Liouville Fock state lattices and potential simulators

We introduce Liouville Fock state lattices (LFSLs) as a framework for visualizing open quantum systems through matrix representations of the Lindblad master equation (LME). By vectorizing the LME, the state evolves in a doubled Hilbert space, naturally forming a synthetic lattice. Unlike the unitary evolution of pure states, LFSL states exhibit nontrivial dynamics due to the non-Hermitian Liouvillian, featuring population drifts, sources, and sinks--paralleling stochastic classical lattices. We explore these "classical simulators" in both the Fock representation and alternative positive semidefinite representations, which more closely resemble classical probability distributions. We further demonstrate how infinite steady state manifolds can derive from frustration in the LFSL.