If one wants to solve the conditional wave function to get a guiding field for the Bohmian trajectory one needs an approximated expression for the effective potential originating from the potential network $\phi_i$.

In the case of an unentangled state that factorizes as $\Psi(t,x,y) = \alpha(t,x)\beta(t,y)$ we see that $\phi_i(t) = \nabla_y^i \beta(t,y)/\beta(t,y)|_{y=Y(t)}$ with no $x$-dependence left. Thus we cannot expect any real influence from these auxiliary potentials on the conditional wave function $\psi(t,x)$. This leads us directly to a very rough approximation where all such entanglement effects are ignored and $\phi_i=0$ for all $i \geq 1$ and thus $A=B=0$. This kind of approximation has been used in ^[1] and ^[2] (where it is called “small entanglement approximation”).

A higher order approximation would at try to take $\phi_1$ and $\phi_2$ into account and ignore higher orders in their respective evolution equations. The potentials are then evolved by a time-stepping procedure alongside the conditional wave functions. Note that the effective potential is different for every conditional wave equation. This is in contrast to other effective potential techniques like the Kohn-Sham approach in density functional theory, where the effective potential acts like a usual external potential on non-interacting particles.