Local Complex Dependence and Separability in Madelung Hydrodynamics
For a many-particle pure state in a fixed position representation, we consider the mixed cross-particle derivatives of the logarithm of the wave function. Their real part is one half of the Holland-Wang local dependence function of the configuration density, while their imaginary part is the cross-Jacobian of the Madelung velocity field. On a node-free product region, vanishing of all cross blocks throughout the region is equivalent to local multiplicative separability. Under Schrodinger evolution with a real scalar potential, the initial growth of a cross block from a separable state is sourced by the corresponding mixed Hessian of the potential and is purely imaginary to first order in time. The construction is a local separability and dependence diagnostic, rather than a basis-independent entanglement measure.