Lucy Keer

Interesting thing I found out while writing the post: quantum computing people call the states with negative entries 'magic states'. Because they contain the magic! I.e., potential for speedup over classical computing.

Even the most magic states are nowhere near as negative as the -½ of the toy example. Why not? Well, there are some constraints.

The interesting one is to do with information. For the toy example, we got definite Y/N answers to all 3 questions. This time the rule is you can only get max half the information. In a weird sense of 'information' I'll get to soon.

There's a very influential toy model that also has this 'half the information' property: https://en.wikipedia.org/wiki/Spekkens_toy_model … Very simple model that reproduces some, but not all, features of quantum theory. It uses just the states on the six corners of the octahedron. So no magic.

Now, there's a fascinating paper by van Enk that builds on the Spekkens toy model, by extending the 'half the information' property to more states: https://arxiv.org/abs/0705.2742  To do this, you need a good definition of information.

Most obvious choice is the usual one from information theory, linked to the Shannon entropy: https://en.wikipedia.org/wiki/Entropy_(information_theory) … Turns out this isn't the right choice to reproduce QM, though. Instead you want a weirder entropy.