On the injective norm of random fermionic states and skew-symmetric tensors

• URL: http://arxiv.org/abs/2510.25474v1
• Date: Wed, 29 Oct 2025 12:52:07 GMT
• Title: On the injective norm of random fermionic states and skew-symmetric tensors
• Authors: Stephane Dartois, Parham Radpay,
• Abstract summary: We study the injective norm of random skew-symmetric tensors and the associated fermionic quantum states.<n> Extending recent advances on random quantum states, we analyze both real and complex skew-symmetric ensembles.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the injective norm of random skew-symmetric tensors and the associated fermionic quantum states, a natural measure of multipartite entanglement for systems of indistinguishable particles. Extending recent advances on random quantum states, we analyze both real and complex skew-symmetric Gaussian ensembles in two asymptotic regimes: fixed particle number with increasing one-particle Hilbert space dimension, and joint scaling with fixed filling fraction. Using the Kac--Rice formula on the Grassmann manifold, we derive high-probability upper bounds on the injective norm and establish sharp asymptotics in both regimes. Interestingly, a duality relation under particle--hole transformation is uncovered, revealing a symmetry of the injective norm under the action of the Hodge star operator. We complement our analytical results with numerical simulations for low fermion numbers, which match the predicted bounds.

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