In [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class.In [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class.

arXiv:2201.07377Jan 19, 20220 citations

Local unitary equivalence of the SLOCC class of three qubits

Dafa Li, Yu Jing Guo

Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.

Journal of Physics A Mathematical and TheoreticalAug 19, 20204 citations

Quantification of Entanglement in Three-Qubit Systems

Zikra Aram

This study presents an analytical investigation of entanglement quantification in three-qubit pure states through the Acín canonical representation, which serves as a generalization of the two-qubit Schmidt decomposition. Driven by the intricacies of multipartite entanglement and the shortcomings of current measures, we employ a global concurrence measure derived from the generalized concurrence of various bipartitions within the system. The characteristics of this measure are explored both analytically and numerically across a range of SLOCC entanglement classes, such as product, biseparable, GHZ-type, and W-type states. When compared to the three-tangle and tripartite negativity, our findings indicate that this measure yields complementary insights into multipartite correlations. The outcomes underscore the significance of this measure in elucidating the structure and distribution of entanglement within three-qubit quantum systems.This study presents an analytical investigation of entanglement quantification in three-qubit pure states through the Acín canonical representation, which serves as a generalization of the two-qubit Schmidt decomposition. Driven by the intricacies of multipartite entanglement and the shortcomings of current measures, we employ a global concurrence measure derived from the generalized concurrence of various bipartitions within the system. The characteristics of this measure are explored both analytically and numerically across a range of SLOCC entanglement classes, such as product, biseparable, GHZ-type, and W-type states. When compared to the three-tangle and tripartite negativity, our findings indicate that this measure yields complementary insights into multipartite correlations. The outcomes underscore the significance of this measure in elucidating the structure and distribution of entanglement within three-qubit quantum systems.

quant-pharXiv:2606.04721Jun 4, 20260 citations

Partition Ghz Slocc Class of Three Qubits into Ten Families Under Lu

Dafa Li

Jan 1, 20220 citations

Entanglement of Three-Qubit Random Pure States

Marco Enríquez, Francisco Delgado, Karol Życzkowski

We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.

arXiv:1809.00642EntropySep 29, 201835 citations

Greenberger-Horne-Zeilinger Symmetry in Four Qubit System

DaeKil Park

Like a three-qubit Greenberger-Horne-Zeilinger(GHZ) symmetry we explore a corresponding symmetry in the four-qubit system, which we call GHZ $_4$ symmetry. While whole GHZ-symmetric states can be represented by two real parameters, the whole set of the GHZ $_4$ -symmetric states is represented by three real parameters. In the parameter space all GHZ $_4$ -symmetric states reside inside a tetrahedron. We also explore a question where the given SLOCC class of the GHZ $_4$ -symmetric states resides in the tetrahedron. Among nine SLOCC classes we have examined five SLOCC classes, which results in three linear hierarchies $L_{abc_2} \subset L_{a_4} \subset L_{a_2b_2} \subset G_{abcd}$ , $L_{a_20_{3\oplus\bar{1}}} \subset G_{abcd}$ , and $L_{0_{3\oplus\bar{1}}0_{3\oplus\bar{1}}} \subset G_{abcd}$ which hold, at least, in the whole set of the GHZ $_4$ -symmetric states. Difficulties arising in the analysis of the remaining SLOCC classes are briefly discussed.Like a three-qubit Greenberger-Horne-Zeilinger(GHZ) symmetry we explore a corresponding symmetry in the four-qubit system, which we call GHZ$_4$ symmetry. While whole GHZ-symmetric states can be represented by two real parameters, the whole set of the GHZ$_4$-symmetric states is represented by three real parameters. In the parameter space all GHZ$_4$-symmetric states reside inside a tetrahedron. We also explore a question where the given SLOCC class of the GHZ$_4$-symmetric states resides in the tetrahedron. Among nine SLOCC classes we have examined five SLOCC classes, which results in three linear hierarchies $L_{abc_2} \subset L_{a_4} \subset L_{a_2b_2} \subset G_{abcd}$, $L_{a_20_{3\oplus\bar{1}}} \subset G_{abcd}$, and $L_{0_{3\oplus\bar{1}}0_{3\oplus\bar{1}}} \subset G_{abcd}$ which hold, at least, in the whole set of the GHZ$_4$-symmetric states. Difficulties arising in the analysis of the remaining SLOCC classes are briefly discussed.

arXiv:1611.09999Nov 30, 20160 citations

Like a three-qubit Greenberger-Horne-Zeilinger(GHZ) symmetry we explore a corresponding symmetry in the four-qubit system, which we call GHZ

_4

symmetry. While whole GHZ-symmetric states can be represented by two real parameters, the whole set of the GHZ

_4

-symmetric states is represented by three real parameters. In the parameter space all GHZ

_4

-symmetric states reside inside a tetrahedron. We also explore a question where the given SLOCC class of the GHZ

_4

-symmetric states resides in the tetrahedron. Among nine SLOCC classes we have examined five SLOCC classes, which results in three linear hierarchies

L_{abc_2} \subset L_{a_4} \subset L_{a_2b_2} \subset G_{abcd}

L_{a_20_{3\oplus\bar{1}}} \subset G_{abcd}

, and

L_{0_{3\oplus\bar{1}}0_{3\oplus\bar{1}}} \subset G_{abcd}

which hold, at least, in the whole set of the GHZ

_4

-symmetric states. Difficulties arising in the analysis of the remaining SLOCC classes are briefly discussed.