Approximation theorems for fermionic Gaussian states

Abstract

A recurring principle in many-body physics is that sufficiently distributed interactions or correlations lead to an effectively mean-field description. For fermionic Gaussian states, we establish a quantitative version of this principle directly at the level of Majorana covariance matrices. The main observation is that the admissibility condition for fermionic covariance matrices imposes a quantitative monogamy constraint on two-point correlations: a fixed region has only a bounded covariance budget to distribute among many disjoint regions. We use this principle in three directions. First, for quadratic fermionic Hamiltonians on finite simple graphs of high degree, we obtain product-state approximations to both the ground-state energy and the finite-temperature free energy, with error of order \(D^{-1/2}\) for \(D\)-regular graphs. Second, we prove a finite fermionic Gaussian de Finetti theorem: a \(k\)-copy Gaussian state that is \(n\)-exchangeable under graded fermionic copy permutations is \(O(k/n)\)-close in trace norm to the product of its one-copy marginal. In particular, infinite exchangeability within the Gaussian class implies exact product structure. Third, for spatial fermionic Gaussian states with exponentially clustering mutual information, we prove that the conditional mutual information across a separating buffer decays as \(O(r^{-1})\), and hence, by recoverability, the state admits an \(O(r^{-1/2})\) trace-norm approximation by a recovered state. These results show that product and de Finetti approximations and recoverability estimates for fermionic Gaussian systems can all be derived directly from covariance-level structure.