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:1301.4082 · Physical Review A

Using partial transpose and realignment to generate local unitary invariants

Udaysinh T. Bhosale, K. V. Shuddhodan, Arul Lakshminarayan

Published May 14, 2013• 6 citations
arXivHTMLPDF

Abstract

Motivated by link transformations of lattice gauge theory, a method for generating local unitary invariants, especially for a system of qubits, has been pointed out in an earlier work [M. S. Williamson et al., Phys. Rev. A 83, 062308 (2011)]. This paper first points the equivalence of the so constructed transformations to the combined operations of partial transpose and realignment. This allows construction of local unitary invariants of any system, with subsystems of arbitrary dimensions. Some properties of the resulting operators and consequences for pure tripartite higher-dimensional states are briefly discussed.

DOI
10.1103/physreva.87.052311

Related papers

Geometric local invariants and pure three-qubit states

Mark S. Williamson, Marie Ericsson, Markus Johansson +4 more

We explore a geometric approach to generating local $\text{SU}(2)$ and $\text{SL}(2,\mathbb{C})$ invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or ``gauge'' invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters, and the gauge invariant observable associated with a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three-qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way; however, the framework given here is applicable to generating local unitary invariants for mixed states composed of any number of $d$-level quantum systems. We give an operational interpretation of these invariants in terms of observables.

arXiv:1102.4222•Physical Review A•Jun 7, 2011•17 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

On Local Unitary Equivalence of Two and Three-qubit States

Bao-Zhi Sun, Shao-Ming Fei, Zhi‐Xi Wang

We study the local unitary equivalence for two and three-qubit mixed states by investigating the invariants under local unitary transformations. For two-qubit system, we prove that the determination of the local unitary equivalence of 2-qubits states only needs 14 or less invariants for arbitrary two-qubit states. Using the same method, we construct invariants for three-qubit mixed states. We prove that these invariants are sufficient to guarantee the LU equivalence of certain kind of three-qubit states. Also, we make a comparison with earlier works.

arXiv:1707.03934•Scientific Reports•Jul 3, 2017•15 citations

EQUIVALENCE OF TRIPARTITE QUANTUM STATES UNDER LOCAL UNITARY TRANSFORMATIONS

Sergio A Albeverio, LAURA CATTANEO, Shao-Ming Fei +1 more

The equivalence of tripartite pure states under local unitary transformations is investigated. The nonlocal properties for a class of tripartite quantum states in ℂK ⊗ ℂM ⊗ ℂN composite systems are investigated and a complete set of invariants under local unitary transformations for these states is presented. It is shown that two of these states are locally equivalent if and only if all these invariants have the same values.

arXiv:quant-ph/0512246•International Journal of Quantum Information•Dec 1, 2005•32 citations

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

Matrix Realignment Method for Calculating the Partial Transposes and the Partial Traces of Density Matrices

Wei Cheng, Fang Xu

Motivated by Peres's work [A. Peres, Phys. Rev. Lett. 77: 1413-1415 (1996)] and Horodeckis' work [M, Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223: 1-8 (1996)], we develop a very simple method to directly calculate the partial transposes and the partial traces of density matrices in CMotimes CNand C2otimes C2otimes CNcomposite quantum systems, which is based a realignment matrix constructed from the original density matrices.

International Conference on Natural Computation (ICNC)•Jan 1, 2008•1 citation
Motivated by link transformations of lattice gauge theory, a method for generating local unitary invariants, especially for a system of qubits, has been pointed out in an earlier work [M. S. Williamson et al., Phys. Rev. A 83, 062308 (2011)]. This paper first points the equivalence of the so constructed transformations to the combined operations of partial transpose and realignment. This allows construction of local unitary invariants of any system, with subsystems of arbitrary dimensions. Some properties of the resulting operators and consequences for pure tripartite higher-dimensional states are briefly discussed.
We explore a geometric approach to generating local SU(2)\text{SU}(2)SU(2) and SL(2,C)\text{SL}(2,\mathbb{C})SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or ``gauge'' invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters, and the gauge invariant observable associated with a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three-qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way; however, the framework given here is applicable to generating local unitary invariants for mixed states composed of any number of ddd-level quantum systems. We give an operational interpretation of these invariants in terms of observables.
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.
We study the local unitary equivalence for two and three-qubit mixed states by investigating the invariants under local unitary transformations. For two-qubit system, we prove that the determination of the local unitary equivalence of 2-qubits states only needs 14 or less invariants for arbitrary two-qubit states. Using the same method, we construct invariants for three-qubit mixed states. We prove that these invariants are sufficient to guarantee the LU equivalence of certain kind of three-qubit states. Also, we make a comparison with earlier works.
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.