General nonexistence of LHV dynamics for interacting quantum dynamics

Prove that there exists at least one instance of interacting quantum dynamics for which dynamical local hidden-variable (LHV) dynamics does not exist; specifically, establish the existence of an interacting evolution that cannot be captured by any state-independent evolution of hidden variables on a fixed single-particle hidden-variable space reproducing the time evolution of local correlations.

Background

After constructing dynamical LHV models for noninteracting (separable) unitaries, the authors identify a simple interacting two-qubit example that appears incompatible with LHV dynamics. This motivates a broader conjecture that interactions can obstruct the existence of state-independent hidden-variable evolutions even when correlations remain local at all times.

The conjecture seeks a fundamental limitation: dynamics-induced nonlocality manifested as the impossibility of assigning a single, state-independent hidden-variable evolution to reproduce the full time evolution of otherwise locally simulable correlations.