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In learning device-independent QM from [https://arxiv.org/abs/1303.3081], Scarani tells us that all quantum statistics involving one particle can be reproduced with local (hidden) variables. He then constructs the statistics of a qubit in such a model. Insightful as this is, it is still not clear to me why all single-particle statistics can be reproduced. Is it because the local variable can hold information about the single-particle state? Why can't it do the same in a many-particle system?

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Bohmian mechanics provides a simple way to do this, and in the case of one particle alone, is completely local. It only becomes non-local with multiple particles.

The objective state of a single particle in Bohmian mechanics is described by a pair $(\psi, P)$ consisting of an objective wave function $\psi$ that follows the same Schrodinger equation as the subjective one, and a new position parameter $P$ that is the "true" position one measures. This $P$, then, is governed by the guiding equation

$$\frac{dP}{dt} = \Im\left(\left[\frac{\hbar}{m} \frac{\nabla \psi}{\psi}\right](P)\right)$$

. Since the $\psi$ here only depends on a single point in space, this is a fully local description: it's literally "a particle riding atop a wave", hence the other name, "pilot wave" interpretation. In multiple trials, of course, the initial state of $P$ should be suitably randomized, which completes the theory.

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