Suppose Alice and Bob agree that she will send him a sequence of $4$ bits of data which are not all $1$s. This is a simple example of a communication protocol as defined, for example, in Wikipedia. The result is that Alice can make a free choice among 15 options and send that choice to Bob. So this protocol effectively sends slightly fewer than $4$ bits of information: to be precise, $\log_2 15$ bits. Similarly, for any integer $n >= 1$, there's an obvious protocol which effectively sends $\log_2 n$ bits of information.
Are there any other possibilities here? That is, for what real values $r$ is there a protocol which effectively sends $r$ bits?
In particular, is there a protocol which sends more than zero bits but less than one bit? If there were, then we would expect that Alice cannot use it to send one bit, but can use several repetitions of the same protocol to send one bit.
I suspect the answer is no. I'd appreciate any thoughts on how to prove it.
I don't have a definition of "protocol" or "effectively sends", but the idea seems pretty clear. Here are some more details. (1) We're assuming that the basic communication channel has no noise. (2) If she sends the forbidden $4$-bit sequence, we consider that the protocol was not completed; it's much the same as if she sent $3$ bits and then stopped. (3) We should allow a protocol to specify multiple messages back and forth between Alice and Bob.
Edit: expanded for clarity.