Nonlocal magic of fermions is bounded by entanglement fluctuations and is logarithmic for free fermions

arXiv:2611.07003v1 Announce Type: new Abstract: Free-fermion states are classically simulable, yet, written as qubits, they typically carry extensive magic, the quantum resource that stabilizer circuits lack. Part of this magic is nonlocal: local unitaries acting separately on two parts of the system cannot remove it. If these local unitaries are restricted to Gaussian ones, the nonlocal magic grows linearly with the number $m$ of partially entangled fermion pairs. We show that this growth is an artifact of the Gaussian restriction: local Gaussian unitaries minimize the nonlocal stabilizer R\'enyi entropy for up to three pairs, but not from four pairs on. General local non-Gaussian unitaries that respect fermion parity take $m$ free-fermion pairs to fermionic Bell pairs with a fidelity that decays at most as $1/\sqrt m$ compared to exponential decay with Gaussian unitaries, so the nonlocal magic drops from linear to logarithmic in $m$. The reason is statistical --- Bell pairs have equal Schmidt weights, that is, a fully degenerate entanglement spectrum, and a state can stay far from them only if its Schmidt weights are spread over many orders of magnitude. This spread is measured by the capacity of entanglement $C_E$, the variance of the entanglement Hamiltonian, and the nonlocal magic of any fermionic state grows at most logarithmically with $C_E$. Free-fermion pairs add to $C_E$ independently, so $C_E$ grows only linearly in $m$, and this leads to the stabilizer fidelity decreasing at most as $1/\sqrt m$. Almost all the magic of free fermions, including Hartree--Fock and BCS mean-field states, is therefore local.