Extracting Universal Entanglement Scaling from Mixed Fermionic Gaussian States via Entanglement Projected Entropy

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 $\pi$-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.