We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
43/2024
Local unitary equivalence of arbitrary dimensional multipartite mixed quantum states
Qing Zhou, Yi-Zheng Zhen, Xin-Yu Xu, Shuai Zhao, Wen-Li Yang, Shao-Ming Fei, Li Li, Nai-Le Liu and Kai Chen
Abstract
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. In this work, we address the question of verifying whether two quantum states are local unitary equivalent or not. This has been partially answered in multipartite pure states, while it is device-dependent and not operational. Unfortunately, for mixed states, the verification of local unitatry equivalence is still a difficult problem, especially for degenerate density matrices. Here, based on the coefficient matrices in generalized Bloch representations, we find operational necessary conditons for the local unitary equivalence of two arbitrary dimensional bipartite quantum states for given density matrices. Furthermore, we extend our method to three-qudit case, necessary conditions for local unitary equivalence of two three-qudit quantum states are also presented and this result can also be extended to arbitrary dimensional multipartite case. Compared with the existing results, our method can present simple invariants of degenerate density matrices under local unitary transformations and is an effective way to construct necessary conditions for local unitary equivalence of multipartite case. In the semi-device-independent scenario, we present a necessary and sufficient criterion for local unitary equivalence of two special pure entangled states based on self-testing scheme without tomography, which is low cost and robust.
