When discussing ways that quantum computing may fail, a common idea is that it may turn out that the linearity of quantum theory fails. Since no one has seen any evidence of nonlinearity in quantum theory, and it is hard to hide this nonlinearity at small scales, it is usually reasoned that these nonlinearities would arise for large quantum systems. Which got me thinking about how to well we know that quantum theory is linear, which in turn got me thinking about something totally wacko.
For you see I’m of the school which notes that the linearity of quantum theory, if broken, almost always leads to effects which don’t just do crazy things like superluminal signaling, but also have effects, like, say radically mucking with traditional probability theory. That is most nonlinear theories lead to things like the ability to amplify probability distributions: there become allowed physical process which take a mixture of 51 percent 0 and 49 percent 1, and without running repeated experiments, turn this into, say, a mixture of 99 percent 0 and 1 percent 1, but don’t do anything to a mixture of 50 percent 0 and 50 percent 1. This, seems a bit radical to me, but what do I know?
But what this made me wonder was: how well has classical probability theory been tested? Okay I know this sounds really silly, right? Classical probability is a diety given entity sent down from the heavens to mollify those in need of learning measure theory and arguing over frequentest and Baysian interpretations. But if there is anything that we should learn from quantum theory, it is that the universe seems to obey our normal ideas about probabilities, and yet, on a deeper layer, our description is a departure from this in some strange manner. So how confident are we that that classical probability holds? And if you believe in nonlinear quantum theories at some large scale, do you believe that these effects won’t have a large effect on classical probability?