Entangled Subspaces through Algebraic Geometry

Masoud Gharahi; Stefano Mancini

doi:10.22331/q-2025-12-15-1947

Entangled Subspaces through Algebraic Geometry

Masoud Gharahi^1,2 and Stefano Mancini^3,4

¹Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, 30-348 Kraków, Poland
²Department of Physics, University of Trieste, Strada Costiera 11, 34151 Trieste, Italy
³School of Science and Technology, University of Camerino, 62032 Camerino, Italy
⁴INFN Sezione di Perugia, 06123 Perugia, Italy

Published:	2025-12-15, volume 9, page 1947
Editor:	Carlo Beenakker
Eprint:	arXiv:2504.11525v2
Doi:	https://doi.org/10.22331/q-2025-12-15-1947
Citation:	Quantum 9, 1947 (2025).

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Abstract

We propose an algebraic geometry-inspired approach for constructing entangled subspaces within the Hilbert space of a multipartite quantum system. Specifically, our method employs a modified Veronese embedding, restricted to the conic, to define subspaces within the symmetric part of the Hilbert space. By utilizing this technique, we construct the minimal-dimensional, non-orthogonal yet Unextendible Product Basis (nUPB), enabling the decomposition of the multipartite Hilbert space into a two-dimensional subspace, complemented by a Genuinely Entangled Subspace (GES) and a maximal-dimensional Completely Entangled Subspace (CES). In multiqudit systems, we determine the maximum achievable dimension of a symmetric GES and demonstrate its realization through this construction. Furthermore, we systematically investigate the transition from the conventional Veronese embedding to the modified one by imposing various constraints on the affine coordinates, which, in turn, increases the CES dimension while reducing that of the GES.

► BibTeX data

@article{Gharahi2025entangledsubspaces,
  doi = {10.22331/q-2025-12-15-1947},
  url = {https://doi.org/10.22331/q-2025-12-15-1947},
  title = {Entangled {S}ubspaces through {A}lgebraic {G}eometry},
  author = {Gharahi, Masoud and Mancini, Stefano},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1947},
  month = dec,
  year = {2025}
}

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[1] Guillem Müller-Rigat, "Charting separability and entanglement of diagonal Dicke states", Quantum Views 10, 87 (2026).

[2] Seiseki Akibue, Jisho Miyazaki, and Hiroyuki Osaka, "Optimizing Entanglement Manipulation via Algebraic-Geometric Decompositions and Semidefinite Programming Hierarchies", arXiv:2501.17394, (2025).

[3] Rathika V and Rammohan A, "Quantum information meets algebraic geometry: A python framework for CES and UPBs", European Physical Journal Web of Conferences 360, 01006 (2026).

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