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Peer-reviewedCC BY 4.0Mathematics

Extensions of Completely Entangled Subspaces in C2⊗CN

  • Yunlang Li1
  • Ziwei Chen2
  1. 1Department of Mathematics, Imperial College London, London SW7 2AZ, UK
  2. 2Department of Electronics, Beijing Jiaotong University, Beijing 100044, China
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Abstract

In the Kronecker product C2⊗CN, completely entangled subspaces (CESs) are subspaces containing no nonzero product vectors. In this paper, we study the extension of a given M-dimensional CES to a higher-dimensional CES. We first prove an existence result using the geometry of the Segre variety and an incidence-variety dimension-counting argument. In particular, an M-dimensional CES can be extended to an (M+1)-dimensional CES whenever M<N−1. We then introduce a constructive method based on matrix representations and invertible local transformations. We show that if the corresponding matrix Ψ satisfies rank(Ψ)≤N−2, an explicit extension can be constructed. In particular, the condition 2M≤N−2 is sufficient for this construction. Finally, we give a concrete example in C2⊗C8, where a three-dimensional CES is extended to a four-dimensional CES.

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Cite

BibTeXRIS
BibTeX
@article{li2026extensions,
  title = {{Extensions of Completely Entangled Subspaces in C2⊗CN}},
  author = {Yunlang Li and Ziwei Chen},
  journal = {Mathematics},
  year = {2026},
  month = nov,
  doi = {10.3390/math14214050},
  url = {https://www.mdpi.com/2227-7390/14/21/4050},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Extensions of Completely Entangled Subspaces in C2⊗CN
AU  - Yunlang Li
AU  - Ziwei Chen
PY  - 2026
DA  - 2026/11/05
JO  - Mathematics
DO  - 10.3390/math14214050
UR  - https://www.mdpi.com/2227-7390/14/21/4050
AB  - In the Kronecker product C2⊗CN, completely entangled subspaces (CESs) are subspaces containing no nonzero product vectors. In this paper, we study the extension of a given M-dimensional CES to a higher-dimensional CES. We first prove an existence result using the geometry of the Segre variety and an incidence-variety dimension-counting argument. In particular, an M-dimensional CES can be extended to an (M+1)-dimensional CES whenever M<N−1. We then introduce a constructive method based on matrix representations and invertible local transformations. We show that if the corresponding matrix Ψ satisfies rank(Ψ)≤N−2, an explicit extension can be constructed. In particular, the condition 2M≤N−2 is sufficient for this construction. Finally, we give a concrete example in C2⊗C8, where a three-dimensional CES is extended to a four-dimensional CES.
SN  - 2227-7390
ER  -