Maximally antiflat fixed-particle-number fermionic states

arXiv:2610.13903v1 Announce Type: new Abstract: Fermionic Gaussian states are completely characterized by their two-point correlations, making the covariance matrix a natural reference for quantifying non-Gaussian structure. Motivated by fermionic antiflatness, we study its extremal form for pure number-conserving states and ask which states can be maximally antiflat and how they can be constructed. We show that, at fixed particle number, maximal first antiflatness is equivalent to a completely flat one-body reduced density matrix, and we obtain a complete existence classification together with explicit block and cyclic-orbit constructions of all allowed sectors. We then investigate the structure that appears beyond the covariance level. Maximally antiflat states with identical one-body data can exhibit distinct connected four-point correlations, which we characterize through the four-point Majorana Wick error and, equivalently, the connected two-body reduced density matrix cumulant. We show that this quartic structure is necessarily nonzero and formulate its extremization as a two-body reduced density matrix (2RDM) purity problem. Combining this framework with a Hilbert--Schmidt estimate of Christiansen for truncated two-body correlations, we obtain an upper bound on the connected quartic structure in terms of first antiflatness. Our results establish both a complete characterization of maximal fermionic antiflatness at fixed particle number and a constrained hierarchy linking covariance-level flatness to higher-order non-Gaussian correlations.

Abstract:Fermionic Gaussian states are completely characterized by their two-point correlations, making the covariance matrix a natural reference for quantifying non-Gaussian structure. Motivated by fermionic antiflatness, we study its extremal form for pure number-conserving states and ask which states can be maximally antiflat and how they can be constructed. We show that, at fixed particle number, maximal first antiflatness is equivalent to a completely flat one-body reduced density matrix, and we obtain a complete existence classification together with explicit block and cyclic-orbit constructions of all allowed sectors. We then investigate the structure that appears beyond the covariance level. Maximally antiflat states with identical one-body data can exhibit distinct connected four-point correlations, which we characterize through the four-point Majorana Wick error and, equivalently, the connected two-body reduced density matrix cumulant. We show that this quartic structure is necessarily nonzero and formulate its extremization as a two-body reduced density matrix (2RDM) purity problem. Combining this framework with a Hilbert--Schmidt estimate of Christiansen for truncated two-body correlations, we obtain an upper bound on the connected quartic structure in terms of first antiflatness. Our results establish both a complete characterization of maximal fermionic antiflatness at fixed particle number and a constrained hierarchy linking covariance-level flatness to higher-order non-Gaussian correlations.

arXiv:2610.13903 [quant-ph]

(or arXiv:2610.13903v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.13903 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Maximally antiflat fixed-particle-number fermionic states

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