The parametric symmetry and numbers of the entangled class of 2 × M × N system

Li, XiKun; Li, Junli; Liu, Bin; Qiao, C. F.

doi:10.1007/s11433-011-4395-9

Cited by 14 publications

(17 citation statements)

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“…First, we show how the present method works by comparing it with the previous result [4] for a 2 × N × N system (Appendix A in [4]); then we use the present method to construct the canonical form for the specific 3 × N × N system discussed above.…”

Section: Examplesmentioning

confidence: 98%

“…The complete entanglement classification of a 2 × M × N pure system under stochastic local operation and classical communication (SLOCC) has been done in Refs. [3,4], where canonical forms are constructed via Jordan decomposition of the matrices. There the eigenvalues of the Jordan forms serve as the continuous parameters in the entanglement classes [3].…”

Section: Introductionmentioning

confidence: 99%

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Classification of the entangled statesL×N×N

Qiao

2012

Phys. Rev. A

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We presented a general classification scheme for the tripartite L × N × N entangled system under stochastic local operation and classical communication. The whole classification procedure consists of two correlated parts: the simultaneous similarity transformation of a commuting matrix pair into a canonical form and the study of internal symmetry of parameters in the canonical form. Based on this scheme, a concrete example of entanglement classification for a 3 × N × N system is given.

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Section: Examplesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Classification of the entangled statesL×N×N

Qiao

2012

Phys. Rev. A

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“…Quantum entanglement led to the experimental verification for Bell inequality which confirmed quantum mechanics [3]. In recent years, quantum entanglement has been one of the key driving forces that advanced the explosive development of quantum information and quantum computation [4][5][6][7][8][9][10][11][12]. Entanglement has developed from a philosophical concept into a frontier research paradigm, and much more attention has been paid to the quantification of entanglement.…”

mentioning

confidence: 99%

“…From the classification, we got the rules of S 3 and S 4 , which are listed in eqs. (11) and (12), respectively. The number of E = 1 is always 2.…”

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confidence: 99%

Entanglement of linear cluster states in terms of averaged entropies

Cao

Long

2013

Chin. Sci. Bull.

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An entanglement measure, multiple entropy measures (MEMS) was proposed recently by using the geometric mean of partial entropies over all possible i-body combinations of the quantum system. In this work, we study the average subsystem von Neumann entropies of the linear cluster state and investigated the quantum entanglement of linear cluster states in terms of MEMS. Explicit results with specific particle numbers are calculated, and some analytical results are given for systems with arbitrary particle numbers. Compared with other example quantum states such as the GHZ states and W states, the linear cluster states are "more entangled" in terms of MEMS, namely their averaged entropies are larger than the GHZ states and W states. Entanglement [1, 2] is one of the most salient properties of quantum systems. Quantum entanglement led to the experimental verification for Bell inequality which confirmed quantum mechanics [3]. In recent years, quantum entanglement has been one of the key driving forces that advanced the explosive development of quantum information and quantum computation [4][5][6][7][8][9][10][11][12]. Entanglement has developed from a philosophical concept into a frontier research paradigm, and much more attention has been paid to the quantification of entanglement. Two measures of entanglement, formation and distillation [15,16] were also studied and recognized as "good" entanglement measures.In studying quantum entanglement, the density matrices are often used. It should be mentioned that density matrix can represent different physical quantities. In the case of proper mixture, the density matrix represents the averaged "state" of an ensemble of "molecules", while the improper mixture describes the averaged "state" of a subsystem in a coupled quantum system. Though the mathematical expressions are the same, their physical properties are quite differ-*Corresponding author (email: gllong@mail.tsinghua.edu.cn) ent [49,50]. In studying quantum entanglement, we are actually using the improper mixture meaning of the density matrix. The reduced density matrix is obtained by tracing out other degrees of freedom of a composite quantum system. Based on separating the correlations encoded by a density matrix into a common set of marginals, Partovi [37] proposed a measurement in which N!/2 quantities are used to quantify an N-qubit system. In the case of N 3, a "good" measurement [38] can be constructed from the arithmetic average entropy of single reduced density matrices. Realizing the lack of one-qubit reduction, Higuchi et al. [51,52] proposed using the arithmetic mean of two-particle entropies as a measure of entanglement, and reported on a four-qubit entangled state:which is more entangled than the four-qubit GHZ state.

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“…Derivations of new and stronger Bell inequalities are one of the most important and challenging subjects in quantum information processing. Since Bell's work, there were many important generalizations such as [2][3][4][5][6][7][8][9][10][11] and references therein.…”

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confidence: 99%

Characterization of four-qubit states via Bell inequalities

2013

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A set of Bell inequalities classifying the quantum entanglement of four-qubit states is presented. These inequalities involve only two measurement settings per observer and can characterize fully separable, bi-separable and tri-separable quantum states. In addition, a quadratic inequality of the Bell operators for four-qubit systems is derived. Bell inequalities, separability, Bell operatorsCitation: Zhao H, Zhang X H, Fei S M, et al. Characterization of four-qubit states via Bell inequalities. Chin Sci Bull, 2013Bull, , 58: 2334Bull, -2339Bull, , doi: 10.1007 The Bell inequality [1] provided the first possibility to distinguish experimentally between quantum-mechanical predictions and those of local realistic models. Derivations of new and stronger Bell inequalities are one of the most important and challenging subjects in quantum information processing. Since Bell's work, there were many important generalizations such as [2-11] and references therein. The Bell inequalities presented in [12] involve only two measurement settings per observer and can detect perfectly the quantum entanglement of the generalized GHZ states. By using the idea in constructing Bell operators [12], a set of new Bell inequalities are given in [13], which gives rise to a finer classification of the entanglement for three-qubit systems.The entanglement of four-qubit systems has been treated in terms of Bell inequalities of Mermin-Klyshko type. In [14] the quantum nonlocality of some four-qubit states, the GHZ state, W state, cluster state and the state proposed in [15], has been investigated, towards the optimal violations of the Bell inequality for these states. The classification of entanglement has been also studied in such as [16][17][18][19] with linear inequalities for qubit systems and [20] with non-linear inequalities for detecting bi-separable states in arbitrary dimensional quantum systems.In this work, we study the quantum entanglement of four-*Corresponding author (email: zhaohui@bjut.edu.cn) qubit systems by using the idea in constructing Bell operators in [12]. We generalize the results of three-qubit systems in [13] to four-qubit systems. It has been shown that the standard Werner-Wolf-Żukowski-Brukner (WWZB) inequalities cannot detect the entanglement of the generalized Greenberger-Horne-Zeilinger (GHZ) states given by |ψ = cos α|0, ..., 0 + sin α|1, ..., 1 with 0 α π/4 [21, 22]. However, the Bell operators constructed in the way provided in [12] can detect the entanglement of the generalized GHZ state wholly. Our Bell operators are constructed by using the idea in [12]. The resulted Bell inequalities can distinguish fully separable, bi-separable and tri-separable states of a four-qubit system. Moreover, these linear Bell inequalities involve only two measurement settings per observer. Analytical formulas of the average values of the Bell operators for four-qubit systems are also derived. And a quadratic inequality of the Bell operators for all four-qubit systems has been presented. Explicit geometrical pictures show t...

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The parametric symmetry and numbers of the entangled class of 2 × M × N system

Cited by 14 publications

References 20 publications

Classification of the entangled statesL×N×N

Classification of the entangled statesL×N×N

Entanglement of linear cluster states in terms of averaged entropies

Characterization of four-qubit states via Bell inequalities

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