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This paper offers a geometric reframing of pure two-qubit entanglement, mapping maximally entangled states to improper orthogonal maps (roto-reflections) between two Bloch spheres. Its practical value is mainly theoretical: it gives a covariant complement to concurrence and a direct construction of the maximally entangled negative eigenvector under partial transposition. Readers working on quantum information theory or entanglement diagnostics may find the framework useful, but it is foundational work with no immediate hardware or algorithmic implications.
Roto-Reflection Geometry of Pure Two-Qubit Entanglement
arXiv:2606.12637v2 Announce Type: replace 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.
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.
arXiv:2606.12637 [quant-ph]
(or arXiv:2606.12637v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2606.12637 Focus to learn more arXiv-issued DOI via DataCite
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