Multinegativity and single-letter formulas for asymptotic entanglement
Abstract
The study of entanglement inevitably leads to the concept of regularization, where measures are evaluated on infinitely many copies of a state. Single-letter formulas try to make these asymptotic entanglement quantities accessible through a calculation on one copy of a state, but are rarely available. We introduce a decreasing hierarchy of computable upper bounds on the asymptotic relative entropy of entanglement with respect to positive-partial-transpose (PPT) states. The regularization of every fixed hierarchy level equals this asymptotic quantity. Additivity at any level therefore yields a single-letter formula, even when the usual one-copy relative entropy is nonadditive. We establish such additivity at the first nontrivial level for two broad multiparameter families, both containing all Werner states. The upper-bound construction also extends to sandwiched Renyi divergences. The hierarchy motivates k-multinegative states, a generalization of binegative states and the associated k-multinegativity, which gives explicit upper bounds on both the asymptotic relative entropy and exact PPT entanglement cost. We construct states which are k-multinegative at arbitrarily large depths k that separate consecutive levels of the previously introduced entanglement-cost hierarchy, disproving its conjectured finite collapse. These results provide state-dependent single-letter formulas while identifying a limitation of universal finite-level characterizations.
Related
- Static classical-quantum-entanglement trade-offs: an entropic converse and single-letter characterizations
- Entanglement Catalysis Beyond the Asymptotic Rate
- Asymptotic Entanglement Hiding under Stabilizer Restrictions
- Asymptotic Entanglement across Dynamical Regimes of a Dissipative Bosonic Dimer
- Long-distance device-independent quantum key distribution using single-photon entanglement
BibTeX
@misc{brinster2026multinegativity,
title = {{Multinegativity and single-letter formulas for asymptotic entanglement}},
author = {Raphael Brinster and Tulja Varun Kondra and Hermann Kampermann and Dagmar Bruß and Nikolai Wyderka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20698},
eprint = {2609.20698},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20698},
}
RIS
TY - UNPB
TI - Multinegativity and single-letter formulas for asymptotic entanglement
AU - Raphael Brinster
AU - Tulja Varun Kondra
AU - Hermann Kampermann
AU - Dagmar Bruß
AU - Nikolai Wyderka
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20698
UR - https://arxiv.org/abs/2609.20698
AB - The study of entanglement inevitably leads to the concept of regularization, where measures are evaluated on infinitely many copies of a state. Single-letter formulas try to make these asymptotic entanglement quantities accessible through a calculation on one copy of a state, but are rarely available. We introduce a decreasing hierarchy of computable upper bounds on the asymptotic relative entropy of entanglement with respect to positive-partial-transpose (PPT) states. The regularization of every fixed hierarchy level equals this asymptotic quantity. Additivity at any level therefore yields a single-letter formula, even when the usual one-copy relative entropy is nonadditive. We establish such additivity at the first nontrivial level for two broad multiparameter families, both containing all Werner states. The upper-bound construction also extends to sandwiched Renyi divergences. The hierarchy motivates k-multinegative states, a generalization of binegative states and the associated k-multinegativity, which gives explicit upper bounds on both the asymptotic relative entropy and exact PPT entanglement cost. We construct states which are k-multinegative at arbitrarily large depths k that separate consecutive levels of the previously introduced entanglement-cost hierarchy, disproving its conjectured finite collapse. These results provide state-dependent single-letter formulas while identifying a limitation of universal finite-level characterizations.
ER -