Williamson majorization theory of fermionic non-Gaussianity

• URL: http://arxiv.org/abs/2608.10140v2
• Date: Mon, 17 Aug 2026 18:31:02 GMT
• Title: Williamson majorization theory of fermionic non-Gaussianity
• Abstract summary: We establish a corresponding majorization law for fermionic non-Gaussianity.<n>We turn computable non-Gaussianity quantifiers into strong monotones for fermionic non-Gaussianity.<n>All relevant quantities are accessible from two-point Majorana coherenceors.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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