Roto-Reflection Geometry of Pure Two-Qubit Entanglement
Abstract
Pure two-qubit entanglement is usually characterized by scalar quantities such as concurrence. Here we show that it also has a natural geometric form. In the Pauli correlation tensor, maximally entangled states appear as improper orthogonal maps between two local Bloch spheres. These maps are roto-reflections. For partially entangled pure states, the same roto-reflection geometry is recovered after separating the contraction associated with concurrence. We call the corresponding geometric object the Entanglement Roto-Reflection Plane (ERRP). It organizes the maximally correlated directions of the two-qubit state and provides a covariant geometric complement to the scalar magnitude of entanglement. We further illustrate the usefulness of this geometry through partial transposition, where it provides a direct construction of the maximally entangled negative eigenvector and a geometric interpretation of negativity.
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BibTeX
@misc{filatov2026rotoreflection,
title = {{Roto-Reflection Geometry of Pure Two-Qubit Entanglement}},
author = {Stanislav Filatov and Marcis Auzinsh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.12637},
eprint = {2606.12637},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.12637},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Roto-Reflection Geometry of Pure Two-Qubit Entanglement
AU - Stanislav Filatov
AU - Marcis Auzinsh
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2606.12637
UR - https://arxiv.org/abs/2606.12637
AB - Pure two-qubit entanglement is usually characterized by scalar quantities such as concurrence. Here we show that it also has a natural geometric form. In the Pauli correlation tensor, maximally entangled states appear as improper orthogonal maps between two local Bloch spheres. These maps are roto-reflections. For partially entangled pure states, the same roto-reflection geometry is recovered after separating the contraction associated with concurrence. We call the corresponding geometric object the Entanglement Roto-Reflection Plane (ERRP). It organizes the maximally correlated directions of the two-qubit state and provides a covariant geometric complement to the scalar magnitude of entanglement. We further illustrate the usefulness of this geometry through partial transposition, where it provides a direct construction of the maximally entangled negative eigenvector and a geometric interpretation of negativity.
ER -