Lucy Keer

Instead I'm going to go through an example that shows *how* they work. I got this one from another excellent blog post, by @sigfpe: http://blog.sigfpe.com/2008/04/negative-probabilities.html … I like it because the numbers are nice and simple.

So we have the following setup (from that post): "a machine produces boxes with (ordererd) pairs of bits in them, each bit viewable through its own door"

Now we ask 3 questions: 1. Is the first box in state 0? 2. Is the second box in state 0? 3. Are the boxes both in the same state? Obviously these overlap: e.g. to be consistent, if you answered Y to 1. and N to 2., you'd also answer N to 3.

Given this, it all works. E.g. for question 1 P(first 0) = P(first 0, second 0) + P(first 0, second 1) = -½ + ½ = 0 So answer is N. Same for the others.

Now, where that -½ comes from was a bit opaque to me. I mean I could follow the algebra, just didn't have much intuition for how it got there. So I played around for a while and came up with the following.

Motivating idea: NNN is inconsistent, so all four possible assignments don't work. But {first box 0, second box 0} is *especially* bad. It's the wrong answer to all three questions. Other three boxes are only the wrong answer to one question.

So... we penalise bad answers, and, because {first box 0, second box} is particularly bad, it gets clobbered three times instead of just once, driving it negative. That's vague. But I'll outline a precise version.

Is this trick any use? I really don't know! But I definitely find this breakdown more illuminating than just plugging through the algebra.

Now the reason I'm interested in this is the link to quantum physics. This toy model is very similar to phase space for a qubit. In some sense it's slightly 'worse', as -½ is more negative than anything that comes up there. But the simple numbers make it easy to play with.