Extendibility of fermionic states and rigorous ground state approximations of interacting fermionic systems

Solving interacting fermionic quantum many-body problems as they are ubiquitous in quantum chemistry and materials science is a central task of theoretical and numerical physics, a task that can commonly only be addressed in the sense of providing approximations of ground states. For this reason, it is important to have tools at hand to assess how well simple ansatzes would fare. In this work, we provide rigorous guarantees on how well fermionic Gaussian product states can approximate the true ground state, given a weighted interaction graph capturing the interaction pattern of the systems. Our result can be on the one hand seen as a extendibility result of fermionic quantum states: It says in what ways fermionic correlations can be distributed. On the other hand, this is a non-symmetric de-Finetti theorem for fermions, as the direct fermionic analog of a theorem due to Brandao and Harrow. We compare the findings with the distinctly different situation of distinguishable finite-dimensional quantum systems, comment on the approximation of ground states with Gaussian states and elaborate on the connection to the no low-energy trivial state conjecture.