bibkey: denkerimaiguhne2025chiral
authors: Sophia Denker; Satoya Imai; Otfried Gühne
year: 2025
title: "Chiral Symmetries and Multiparticle Entanglement"
doi: null
url: https://arxiv.org/abs/2506.15609v1
claim: "The paper constructs highly entangled three-qudit subspaces from chiral and flip-conjugate symmetries, shows that all states of the flip-conjugate symmetric subspace are local-unitary equivalent, proves the bound G >= 1 - d/(d^2-1) for their geometric measure, and conjectures from numerics for d <= 12 that G = 1 - d^2/((d^2-1)(d+1)) for every state of the subspace."
strata_touched:
  - D5/S3/Quantum/Entanglement/GeometricMeasure/FlipConjugateGeometricMeasure
license: citation-only
triage: anchor

Chiral Symmetries and Multiparticle Entanglement

S. Denker, S. Imai and O. Gühne, arXiv:2506.15609v1 (2025-06-18). Quotations are from the v1 TeX source.

The geometric measure:

The quantity $G(|\psi ⟩)$ is known as the \textit{geometric measure of entanglement} and can be directly extended to the multiparticle case

particles defined as $G(|\psi ⟩)=1-{\max }_{|abc⟩}|⟨abc⟩\psi {|}^2$.

The flip-conjugate symmetric subspace:

\ket{\phi_n} &= \frac{1}{\sqrt{d+1}} \left (z_d^*\sum_{i=0}^{d-1}\ket{ini}+z_d\sum_{i=0}^{d-1}\ket{iin} \right)

with $z_d={}^1/{}_2(1+\mathbf{i}\sqrt{(d+1)/(d-1)})$. Analogously, the complex conjugate vectors span a flip-conjugate symmetric subspace as well (denoted ${\mathcal{H}}_{\bar{\mathcal{I}}}$).

The conjecture:

\textbf{Conjecture:}\textit{ The geometric measure of entanglement for any state in the flip-conjugate symmetric subspace ${\mathcal{H}}_{\mathcal{I}}$ (and respectively ${\mathcal{H}}_{\bar{\mathcal{I}}}$) is given by G(\ket{\psi}) = 1-\frac{d^2}{(d^2-1)(d+1)}, where $d$ denotes the local dimension, which coincides with the dimension $\dim ({\mathcal{H}}_{\mathcal{I}})$ of the subspace. }

Verified locator

Source: https://arxiv.org/abs/2506.15609v1

• URL: https://arxiv.org/abs/2506.15609v1 (source paper-plan.tex, md5 2340b0f14e9e557cbd8cae8020fe909a):
  • the geometric measure, l. 196–206 and l. 262–265;
  • the subspace, l. 625–638 and l. 1426–1431;
  • the local-unitary proposition, l. 1435–1449;
  • the conjecture, l. 1451–1460.