Normal forms and entanglement measures for multipartite quantum states

Verstraete, Frank; Dehaene, Jeroen; Moor, Bart De

doi:10.1103/physreva.68.012103

Cited by 258 publications

(415 citation statements)

References 17 publications

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“…(3) too. Therefore, if we take into account the normalization factor, E 1234 is SLOCC invariant [33]. We now prove that E 1234 satisfies the above condition (2), i.e.…”

mentioning

confidence: 98%

“…After simple algebra calculation and using the fact that the arithmetic mean always exceeds the geometric mean [8,32,33], we can see that…”

mentioning

confidence: 99%

“…E 1234 ≥ 0. It has been shown that [33] all pure multipartite states can be transformed into a normal form, all local density operators of which are proportional to the identity, by the determinant 1 SLOCC operations. A generic pure state of four qubits can always be transformed to the normal form state [34] …”

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confidence: 99%

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Information-theoretic measure of genuine multiqubit entanglement

Cai¹,

Zhou²,

Zhou³

et al. 2006

Phys. Rev. A

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We consider pure quantum states of N qubits and study the genuine N −qubit entanglement that is shared among all the N qubits. We introduce an information-theoretic measure of genuine N -qubit entanglement based on bipartite partitions. When N is an even number, this measure is presented in a simple formula, which depends only on the purities of the partially reduced density matrices. It can be easily computed theoretically and measured experimentally. When N is an odd number, the measure can also be obtained in principle.PACS numbers: 03.67.-a, 03.65. Ud, 73.43.Nq, 89.70.+c The nature of quantum entanglement is a fascinating topic in quantum mechanics since the famous EinsteinPodolsky-Rosen paper [1] in 1935. Recently, much interest has been focused on entanglement in quantum systems containing a large number of particles. On one hand, multipartite entanglement is valuable physical resource in large-scale quantum information processing [2,3]. On the other hand, multipartite entanglement seems to play an important role in condensed matter physics [4], such as quantum phase transitions (QPT) [5,6] and high temperature superconductivity [7]. Therefore, how to characterize and quantify multipartite entanglement remains one of the central issues in quantum information theory.In the present literature, there exist very few measures of multipartite entanglement with a clear physical meaning [8,9,10,11,12,13,14,15]. Because of this, most research in quantum entanglement and QPT focused on bipartite entanglement, for which there have been several well defined measures [16,17,18,19,20,21,22]. However, bipartite entanglement can not characterize the global quantum correlations among all parties in a multiparticle system. Since the correlation length diverges at the critical points, multipartite entanglement plays an essential role in QPT. Though localizable entanglement (LE) [23] can be used to describe long-range quantum correlations, its determination is a formidable task for generic pure states. Therefore, computable measure of multipartite entanglement with clear physical meanings are highly desired [24,25].In this paper, we define a new measure of genuine N −qubit entanglement based on different bipartite partitions of the qubits and existing measures for mutual information. The central idea is that, through bipartite partitions, we can get information about the genuine multi-qubit entanglement. Our measure is a polynomial SLOCC (stochastic local operations and classical communication) invariant [26] and is unchanged under permutations of qubits. When N is an even number, we derive * Electronic address: zwzhou@ustc.edu.cn a simple formula for this measure, which is determined by the purity of partially reduced density matrices only. It can be computed through experimentally observable quantities [10,27]. Therefore, it is easy to obtain not only theoretically but also experimentally. For N = 4, we show that this measure satisfies all the necessary conditions required for a natural entanglement measure [20] exactly....

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“…(3) too. Therefore, if we take into account the normalization factor, E 1234 is SLOCC invariant [33]. We now prove that E 1234 satisfies the above condition (2), i.e.…”

mentioning

confidence: 98%

“…After simple algebra calculation and using the fact that the arithmetic mean always exceeds the geometric mean [8,32,33], we can see that…”

mentioning

confidence: 99%

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Information-theoretic measure of genuine multiqubit entanglement

Cai¹,

Zhou²,

Zhou³

et al. 2006

Phys. Rev. A

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“…The three tangle, which is polynomial invariant [40,41], is a good measure for tripartite entanglement of mixed density matrices. However, it needs an optimal decomposition of a mixed density matrix which, in general, is a tough enough task except in a few rare cases [42].…”

Section: Entanglement Measurement Of Tripartite Statesmentioning

confidence: 99%

Tripartite entanglement of fermionic system in accelerated frames

Khan

2014

Annals of Physics

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The dynamics of tripartite entanglement of fermionic system in noninertial frames through linear contraction criterion when one or two observers are accelerated is investigated. In one observer accelerated case the entanglement measurement is not invariant with respect to the partial realignment of different subsystems and for two observers accelerated case it is invariant. It is shown that the acceleration of the frame does not generate entanglement in any bipartite subsystems. Unlike the bipartite states, the genuine tripartite entanglement does not completely vanish in both one observer accelerated and two observers accelerated cases even in the limit of infinite acceleration. The degradation of tripartite entanglement is fast when two observers are accelerated than when one observer is accelerated. It is shown that tripartite entanglement is a better resource for quantum information processing than the bipartite entanglement in noninertial frames .

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“…al., [29] and Verstraete, et. al., [40] which state that the scale chosen for a measure of entanglement must be invariant under the addition of extra, but unused, Hilbert space dimensions. The two-atom TCM provides one example of the relevant physics underlying these ideas.…”

Section: Entanglement Sharing and The Residual Tanglementioning

confidence: 99%

Entanglement sharing in the two-atom Tavis-Cummings model

et al. 2003

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Individual members of an ensemble of identical systems coupled to a common probe can become entangled with one another, even when they do not interact directly. We investigate how this type of multipartite entanglement is generated in the context of a system consisting of two two-level atoms resonantly coupled to a single mode of the electromagnetic field. The dynamical evolution is studied in terms of the entanglements in the different bipartite partitions of the system, as quantified by the I-tangle. We also propose a generalization of the so-called residual tangle that quantifies the inherent three-body correlations in our tripartite system. This enables us to completely characterize the phenomenon of entanglement sharing in the case of the two-atom Tavis-Cummings model, a system of both theoretical and experimental interest.

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Normal forms and entanglement measures for multipartite quantum states

Cited by 258 publications

References 17 publications

Information-theoretic measure of genuine multiqubit entanglement

Information-theoretic measure of genuine multiqubit entanglement

Tripartite entanglement of fermionic system in accelerated frames

Entanglement sharing in the two-atom Tavis-Cummings model

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