Even by the normal standards of quantum weirdness, this one is really out there. You can take a piece of quantum information and only teach half to another person...but you'll never be able to figure out which half they learned.
To understand just what this phenomenon entails, let's imagine a book with two pages. In quantum thought experiments, we turn to two players, Alice and Bob. Alice is given the book to read and told that she can provide exactly one page's worth of notes to Bob, who is not allowed to read the book. In other words, Bob will necessarily only have 50% of the total information from this book.
Now, in the classical world, it's pretty obvious what would happen if an instructor gave Bob a test on the book. If the test questions covered the 50% of the book Bob had notes on, then he would do quite well, but if they focused on the other half about which Bob had no information, then he would do quite poorly. What's more, the instructor could purposefully design the test so that it only tested the half of the book of which Bob was ignorant. In that way, the instructor could easily figure out what Bob did and didn't learn from Alice's notes. Simple right?
Time to buckle up for the quantum weirdness. You see, if Alice gave a page's worth of quantum information, there would be no way for the instructor to pinpoint what Bob did and didn't know. As long as the instructor asked Bob questions about 50% of the book, Bob would almost certainly be able to answer, even if the instructor was able to inspect the notes Alice gave Bob and designed the test accordingly. The only limit to this is that the quiz has to cover the same amount of knowledge as what Bob initially received - Bob isn't able to produce new information.
That doesn't make any sense at all, right? It's one of the many, many non-intuitive results that we get from probing into the quantum world...but that doesn't mean that it's false. The researchers behind the finding freely admit that they don't understand the result, and it's likely we'll never be able to really comprehend it.
But this issue isn't entirely academic - having at least a working knowledge of this strange information principle could be crucial for quantum cryptography and for quantum computers. Indeed, there could be some tremendous use for only have to encoding an entire block of information in only half the space - as long as we never accessed more than half of it at a time, that is. Now we just need to figure out whether that's really how this strange phenomenon behaves.