# A QM–Anthropic Paradox

Consider the following thought experiment.

Flip a quantum coin. On heads, stop; on tails, flip again and note the result. This yields three possible outcomes: 50% heads, 25% tails-heads, 25% tails-tails.

But this -- if not *implies*, exactly, then *encourages* -- a misunderstanding of MWI. The Born probabilities *square* the *complex* amplitudes. Quantum coins aren't splitting a fixed allocation of probability-mass. They're *transforming* an amplitude, multiplying it by complex factors.

Squaring a big number makes it bigger; squaring a small number makes it smaller. If you could split the universe, the probabilities of the pieces should add up to less than the probability of the whole.

There's a "Universe Splitter" app.

By the logic above, this app should be a Regret Button -- Everett branches where it's used are, all else being equal, less probable than branches where it's not. It may not be a very *strong* Regret Button, comparatively, but it seems to have, at a conservative estimate, several users who have used it several times each.

Since the anthropic principle provides (the appearance of) protection against destroying the universe, we probably ought to find ourselves in a world where this app had not been created.

What about that "multiplying by complex factors" bit above? Suppose that each of the split branches has the same absolute value as the pre-split parent branch?

In that case we should find ourselves in a world where the app is a smash hit and everyone uses it obsessively.

A *modestly* successful app makes *no sense*. It could only happen if splitting the universe has, on average, *no net effect* on the overall probability of the branch-family. This is what we should see if the probabilities were directly proportional to positive real amplitudes -- a hypothesis thoroughly disproven by more rigorous experiments.

Hopefully, it should be obvious (by modus tollens) that I don't actually know what I'm talking about and this argument has at least one serious flaw of some sort. But I'm not able to identify exactly where the flaw is, so I'm posting in the hope that someone can explain it.