Complete Classification of Reversible Entanglement Transformations of Two Qutrits
Abstract
Suppose Alice and Bob can transform one shared mixed state into another by local operations and classical communication, and can also recover the first state from the second. What can change in such a conversion? We give a complete classification of reversible conversions between two-qutrit states. We consider exact deterministic LOCC transformations of a single copy, allowing finite protocols and the countable protocols specified here. Local flags are known to allow a separable part to be replaced by local preparation while an entangled block is retained. We prove that such a decomposition is necessary for two mutually convertible entangled states that are not locally unitarily equivalent. The two parts are distinguishable by local measurements. The entangled blocks have the same weight and are locally unitarily equivalent after normalization. Each entangled state has a unique smallest retained block. Two entangled states are reversibly convertible if and only if these unnormalized blocks are locally unitarily equivalent. For a state with negative partial transpose, the smallest block can be found using the negative eigenspace of its partial transpose and at most three matrix zero tests.
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BibTeX
@misc{song2026complete,
title = {{Complete Classification of Reversible Entanglement Transformations of Two Qutrits}},
author = {Wei Song and Xiao-Lan Zong and Ming Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05234},
eprint = {2610.05234},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05234},
}
RIS
TY - UNPB
TI - Complete Classification of Reversible Entanglement Transformations of Two Qutrits
AU - Wei Song
AU - Xiao-Lan Zong
AU - Ming Yang
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05234
UR - https://arxiv.org/abs/2610.05234
AB - Suppose Alice and Bob can transform one shared mixed state into another by local operations and classical communication, and can also recover the first state from the second. What can change in such a conversion? We give a complete classification of reversible conversions between two-qutrit states. We consider exact deterministic LOCC transformations of a single copy, allowing finite protocols and the countable protocols specified here. Local flags are known to allow a separable part to be replaced by local preparation while an entangled block is retained. We prove that such a decomposition is necessary for two mutually convertible entangled states that are not locally unitarily equivalent. The two parts are distinguishable by local measurements. The entangled blocks have the same weight and are locally unitarily equivalent after normalization. Each entangled state has a unique smallest retained block. Two entangled states are reversibly convertible if and only if these unnormalized blocks are locally unitarily equivalent. For a state with negative partial transpose, the smallest block can be found using the negative eigenspace of its partial transpose and at most three matrix zero tests.
ER -