Free fermion dynamics is classically simulable, and a single non-Gaussian state makes it universal. We show that the same boundary governs quantum nonlocality. Measured as fermionic qubits, fermionic Gaussian states reach at most the biseparable bound of the Mermin inequality for three parties and, for any number of parties, violate the Mermin family by exactly the factor of a single entangled pair. They are never genuinely multipartite nonlocal. Genuine multipartite nonlocality and computational universality are thus one resource for fermions, the opposite of the stabilizer case. The mechanism is fermion parity. The bound is monogamous and certifies non-Gaussianity from measured correlations alone. On superconducting processors we watch nonlocality beyond free fermions switch on under an interaction between densities, confirm the monogamy law on 96 qubits, and measure the ground state of twelve interacting fermions on a ring cross the bound as the interaction grows, detecting the charge density wave transition.
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.
Identifying spatial quantum correlations in mixed states is challenging because thermal mixed-state contributions obscure the entanglement encoded in subsystem entropy. Here, we introduce the entanglement projected entropy (EPE), a purification-independent Gaussian spatial filter for mixed fermionic states. By resolving subsystem entropy into Gaussian entropy channels and projecting their purification partners onto the physical complement, we obtain a closed-form expression in terms of the physical covariance matrix. In a one-dimensional free-fermion chain, it removes the volume-law mixed-state background and recovers the zero-temperature conformal scaling with the
c/3 coefficient. In a two-dimensional half-filled
π-flux model, it reveals a universal finite-temperature scaling collapse governed by a Dirac infrared length fixed by the low-energy velocity. These results establish EPE as an entropy-channel filter that exposes boundary-sensitive universal scaling hidden beneath mixed-state entropy.