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Entangled Subspaces through Algebraic Geometry

Masoud Gharahi, Stefano Mancini

TL;DR

This work develops an algebraic-geometry framework for constructing entangled subspaces in multipartite quantum systems by embedding non-orthogonal UPBs (nUPBs) into Veronese- and Segre-Veronese-type varieties. By enforcing maximal coordinate constraints (via a modified embedding), the authors minimize the nUPB size to maximize the CES, while creating explicit symmetric GESs in the symmetric subspace, especially for multiqubit and multiqudit scenarios. They provide concrete decompositions of the full Hilbert space into a two-dimensional product subspace, a GES, and a maximal CES, with formulas for dimensions in both uniform-qudit and heterogeneous multipartite settings, and show how to interpolate between embeddings to tune CES and GES dimensions. The approach yields explicit bases for GESs, demonstrates the extraction of multiple orthogonal GESs from CES, and points toward deeper connections with algebraic geometry and representation theory, offering a structured path to analyzing multipartite entanglement geometry and its applications.

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.

Entangled Subspaces through Algebraic Geometry

TL;DR

This work develops an algebraic-geometry framework for constructing entangled subspaces in multipartite quantum systems by embedding non-orthogonal UPBs (nUPBs) into Veronese- and Segre-Veronese-type varieties. By enforcing maximal coordinate constraints (via a modified embedding), the authors minimize the nUPB size to maximize the CES, while creating explicit symmetric GESs in the symmetric subspace, especially for multiqubit and multiqudit scenarios. They provide concrete decompositions of the full Hilbert space into a two-dimensional product subspace, a GES, and a maximal CES, with formulas for dimensions in both uniform-qudit and heterogeneous multipartite settings, and show how to interpolate between embeddings to tune CES and GES dimensions. The approach yields explicit bases for GESs, demonstrates the extraction of multiple orthogonal GESs from CES, and points toward deeper connections with algebraic geometry and representation theory, offering a structured path to analyzing multipartite entanglement geometry and its applications.

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.

Paper Structure

This paper contains 12 sections, 9 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiqubit case
  4. Multiqudit case
  5. Veronese embedding
  6. Modified Segre-Veronese embedding
  7. From Veronese to modified Segre-Veronese embedding
  8. Multipartite case
  9. Conclusions and outlook
  10. Normalization factor
  11. Alternative algorithm
  12. Decomposition of three-qubit Hilbert space into GESs

Key Result

Theorem 1

In a Hilbert space $\mathcal{H}_{\delta}=\otimes_{j=1}^n\mathbbm{C}^{d_j}$, the maximum dimension of a CES is given by

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9