On the coincidence of discrete and continuous definitions of Entanglement of Formation
Abstract
It is proved that every Borel ensemble of pure states on a separable bipartite Hilbert space with finite average entanglement admits a countable ensemble with the same barycenter and arbitrarily small additional average entanglement. Consequently, the discrete and continuous versions of the Entanglement of Formation coincide on the whole space of states of infinite-dimensional bipartite quantum system (with infinite values allowed). The main idea of the proof (realized with the AI-assistance) is the lifting a Borel ensemble of pure bipartite states to a Borel probability measure on the unit sphere in the space of Hilbert--Schmidt operators followed by dividing the support of this measure into appropriate cells and replacing each cell by its conditional barycenter. The resulting countable ensemble is obtained by a recursive construction.
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BibTeX
@misc{ma2026coincidence,
title = {{On the coincidence of discrete and continuous definitions of Entanglement of Formation}},
author = {Haobo Ma and Maksim E. Shirokov and Wenlin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18816},
eprint = {2610.18816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18816},
}
RIS
TY - UNPB
TI - On the coincidence of discrete and continuous definitions of Entanglement of Formation
AU - Haobo Ma
AU - Maksim E. Shirokov
AU - Wenlin Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.18816
UR - https://arxiv.org/abs/2610.18816
AB - It is proved that every Borel ensemble of pure states on a separable bipartite Hilbert space with finite average entanglement admits a countable ensemble with the same barycenter and arbitrarily small additional average entanglement. Consequently, the discrete and continuous versions of the Entanglement of Formation coincide on the whole space of states of infinite-dimensional bipartite quantum system (with infinite values allowed). The main idea of the proof (realized with the AI-assistance) is the lifting a Borel ensemble of pure bipartite states to a Borel probability measure on the unit sphere in the space of Hilbert--Schmidt operators followed by dividing the support of this measure into appropriate cells and replacing each cell by its conditional barycenter. The resulting countable ensemble is obtained by a recursive construction.
ER -