Nonexistence of LHV dynamics for the Bell-LHV base model under two-qubit Heisenberg interaction

Prove that for two qubits evolving under the Heisenberg Hamiltonian, there exists a set S of local two-qubit states that remain local at all times such that, although the Bell-LHV base model (with single-particle hidden-variable space S^2 and measurement rule q(a|n,λ)=Θ(n·λ)) reproduces the instantaneous statistics at each time, no choice of gauge yields a state-independent hidden-variable velocity field that makes this base model into a dynamical LHV model.

Background

Using the Bell-LHV base model for projective qubit measurements, the authors explicitly construct time-dependent hidden-variable distributions reproducing the quantum correlations of separable two-qubit states undergoing Heisenberg interaction. They then show that no state-independent velocity field can satisfy the associated continuity equation for all such states, indicating incompatibility with LHV dynamics for the constructed gauge.

They conjecture this obstruction persists for any gauge: even if alternative valid hidden-variable distributions are chosen at each time, there should still be no state-independent velocity field turning the Bell-LHV base model into a dynamical LHV model for this interacting scenario.