Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0602136 · Communications in Theoretical Physics

Matrix Tensor Product Approach to the Equivalence of Multipartite States under Local Unitary Transformations

Xiu-Hong Gao, S Alberverio, Shao-Ming Fei, Zhi‐Xi Wang

Published Feb 1, 2006• 1 citation
arXivHTMLPDF

Abstract

The equivalence of multipartite quantum mixed states under local unitary transformations is studied. A criterion for the equivalence of non-degenerate mixed multipartite quantum states under local unitary transformations is presented.
DOI
10.1088/0253-6102/45/2/015

Related papers

Criterion of local unitary equivalence for multipartite states

Ting-Gui Zhang, Ming‐Jing Zhao, Ming Li +2 more

We study the local unitary equivalence of arbitrary dimensional multipartite quantum mixed states. We present a necessary and sufficient criterion of the local unitary equivalence for general multipartite states based on matrix realignment. The criterion is shown to be operational, even for particularly degenerated states, by detailed examples. In addition, explicit expressions of the local unitary operators are constructed for locally equivalent states. Complementary to the criterion, an alternative approach based on the partial transposition of matrices is also given, which makes the criterion more effective in dealing with generally degenerated mixed states.

arXiv:1310.2042•Physical Review A•Oct 3, 2013•23 citations

Multipartite states under local unitary transformations

Sergio A Albeverio, Cattaneo, L, Fei, S M +1 more

The equivalence problem under local unitary transformation for $n$--partite pure states is reduced to the one for $(n-1)$--partite mixed states. In particular, a tripartite system $\\mathcal{H}_A\\otimes\\mathcal{H}_B\\otimes\\mathcal{H}_C$, where $\\mathcal{H}_j$ is a finite dimensional complex Hilbert space for $j=A,B,C$, is considered and a set of invariants under local transformations is introduced, which is complete for the set of states whose partial trace with respect to $\\mathcal{H}_A$ belongs to the class of generic mixed states.

•Jan 1, 2005•17 citations

Local Unitary Invariants for Multipartite Quantum Systems

Jing Wang, Ming Li, Shao-Ming Fei +1 more

We present an approach of constructing invariants under local unitary transformations for multipartite quantum systems. The invariants constructed in this way can be complement to that in [Science 340 (2013) 1205–1208]. Detailed examples are given to compute such invariant in detail. It is shown that these invariants can be used to detect the local unitary equivalence of degenerated quantum states.

arXiv:1410.6254•Communications in Theoretical Physics•Nov 1, 2014•1 citation

Local Unitary Invariants for Multipartite Quantum Systems

Wang Jing, Li Zhi ming, 费少明 +1 more

We present an approach of constructing invariants under local unitary transformations for multipartite quantum systems. The invariants constructed in this way can be complement to that in [Science 340(2013) 1205-1208].Detailed examples are given to compute such invariant in detail. It is shown that these invariants can be used to detect the local unitary equivalence of degenerated quantum states.

理论物理通讯:英文版•Jan 1, 2014•0 citations

Characterizing local unitary equivalence of multipartite states by local unitary Bargmann invariants

Yi Shen, Lin Chen, Shao-Ming Fei

Local unitary (LU) equivalent states are crucial for understanding the structure of quantum entanglement. We investigate the LU equivalence based on the LU Bargmann invariants (LUBIs). We consider for which states the LUBIs are sufficient to ensure the LU equivalence. We focus on a class of multipartite states with complete local commutativity, called CLC states. We first show that for multipartite CLC states, all the LUBIs are sufficient to ensure the local permutation equivalence, if their single-party marginals are diagonal and non-degenerate. By virtue of the fact that the CLC property remains unchanged under local operations, we verify that the LUBIs are also sufficient to ensure the LU equivalence of multipartite CLC states whose single-party marginals are non-degenerate. Furthermore, when the single-party marginals are degenerate, we propose concrete examples of two-qudit and three-qubit systems, showing that there are states which share all the LUBIs but are not LU equivalent.

quant-pharXiv:2609.22902•Sep 22, 2026•0 citations

Local Unitary Equivalence of Quantum States Based on the Tensor Decompositions of Unitary Matrices

Jing Wang, Xiaoqi Liu, Li Xu +3 more

Since two quantum states that are local unitary (LU) equivalent have the same amount of entanglement, it is meaningful to find a practical method to determine the LU equivalence of given quantum states. In this paper, we present a valid process to find the unitary tensor product decomposition for an arbitrary unitary matrix. Then, by using this process, the conditions for determining the local unitary equivalence of quantum states are obtained. A numerical verification is carried out, which shows the practicability of our protocol. We also present a property of LU invariants by using the universality of quantum gates which can be used to construct the complete set of LU invariants.

Entropy•Jul 29, 2023•4 citations
We study the local unitary equivalence of arbitrary dimensional multipartite quantum mixed states. We present a necessary and sufficient criterion of the local unitary equivalence for general multipartite states based on matrix realignment. The criterion is shown to be operational, even for particularly degenerated states, by detailed examples. In addition, explicit expressions of the local unitary operators are constructed for locally equivalent states. Complementary to the criterion, an alternative approach based on the partial transposition of matrices is also given, which makes the criterion more effective in dealing with generally degenerated mixed states.
The equivalence problem under local unitary transformation for nnn--partite pure states is reduced to the one for (n−1)(n-1)(n−1)--partite mixed states. In particular, a tripartite system mathcalHAotimesmathcalHBotimesmathcalHC\\mathcal{H}_A\\otimes\\mathcal{H}_B\\otimes\\mathcal{H}_CmathcalHA​otimesmathcalHB​otimesmathcalHC​, where mathcalHj\\mathcal{H}_jmathcalHj​ is a finite dimensional complex Hilbert space for j=A,B,Cj=A,B,Cj=A,B,C, is considered and a set of invariants under local transformations is introduced, which is complete for the set of states whose partial trace with respect to mathcalHA\\mathcal{H}_AmathcalHA​ belongs to the class of generic mixed states.
Local unitary (LU) equivalent states are crucial for understanding the structure of quantum entanglement. We investigate the LU equivalence based on the LU Bargmann invariants (LUBIs). We consider for which states the LUBIs are sufficient to ensure the LU equivalence. We focus on a class of multipartite states with complete local commutativity, called CLC states. We first show that for multipartite CLC states, all the LUBIs are sufficient to ensure the local permutation equivalence, if their single-party marginals are diagonal and non-degenerate. By virtue of the fact that the CLC property remains unchanged under local operations, we verify that the LUBIs are also sufficient to ensure the LU equivalence of multipartite CLC states whose single-party marginals are non-degenerate. Furthermore, when the single-party marginals are degenerate, we propose concrete examples of two-qudit and three-qubit systems, showing that there are states which share all the LUBIs but are not LU equivalent.
Since two quantum states that are local unitary (LU) equivalent have the same amount of entanglement, it is meaningful to find a practical method to determine the LU equivalence of given quantum states. In this paper, we present a valid process to find the unitary tensor product decomposition for an arbitrary unitary matrix. Then, by using this process, the conditions for determining the local unitary equivalence of quantum states are obtained. A numerical verification is carried out, which shows the practicability of our protocol. We also present a property of LU invariants by using the universality of quantum gates which can be used to construct the complete set of LU invariants.