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 $A=B=0$. Another reasoning would not set them to zero but take the zero-order Taylor expansion in $x$ at $X(t)$ thus simply setting $x=X(t)$. This means the auxiliary potentials are time-dependent but constant in space and only contribute a global phase to the conditional wave function.¹⁾

This kind of approximation has been used in ^[1] and ^[2] (where it is called “small entanglement approximation”). Note that even though the effective potential is reduced to the pure external potential this quantity enters the conditional evolution equation as $V(t,x,Y(t))$ and thus depends on the path $Y(t)$ of the remaining particles.

A higher order approximation could take $\phi_1$ and $\phi_2$ into account and ignore higher orders in the respective evolution equations. The potentials are then evolved by a time-stepping procedure alongside the conditional wave functions. Note however that the effective potential is different for every conditional wave function. 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.