. Mathematically, that's about as trivial as it gets.
I have a 15 gallon gas tank and I want to know how much gas is in it with 1 gallon resolution and all you get is 1 bit to tell me. It works great if it's full or empty but how do you tell me I have 5 gallons? What kind of decompression algorithm is going to take 1 bit and tell me that?
That's not even the same question at all - perhaps that explains your non-comprehension. Data compression is a technique of representing a set of information ( a given length of data bits is a simple example) with a smaller length of bits using a specified coding/decoding algorithm (or an embedded decoding algorithm).
If you need to distinguish between 16 arbitrary states of a variable (gallons in your gas tank), it takes four bits. The point is that those states can be 1 bit of information or 100 GB's since their definition doesn't matter to the coding/decoding algorithm.
For arbitrary length strings of a fixed set of states (say gallons of gas in the tanks of cars that drive by on a highway or ASCII bytes), you can use a different class of coding/decoding algorithms based on random statistics for brute force techniques or tuned to known past samples and, perhaps, even self refining. There's plenty of elementary discussion about this on the web although most cases assume the case of large enough states to be useful (such 7- or 8-bit words).
I'm biting my tongue here.
You should be.
That's not even the same question at all.
If you need to distinguish between 16 arbitrary states of a variable (gallons in your gas tank), it takes four bits.
That's what I've been saying, you can't squeeze 4 bits worth of data into 1 bit. Nice try. If a 1 state means read the bible and a 0 state means read the Satanic Bible, the bit is still only a bit. You somehow believe you have stored the two books in the bit.