One of my favorite information theory things is parallel construction.
Let's say you have a piece of information, but either you're not allowed to have that information, or you don't want to reveal your sources. Well, sometimes you can just say "I'm not telling you" and that works. But in certain circumstances, you can instead produce an alternate explanation for how you might have gotten this information, now that you have it.
So, let's say that you're a private detective in possession of mind-reading abilities, but you don't want people to know that you have mind-reading abilities. This means that you can't act on what you seeing when mind-reading ... except that you can engage in parallel construction. If you're trying to catch a cheating spouse, you read the mind in order to know when and where to be, and then everyone will just assume that you were doing a normal tail and stakeout. There's no reason to suspect you of being a mind-reader.
It's a really interesting problem, trying to construct a plausible way that you could have obtained this information, and it gets harder depending on what the parameters are. Are you trying for a fig leaf? Or does it need to be rock solid, standing up to scrutiny?
This is one of the things the Allies had to think about after the Enigma code was broken. If they acted without any obvious reason to act, they'd be found out, so they would e.g. send out a scout plane to spot a ship whose location they already knew. That way the Axis would report being spotted by a plane, and it would just seem like luck.
This is also one of the things that's done by our pervasive surveillance state. NSA intercepts point DEA (or whomever) in the right direction, then they do a "random" traffic stop or "anonymous tip" as justification. The defendant and their attorney don't ever find out the real origins of case. Which is, in my opinion, less fun than the Enigma example.
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