bibkey: rouleux2019multimodeentanglement
authors: Michel Rouleux
year: 2019
title: "Multimode entanglement for fermions"
doi: 10.48550/arXiv.1910.04790
url: https://arxiv.org/abs/1910.04790v1
claim: "The affine determinant det(x1 − x0, …, xd − x0) of d+1 points of C^d is an antisymmetric, non-multilinear form of degree d in its d+1 variables. The paper conjectures that it is the only non-degenerate antisymmetric tensor of degree d on C^d depending on d+1 variables, and that no nontrivial antisymmetric tensor of degree d depends on more than d+1 variables."
strata_touched:
  - D5/S3/Quantum/Fermionic/AffineDeterminantAntisymmetricForms
license: citation-only
triage: anchor

Rouleux, Multimode entanglement for fermions

M. Rouleux, arXiv:1910.04790v1 [quant-ph] (10 October 2019), contribution to the conference ISQS26 (Prague, July 2019); HAL hal-02323515. Quotations are from the v1 TeX source, with the source’s macros \vec and {\bf C} kept as written.

Section 3 treats antisymmetric tensors $x_1\wedge \cdots \wedge x_p$ in ${\Lambda }^p\mathcal{H}\subset {⨂}^p\mathcal{H}$ for $\mathcal{H}={\mathbb{C}}^d$, “degree being understood as degree of homogeneity”, and antisymmetric forms that are not multilinear, such as the 3-fermion $a\wedge b+b\wedge c+c\wedge a$. Definition 3:

The affine determinant of $x_0,x_1,\cdots ,x_d$ in ${\mathbf{C}}^d$ is the antisymmetric 2-form in variables $x_0,x_1\cdots ,x_d$ $$\tilde{\omega }(x_0,x_1,\cdots ,x_d)=\det (\overrightarrow{x_0{x}_1},\overrightarrow{x_0{x}_2},\cdots ,\overrightarrow{x_0{x}_d})$$

Such forms arise by antisymmetrizing monomial generators, and $\omega$ is called non-degenerate when its vanishing for every $x_0$ forces $x_1,\dots ,x_d$ to lie in an affine subspace of dimension $d-2$. The section states:

We conjecture that: (1) the affine determinant is the only non degenerate antisymmetric tensor $\omega$ on ${\mathbf{C}}^d$ of degree $d$, depending on $d+1$ variables, and: (2) there is no non trivial antisymmetric tensor on ${\mathbf{C}}^d$ of degree $d$, depending on more than $d+1$ variables.

Verified locator

Source: https://arxiv.org/abs/1910.04790v1

• DOI: https://doi.org/10.48550/arXiv.1910.04790 (arXiv-issued DataCite DOI; the arXiv record lists only version 1 and no journal reference).
• URL: https://arxiv.org/abs/1910.04790v1. The e-print member 3fermiProc.tex has 425 lines and md5 8a11de70d5f9bebe5e58284b95a44d75. Locations:
  • antisymmetric tensors and the degree convention, l. 233–248;
  • Definition 3 (the affine determinant), l. 251–253; generators, l. 255–260;
  • non-degeneracy, l. 262–263; the conjecture, l. 264–269.