Pure-state entanglement rests on a single algebraic backbone: majorization of the Schmidt spectrum governs state conversion under local operations and classical communication, and constrains entanglement monotones. Here we establish a corresponding majorization law for fermionic non-Gaussianity, the resource that elevates free fermions to universal quantum computation. Under any fermionic Gaussian protocol with pure state outcomes, the Williamson spectrum of a pure state's Majorana covariance matrix is weakly majorized by its ensemble average. This spectral law mirrors that of entanglement theory. It turns computable non-Gaussianity quantifiers such as fermionic antiflatness and occupation entropies into strong monotones for fermionic non-Gaussianity, and delivers necessary conditions and converse bounds on state conversion under Gaussian protocols. When fermion parity is conserved, no catalyst can remove a majorization obstruction---unless it carries parity coherence---and asymptotic interconversion is irreversible already for pure states. All relevant quantities are accessible from two-point Majorana correlators, turning the theory developed here into experimentally observable properties of quantum matter, testable on present-day quantum devices.
X. Turkeshi, P. Sierant, P. S. Tarabunga· 0 citations
Fermionic non-Gaussianity, or fermionic magic, is a key resource underlying the computational complexity of fermionic quantum systems, yet tractable and operationally meaningful ways to quantify it remain limited. We address this challenge by developing a convex resource theory of fermionic non-Gaussianity and introducing two families of computable quantities for pure fermionic states, both derived from the Williamson normal form of the covariance matrix. The first family, occupation number entropies, is defined as the Tsallis-$\alpha$ entropy of the occupation numbers. We prove that one member of this family is monotonic under Gaussian protocols, establishing it as a computable convex resource monotone. It consequently lower bounds the number of non-Gaussian gates needed for state preparation. The second family, natural-orbital participation entropies, is given by the R\'enyi-$\alpha$ entropy of the squared amplitudes of the state in the natural-orbital basis, defined by the eigenvectors of the covariance matrix. They quantify state compressibility in this basis and thus upper bound the classical simulation cost in an orthonormal Gaussian basis. We analyze both families for stabilizer and translation-invariant states, where they simplify and reveal additional structure. We further study representative examples, including random SWAP-doped matchgate circuits and the bond-modulated XXZ model, highlighting the role of non-Gaussianity in many-body phenomena. Our work establishes a resource-theoretic framework for computable fermionic non-Gaussianity that unifies notions arising across quantum information, condensed-matter physics, and quantum chemistry, opening new directions for studying the complexity of quantum many-body systems and providing practical tools to assess the classical simulability of fermionic states relevant for quantum advantage.
P. S. Tarabunga, B. Jobst, Ra'ul Morral-Yepes et al.· 4 citations· ⚡1