(Preamble: I apologize in advance for cluttering TSZ with these three posts. There are very few people on either side of the debate that actually care about the details of this “conservation of information” stuff, but these posts make good on some claims I made at UD.)
To see that active information can easily be created by chance, even when the LCI holds, we’ll return to the Bertrand’s Box example. Recall that the LCI holds for this example, and all choices are strictly random. Recall further that choosing the GG box gives us 1 bit of active information since it doubles our chance of getting a gold coin. If we conduct 100 trials, we expect to get the GG box about 33 times, which means we expect 33 bits of active information to be generated by nothing but chance.
But before we say QED, we should note a potential objection, namely that we also expect to get SS about 33 times, and each such outcome gives us negative infinity bits of active information. So if we include the SS outcomes in our tally of active information, the total is negative infinity. Be that as it may, the fact remains that in 33 of the trials, 1 bit of information was generated. This fact is not rendered false by the outcomes of other trials, so those 33 trials produced 33 bits of information.