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

TL;DR

This paper investigates the injective norm of random fermionic states and skew-symmetric tensors, revealing asymptotic behaviors, dualities, and symmetries relevant to multipartite entanglement in quantum systems of indistinguishable particles.

Contribution

It extends analysis of random quantum states to fermionic systems, deriving bounds and asymptotics for the injective norm in different scaling regimes, and uncovers a particle-hole symmetry.

Findings

Derived high-probability bounds on the injective norm.

Established sharp asymptotics in two asymptotic regimes.

Discovered a duality symmetry under particle--hole transformation.

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.