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arXiv:0905.3791 · Physical Review A

Maximally entangled three-qubit states via geometric measure of entanglement

Sayatnova Tamaryan, Tzu-Chieh Wei, DaeKil Park

Published Nov 11, 2009• 48 citations
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Abstract

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.

DOI
10.1103/physreva.80.052315

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arXiv:0909.1077•Sep 6, 2009•1 citation
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.
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].
The bipartite entanglement of a pure quantum state is known to be characterized by its Schmidt decomposition. In particular the state is maximally entangled when all the Schmidt coefficients are equal. We point out a convenient method which always yields a single analytical condition for the state to be maximally entangled, in terms of its expansion coefficients in any basis. The method works even when the Schmidt coefficients cannot be calculated analytically, and does not require their calculation. As an example this technique is used to derive the Bell basis for a system of two qubits. In a second example the technique shows a particular state to \textit{never} be maximally entangled, a general conclusion that cannot be reached using the Schmidt decomposition.
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.
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.
For two qubits in a pure state there exists a one-to-one relation between the entanglement measure (the concurrence C\mathcal{C}C) and the maximal violation M\mathcal{M}M of a Bell inequality. No such relation exists for the three-qubit analog of C\mathcal{C}C (the tangle $\ensuremath{\tau}$), but we have found that numerical data is consistent with a simple set of upper and lower bounds for $\ensuremath{\tau}$ given M\mathcal{M}M. The bounds on $\ensuremath{\tau}$ become tighter with increasing M\mathcal{M}M, so they are of practical use. The Svetlichny form of the Bell inequality gives tighter bounds than the Mermin form. We show that the bounds can be tightened further if the tangle is replaced by an entanglement monotone that can identify both the W state and the Greenberger-Horne-Zeilinger state.
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.