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**, which is in the realm of Bohmian mechanics, Antony Valentini claims that by having a "non-equilibrium" particle with arbitrarily accurate "known" position, we can measure another particle's position with arbitrary precision, violating Heisenberg's uncertainty principle.**

In summary, the "sub-quantum measurement" method works as follows (see part 4 of the paper): In a simple case, we have

**y(t) = y(0) + a.x(0).t**, where

**x**and

**y**are the positions of the measured and test (pointer) particles, respectively. We have kind of entanglement between the test particle's momentum and the measured particle's position (with Hamiltonian

**H^ = a.x^.p^y**, where

**a**is a coupling constant,

**py**is the momentum canonically conjugate to

**y**, and

**x**is the measured particle's position). Here we have

**y(0)**and

**y(t)**with arbitrary precision, since the test particle is in quantum non-equilibrium.

The point is that, In order to predict the trajectory of the measured particle,

**x(t)**, we need BOTH

**x(0)**and

**Ψ(x,y,t)**(the wavefunction of the ensemble). This wavefunction would be 6-Dimensional in Bohmian mechanics and inaccessible to us in the 3-D world, so actually the theory would in general be useless. It's only in the limit

**a.t -> 0**that

**Ψ(x,y,t) ≈ ψ0(x)g0(y)**, and the total wavefunction would be approximately 3-D.

Now my question comes.

**Ψ(x,y,t) ≈ ψ0(x)g0(y)**is only an approximation, being completely precise only when

**a.t = 0**, in which case

**y(t) = y(0)**and we lose connection to the measured particle's position (

**x(0)**). In other words, the bigger the

**a.x(0).t**, the more the error we have in our approximation of the ensemble's wavefunction as 3D. We can have a large

**a.t,**in which case the wavefunction would be inaccurate and we would not be able to accurately calculate

**x(t)**from

**x(0)**, OR we can have a very small

**a.t**and a precise wavefunction, in which case the parameter

**a.x(0).t**, our tool to determine

**x(0),**would also be very small and maybe comparable to the uncertainty in

**y(0)**, which makes our measurement of

**x(0)**inaccurate. So there doesn't seem to be any way to track the trajectory

**x(t)**precisely, we need to make either

**x(0)**or

**Ψ(x)**inaccurate.

So is my discussion valid? Is the uncertainty in

**x(t)**kind of irreducible, because of the uncertainty in any case we have, either in

**x(0)**or in

**Ψ(x),**which are in a kind of trade-off?