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

Stephane Dartois, Parham Radpay

TL;DR

This work extends the study of the injective norm to random fermionic states represented by skew-symmetric tensors, analyzing real and complex Gaussian ensembles in fixed-$p$ and double-scaling regimes. Leveraging the Kac–Rice formula on Grassmann manifolds, it provides high-probability upper bounds and sharp asymptotics for the injective norm, revealing a particle–hole duality under the Hodge star. The results connect geometric entanglement in fermionic systems with random-tensor landscapes, and are complemented by numerical simulations that confirm the theoretical predictions. Overall, the paper advances understanding of multipartite entanglement for indistinguishable particles and highlights a symmetry structure that could inform future analyses of random tensors across symmetry classes.

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.

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

TL;DR

This work extends the study of the injective norm to random fermionic states represented by skew-symmetric tensors, analyzing real and complex Gaussian ensembles in fixed- and double-scaling regimes. Leveraging the Kac–Rice formula on Grassmann manifolds, it provides high-probability upper bounds and sharp asymptotics for the injective norm, revealing a particle–hole duality under the Hodge star. The results connect geometric entanglement in fermionic systems with random-tensor landscapes, and are complemented by numerical simulations that confirm the theoretical predictions. Overall, the paper advances understanding of multipartite entanglement for indistinguishable particles and highlights a symmetry structure that could inform future analyses of random tensors across symmetry classes.

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.

Paper Structure

This paper contains 14 sections, 25 theorems, 124 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Random matrix models and their properties
  4. Properties of injective norm of fermions and their geometric entanglement
  5. Real Case
  6. Real, fixed $p$, $d \rightarrow \infty$
  7. Real, double scaling
  8. Complex Case
  9. Complex, fixed $p$, $d \rightarrow \infty$
  10. Complex, double scaling
  11. Numerical Simulations
  12. Grassmannian Geometry
  13. Normal coordinates and non-compact Stiefel manifold
  14. Computation of Hessian and other correlations

Key Result

Theorem 1.2

Let $T$ be a skew-symmetric random tensor over the field $\mathbb{K}$, with i.i.d entries up to antisymmetry, with entries distributed as $\mathcal{N}_\mathbb{R}(0,1)$ in the case $\mathbb{K}=\mathbb{R}$ and $\mathcal{N}_\mathbb{C}(0,1)$ in the case $\mathbb{K}=\mathbb{C}$. And let $\ket{\psi_f}:=\f and for a double scaling regime where both $d,p \rightarrow \infty$, with $p =\lfloor \alpha d \rf

Figures (3)

  • Figure 1: Plot of the function $\beta(\alpha)=\frac{3}{4} + \frac{1}{4} \frac{\alpha}{1-\alpha} \log(\alpha)+ \frac{1}{4} \frac{1-\alpha}{\alpha} \log(1-\alpha)+ \frac{1}{2} \log(\alpha(1-\alpha))$ over the interval from 0 to 1. The symmetry of the plot reflects the particle-hole duality of the injective norm of proposition \ref{['prop:particle-hole-duality']}.
  • Figure 2: Comparison of the values of the injective norm for random tensors of order 2, 3 and 4 as a function of the dimension d. The asymptotic analytical upper bound is indicated by the blue line.
  • Figure 3: Comparison between the injective norm of normalized tensors and corresponding ratios.

Theorems & Definitions (46)

  • Remark 1.1
  • Theorem 1.2
  • Definition 2.1: Injective norm of a skew-symmetric tensor
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Proposition 2.5: Geometric entanglement particle-hole duality
  • proof
  • Definition 2.6
  • ...and 36 more