| There is usually no obvious way to convert the initial condition of an arbitrary quantum system to that for a classical system. The other way around, there is some consensus what the quantization of standard classical systems should be, but to define this transition in general is, again, not straightforward. Also, you obviously don't arrive at the same final condition for two different theories, otherwise they would be equivalent. To talk about "number of computational steps" (which I have never seen referred to as "definition of computational complexity") you need to specify a computational model. The number and the comparison between two programs will depend upon it and you can get whatever result you want by choice of the model. The conversion of theory -> model -> program is also not at all straightforward. The complexity strongly depends on the conversion. If you specify "best possible conversion" for model to program (which is probably the easiest part of the huge amount of things you need to define), you cannot prove anything about it anymore. The complexity difference between asymptotic classical and quantum calculations is usually determined by its scaling in the "size of state space". Why do you focus on time and precision? I don't know for sure but it seems reasonable to me that the complexity with respect to those two might actually be the same for common system models. Regardless, you have to choose some parameters and you provided no arguments that this choice is in any way objective. > In fact, if it turns out to be false, it is quite likely that it will also turn out quantum mechanics is in fact reducible to classical mechanics Strongly disagree for any reasonable notion of "reducible". Quantum theory is different from classical in many qualitative and experimentally accessible ways (e.g. Bell inequalities). Computational complexity isn't everything. To end, I want to further contest your defining complexity of a theory according to "length of computation to get model prediction". An interesting alternative would be "length of theory specification under optimal compression " (i.e., "Kolmogorov complexity of a theory"). This notion obviously also isn't rigorously defined yet but I think it's about as hard to definine as your notion. Neither of them seems obviously the "right one" to me in this context. |