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:0909.1077

Effect of Phase Factor in the Geometric Entanglement Measure of Three-Qubit States

Sayatnova Tamaryan, Hung-Soo Kim, MuSeong Kim, Kap Soo Jang, DaeKil Park

Published Sep 6, 2009• 1 citation
arXivHTMLPDF

Abstract

Any pure three-qubit state is uniquely characterized by one phase and four positive parameters. The geometric measure of entanglement as a function of state parameters can have different expressions. Each of expressions has its own applicable domain and thus the whole state parameter space is divided into subspaces that are ranges of definition for corresponding expressions. The purpose of this paper is to examine the applicable domains for the most general qubit-interchange symmetric three-qubit states. First, we compute the eigenvalues of the non-linear eigenvalue equations and the nearest separable states for the permutation invariant three-qubit states with a fixed phase. Next, we compute the geometric entanglement measure, deduce the boundaries of all subspaces, and find allocations of highly and slightly entangled states. It is shown that there are three applicable domains when the phase factor is $π/2$ while other cases have only two domains. The emergence of the three domains is due to the appearance of the additional W-state. We show that most of highly entangled states reside near the boundaries of the domains and states located far from the boundaries become less-entangled and eventually go to the product states. The neighbors of W-state are generally more entangled than the neighbors of Greenberger-Horne-Zeilinger(GHZ) state from the aspect of the geometric measure. However, the range of the GHZ-neighbors is much more wider than the range of the W-neighbors.

DOI
10.48550/arxiv.0909.1077

Related papers

Analytic expressions for geometric measure of three-qubit states

Levon Tamaryan, DaeKil Park, Sayatnova Tamaryan

A method is developed to derive algebraic equations for the geometric measure of entanglement of three-qubit pure states. The equations are derived explicitly and solved in the cases of most interest. These equations allow one to derive analytic expressions of the geometric entanglement measure in a wide range of three-qubit systems, including the general class of W states and states which are symmetric under the permutation of two qubits. The nearest separable states are not necessarily unique, and highly entangled states are surrounded by a one-parametric set of equally distant separable states. A possibility for physical applications of the various three-qubit states to quantum teleportation and superdense coding is suggested from the aspect of entanglement.

arXiv:0710.0571•Physical Review A•Feb 27, 2008•34 citations

Maximally entangled three-qubit states via geometric measure of entanglement

Sayatnova Tamaryan, Tzu-Chieh Wei, DaeKil Park

Bipartite maximally entangled states have the property that the largest Schmidt coefficient reaches its lower bound. However, for multipartite states, the standard Schmidt decomposition generally does not exist. We use a generalized Schmidt decomposition and the geometric measure of entanglement to characterize three-qubit pure states and derive a single-parameter family of maximally entangled three-qubit states. The paradigmatic Greenberger-Horne-Zeilinger (GHZ) and $W$ states emerge as extreme members in this family of maximally entangled states. This family of states possesses different trends of entanglement behavior: in going from GHZ to $W$ states, the geometric measure, the relative entropy of entanglement, and the bipartite entanglement all increase monotonically whereas the three-tangle and bipartition negativity both decrease monotonically.

arXiv:0905.3791•Physical Review A•Nov 11, 2009•48 citations

Geometric measure of entanglement and shared quantum states

Levon Tamaryan, DaeKil Park, Jin-Woo Son +1 more

We give an explicit expression for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space. Each expression of the measure has its own geometrically meaningful interpretation. Such an interpretation allows oneself to take one step toward a complete understanding for the general properties of the entanglement measure. The states that lie on joint surfaces separating different ranges of definition, designated as shared states, seem to have particularly interesting features. The properties of the shared states are fully discussed.

arXiv:0803.1040•Physical Review A•Sep 2, 2008•18 citations

Geometric Measure of Entanglement and Schmidt Decomposition of Multipartite Systems

Levon Tamaryan

The thesis includes the original results of our articles [30, 37, 40, 42, 51, 53, 75]. A method is developed to compute analytically entanglement measures of three-qubit pure states. Owing to it closed-form expressions are presented for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space and the Hilbert space of three-qubits consists of three different entangled regions. The states that lie on joint surfaces separating different entangled regions, designated as shared states, have particularly interesting features and are dual quantum channels for the perfect teleportation and superdense coding [42]. A powerful method is developed to compute analytically multipartite entanglement measures. The method uses the duality concept and creates a bijection between highly entangled quantum states and their nearest separable states. The bijection gives explicitly the geometric entanglement measure of arbitrary generalized W states of n qubits [30, 75]. Additionaly, the behavior of the geometric entanglement measure of many-qubit W states is analyzed and an interpolating formula is derived [51]. Generalized Schmidt decomposition of pure three-qubit states has four positive and one complex coefficients. In contrast to the bipartite case, they are not arbitrary and the largest Schmidt coefficient restricts severely other coefficients. It is derived a non-strict inequality between three-qubit Schmidt coefficients, where the largest coefficient defines the least upper bound for the three nondiagonal coefficients or, equivalently, the three nondiagonal coefficients together define the greatest lower bound for the largest coefficient [53].

arXiv:1311.5994•Nov 23, 2013•0 citations

Hierarchies of geometric entanglement

Massimo Blasone A, Fabio Dell’Anno, Silvio De Siena +1 more

We introduce a class of generalized geometric measures of entanglement. For pure quantum states of $\mathit{N}$ elementary subsystems, they are defined as the distances from the sets of $\mathit{K}$-separable states $(K=2,\dots{},N)$. The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows discrimination among the different contributions. The extended measures are applied to the study of entanglement in different classes of $\mathit{N}$-qubit pure states. These classes include $\mathit{W}$ and Greenberger-Horne-Zeilinger (GHZ) states, and their symmetric superpositions; symmetric multimagnon states; cluster states; and, finally, asymmetric generalized $\mathit{W}$-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between GHZ and $\mathit{W}$ states, clarifying the original result of Wei and Goldbart that $\mathit{W}$ states possess a larger global entanglement than GHZ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.

arXiv:0712.4085•Physical Review A•Jun 3, 2008•114 citations

Entanglement of Three-Qubit Greenberger-Horne-Zeilinger–Symmetric States

Christopher Eltschka, Jens Siewert

The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.

arXiv:1304.6095•Physical Review Letters•Jan 13, 2012•113 citations
Any pure three-qubit state is uniquely characterized by one phase and four positive parameters. The geometric measure of entanglement as a function of state parameters can have different expressions. Each of expressions has its own applicable domain and thus the whole state parameter space is divided into subspaces that are ranges of definition for corresponding expressions. The purpose of this paper is to examine the applicable domains for the most general qubit-interchange symmetric three-qubit states. First, we compute the eigenvalues of the non-linear eigenvalue equations and the nearest separable states for the permutation invariant three-qubit states with a fixed phase. Next, we compute the geometric entanglement measure, deduce the boundaries of all subspaces, and find allocations of highly and slightly entangled states. It is shown that there are three applicable domains when the phase factor is π/2π/2π/2 while other cases have only two domains. The emergence of the three domains is due to the appearance of the additional W-state. We show that most of highly entangled states reside near the boundaries of the domains and states located far from the boundaries become less-entangled and eventually go to the product states. The neighbors of W-state are generally more entangled than the neighbors of Greenberger-Horne-Zeilinger(GHZ) state from the aspect of the geometric measure. However, the range of the GHZ-neighbors is much more wider than the range of the W-neighbors.
A method is developed to derive algebraic equations for the geometric measure of entanglement of three-qubit pure states. The equations are derived explicitly and solved in the cases of most interest. These equations allow one to derive analytic expressions of the geometric entanglement measure in a wide range of three-qubit systems, including the general class of W states and states which are symmetric under the permutation of two qubits. The nearest separable states are not necessarily unique, and highly entangled states are surrounded by a one-parametric set of equally distant separable states. A possibility for physical applications of the various three-qubit states to quantum teleportation and superdense coding is suggested from the aspect of entanglement.
Bipartite maximally entangled states have the property that the largest Schmidt coefficient reaches its lower bound. However, for multipartite states, the standard Schmidt decomposition generally does not exist. We use a generalized Schmidt decomposition and the geometric measure of entanglement to characterize three-qubit pure states and derive a single-parameter family of maximally entangled three-qubit states. The paradigmatic Greenberger-Horne-Zeilinger (GHZ) and WWW states emerge as extreme members in this family of maximally entangled states. This family of states possesses different trends of entanglement behavior: in going from GHZ to WWW states, the geometric measure, the relative entropy of entanglement, and the bipartite entanglement all increase monotonically whereas the three-tangle and bipartition negativity both decrease monotonically.
We give an explicit expression for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space. Each expression of the measure has its own geometrically meaningful interpretation. Such an interpretation allows oneself to take one step toward a complete understanding for the general properties of the entanglement measure. The states that lie on joint surfaces separating different ranges of definition, designated as shared states, seem to have particularly interesting features. The properties of the shared states are fully discussed.
The thesis includes the original results of our articles [30, 37, 40, 42, 51, 53, 75]. A method is developed to compute analytically entanglement measures of three-qubit pure states. Owing to it closed-form expressions are presented for the geometric measure of entanglement for three-qubit states that are linear combinations of four orthogonal product states. It turns out that the geometric measure for these states has three different expressions depending on the range of definition in parameter space and the Hilbert space of three-qubits consists of three different entangled regions. The states that lie on joint surfaces separating different entangled regions, designated as shared states, have particularly interesting features and are dual quantum channels for the perfect teleportation and superdense coding [42]. A powerful method is developed to compute analytically multipartite entanglement measures. The method uses the duality concept and creates a bijection between highly entangled quantum states and their nearest separable states. The bijection gives explicitly the geometric entanglement measure of arbitrary generalized W states of n qubits [30, 75]. Additionaly, the behavior of the geometric entanglement measure of many-qubit W states is analyzed and an interpolating formula is derived [51]. Generalized Schmidt decomposition of pure three-qubit states has four positive and one complex coefficients. In contrast to the bipartite case, they are not arbitrary and the largest Schmidt coefficient restricts severely other coefficients. It is derived a non-strict inequality between three-qubit Schmidt coefficients, where the largest coefficient defines the least upper bound for the three nondiagonal coefficients or, equivalently, the three nondiagonal coefficients together define the greatest lower bound for the largest coefficient [53].
We introduce a class of generalized geometric measures of entanglement. For pure quantum states of N\mathit{N}N elementary subsystems, they are defined as the distances from the sets of K\mathit{K}K-separable states (K=2,…,N)(K=2,\dots{},N)(K=2,…,N). The entire set of generalized geometric measures provides a quantification and hierarchical ordering of the different bipartite and multipartite components of the global geometric entanglement, and allows discrimination among the different contributions. The extended measures are applied to the study of entanglement in different classes of N\mathit{N}N-qubit pure states. These classes include W\mathit{W}W and Greenberger-Horne-Zeilinger (GHZ) states, and their symmetric superpositions; symmetric multimagnon states; cluster states; and, finally, asymmetric generalized W\mathit{W}W-like superposition states. We discuss in detail a general method for the explicit evaluation of the multipartite components of geometric entanglement, and we show that the entire set of geometric measures establishes an ordering among the different types of bipartite and multipartite entanglement. In particular, it determines a consistent hierarchy between GHZ and W\mathit{W}W states, clarifying the original result of Wei and Goldbart that W\mathit{W}W states possess a larger global entanglement than GHZ states. Furthermore, we show that all multipartite components of geometric entanglement in symmetric states obey a property of self-similarity and scale invariance with the total number of qubits and the number of qubits per party.
The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.