Supporting functionals and singular boundary geometry of two-qubit entanglement of formation
Abstract
We give a criterion for the existence of a global supporting affine functional for the entanglement of formation at an entangled two-qubit state. It exists if and only if the spin-flip bilinear form is nondegenerate on the support of the state, or equivalently, if the last concurrence root associated with the support is nonzero. In this case the entanglement of formation satisfies a two-sided Lipschitz bound relative to the reference state. In the degenerate case the state has rank two or three, and its projective support is tangent to the product-state quadric. By suitably choosing a perturbation outside the original support space, the decrease in the entanglement of formation is, to leading order, proportional to the square root of the mixing parameter. We give an explicit expression for the leading coefficient. Such a square-root decrease violates the one-sided Lipschitz bound, and therefore no globally supported affine functional exists at this state.
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BibTeX
@misc{song2026supporting,
title = {{Supporting functionals and singular boundary geometry of two-qubit entanglement of formation}},
author = {Wei Song and Xiao-Lan Zong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.31097},
eprint = {2609.31097},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31097},
}
RIS
TY - UNPB
TI - Supporting functionals and singular boundary geometry of two-qubit entanglement of formation
AU - Wei Song
AU - Xiao-Lan Zong
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2609.31097
UR - https://arxiv.org/abs/2609.31097
AB - We give a criterion for the existence of a global supporting affine functional for the entanglement of formation at an entangled two-qubit state. It exists if and only if the spin-flip bilinear form is nondegenerate on the support of the state, or equivalently, if the last concurrence root associated with the support is nonzero. In this case the entanglement of formation satisfies a two-sided Lipschitz bound relative to the reference state. In the degenerate case the state has rank two or three, and its projective support is tangent to the product-state quadric. By suitably choosing a perturbation outside the original support space, the decrease in the entanglement of formation is, to leading order, proportional to the square root of the mixing parameter. We give an explicit expression for the leading coefficient. Such a square-root decrease violates the one-sided Lipschitz bound, and therefore no globally supported affine functional exists at this state.
ER -