bibkey: ballesterosetal2025qsymmetric
authors: Angel Ballesteros; Ivan Gutierrez-Sagredo; Jose de Ramón; J. Javier Relancio
year: 2025
title: "Deformations of the symmetric subspace of qubit chains"
doi: 10.1088/1751-8121/ae05da
url: https://arxiv.org/abs/2503.23554v3
claim: "The paper defines the q-symmetric subspace of an N-qubit chain through the coproduct of the quantum group U_q(su(2)), constructs q-Dicke states, and in its applications section conjectures that, as it asserts for two qubits, the closest product state to an arbitrary q-Dicke state in the geometric measure of entanglement is q-symmetric."
strata_touched:
  - D5/S3/Quantum/Entanglement/GeometricMeasure/QDickeClosestProductRefutation
license: citation-only
triage: anchor

Deformations of the symmetric subspace of qubit chains

A. Ballesteros, I. Gutierrez-Sagredo, J. de Ramón and J. J. Relancio, arXiv:2503.23554v3 (11 March 2026; first version 30 March 2025); J. Phys. A 58(40) (2025), 405301. Quotations are from the v3 TeX source, with the macros \up and \down written as $\uparrow$ and $\downarrow$.

The q-number:

$[x{]}_q=\frac{q^{\frac{x}{2}}-q^{-\frac{x}{2}}}{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}.$

The q-symmetric subspace:

Similar to the case of the symmetric group, we define the $q$-symmetric subspace ${\text{Sym}}_q\mathcal{H}$ as the subspace carrying the trivial representation of the Hecke algebra.

The q-Dicke states for three qubits (Example ex:qDickeN3), with $|G⟩={|D_3^0⟩}_q=|\downarrow \downarrow \downarrow ⟩$:

${|D_3^1⟩}_q=\frac{1}{\sqrt{[3{]}_q}}\left(q^{1/2}|\downarrow \downarrow \uparrow ⟩+|\downarrow \uparrow \downarrow ⟩+q^{-1/2}|\uparrow \downarrow \downarrow ⟩\right)$, ${|D_3^2⟩}_q=\frac{1}{\sqrt{[3{]}_q}}\left(q^{1/2}|\downarrow \uparrow \uparrow ⟩+|\uparrow \downarrow \uparrow ⟩+q^{-1/2}|\uparrow \uparrow \downarrow ⟩\right)$, ${|D_3^3⟩}_q=|\uparrow \uparrow \uparrow ⟩$, and thus ${\mathrm{Sym}}_q\mathcal{H}=\text{span }\{{|D_3^0⟩}_q,{|D_3^1⟩}_q,{|D_3^2⟩}_q,{|D_3^3⟩}_q\}$.

The geometric measure of entanglement:

$G[|\Psi ⟩]=\underset{\{{\varphi }_i\}}{\max }|⟨\Psi |\{{\varphi }_i\}⟩|$ where $|\{{\varphi }_i\}⟩=|{\varphi }_1⟩\dots |{\varphi }_N⟩$ are product states.

The conjecture:

Therefore, the closest separable state to a $q$-symmetric state for two qubits is also $q$-symmetric. If we conjecture that a similar relation holds for arbitrary $q$-Dicke states, it would mean that we can construct whole new families of states whose multipartite entanglement has an analytic expression.

Verified locator

Source: https://arxiv.org/abs/2503.23554v3

• URL: https://arxiv.org/abs/2503.23554v3 (the e-print is a tar archive; member newsubmission__1_.tex, md5 18fa3b609df74e7f0e866cc4ee871711, 1495 lines):
  • the q-number, l. 469;
  • the q-symmetric subspace, l. 532–535;
  • the three-qubit q-Dicke states and their span, l. 579–586;
  • the geometric measure of entanglement, l. 1232–1243;
  • the two-qubit discussion and the conjecture, l. 1249–1258.
• DOI: 10.1088/1751-8121/ae05da (https://doi.org/10.1088/1751-8121/ae05da; Crossref lists Journal of Physics A: Mathematical and Theoretical, volume 58, issue 40, published 30 September 2025).