Decision problem for existence of dynamical LHV models

Determine, for a given Hamiltonian H(t), a specified set of single-particle measurements M1 (with corresponding local product measurements M), and a set S of local quantum states that remain local for all times under the evolution generated by H(t), whether there exists a dynamical local hidden-variable (LHV) model with local and deterministic hidden-variable dynamics that reproduces the time evolution of all measurement statistics for measurements in M.

Background

The paper defines dynamical LHV models as state-independent evolutions of hidden variables on a fixed single-particle hidden-variable space that reproduce the quantum time evolution of local measurement statistics. The authors distinguish deterministic, local, microscopic, and smooth properties for such dynamics.

They show constructive positive results for noninteracting unitaries and present obstacles for interacting cases, motivating a general decision problem: given a concrete Hamiltonian H(t), measurement set, and a class of states that remain local for all times, decide whether compatible local deterministic LHV dynamics exists.

References

Open Question. Let H(t) be a Hamiltonian, let M1 be a set of measurements and let S be a set of local states that remain local for all times under evolution with respect to H(t). Does there exist a dynamical LHV model for H(t), S, M with local and deterministic LHV dynamics?

— On the Dynamics of Local Hidden-Variable Models  (2512.16682 - Selzam et al., 18 Dec 2025) in Section 5 (A Conjecture), Open Question