Entanglement conditions for tripartite systems via indeterminacy relations

Song, Lei; Wang, Xiaoguang; Yan, Dong; Pu, Zhongsheng

doi:10.1088/0953-4075/41/1/015505

J. Phys. B: At. Mol. Opt. Phys.

2007

DOI: 10.1088/0953-4075/41/1/015505

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Entanglement conditions for tripartite systems via indeterminacy relations

Lei Song¹,

Xiaoguang Wang²,

Dong Yan³

et al.

Abstract: Based on the Schrödinger–Robertson indeterminacy relations in conjugation with the partial transposition, we derive a class of inequalities for detecting entanglement in several tripartite systems, including bosonic, SU(2) and SU(1, 1) systems. These inequalities are in general stronger than those based on the usual Heisenberg relations for detecting entanglement. We also discuss the reduction from SU(2) and SU(1, 1) to bosonic systems and the generalization to a multipartite case.

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Cited by 6 publications

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References 27 publications

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“…For multimode entangled states, some inequalities have also been derived using similar methods [15,16,17,18]. Nevertheless, our capacity for detecting and characterizing multipartite entangled states must be enhanced to a far richer level for further applications.…”

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confidence: 99%

Entanglement criteria and nonlocality for multimode continuous-variable systems

2009

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We demonstrate how to efficiently derive a broad class of inequalities for entanglement detection in multi-mode continuous variable systems. The separability conditions are established from partial transposition (PT) in combination with several distinct necessary conditions for a quantum physical state, which include previously established inequalities as special cases. Remarkably, our method enables us to support Peres' conjecture to its full generality within the framework of CavalcantiFoster-Reid-Drummond multipartite Bell inequality [Phys. Rev. Lett. 99, 210405 (2007)] that the nonlocality necessarily implies negative PT entangled states. Introduction-Entanglement is the key for many applications in quantum informatics; however, there still exist a number of unresolved issues despite the progresses in the past. In particular, the characterization and the detection of entanglement for multipartite quantum systems are far from complete. Peres and Horodecki et al. first proposed the positive partial transpose (PT) criterion as a necessary condition for separability [1]. To apply this criterion in experiments for continuous variables (CVs), inequalities involving observable moments up to second-order have been derived [2,3,4,5]. To detect bipartite entanglement for non-Gaussian CV states, new inequalities using higher order moments have been derived from the Schwarz inequalities [6], uncertainty relations [7,8,9], and determinants of matrices [10]. These inequalities can be directly implemented in optical systems [8,11,12]. It was also pointed out that stronger conditions can be obtained from the Schrödinger-Robertson uncertainty relation (SRUR) than the Heisenberg uncertainty relation (HUR) [13]. Moreover, it was shown that the uncertainty relation approach is sufficient to detect bipartite entanglement for all negative PT states [14].

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confidence: 99%

Entanglement criteria and nonlocality for multimode continuous-variable systems

2009

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“…Other criteria have been proposed so they could be tested experimentally in a direct manner, as the Bell inequalities [3,4] or the entanglement witnesses [5]. More recently, criteria based on variance measurements have been studied for continuous and discrete variable systems [6,7,8,9,10,11,12,13,14].In [11] the Heisenberg relation has been used along with the partial transpose operation to obtain a criterion detecting entanglement condition in bipartite nongaussian states. That idea was generalized in [13,14] with use of the Schrödinger-Robertson relation instead of the Heisenberg inequality.…”

mentioning

confidence: 99%

“…More recently, criteria based on variance measurements have been studied for continuous and discrete variable systems [6,7,8,9,10,11,12,13,14].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

“…That idea was generalized in [13,14] with use of the Schrödinger-Robertson relation instead of the Heisenberg inequality. In this paper, we generalize completely those concepts and prove that the Schrödinger-Robertson type inequality is able to detect entanglement in any pure state of bipartite and tripartite systems.…”

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Multipartite entanglement criterion from uncertainty relations

2008

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We formulate an entanglement criterion using Peres-Horodecki positive partial transpose operations combined with the Schrödinger-Robertson uncertainty relation. We show that any pure entangled bipartite and tripartite state can be detected by experimentally measuring mean values and variances of specific observables. Those observables must satisfy a specific condition in order to be used, and we show their general form in the 2 × 2 (two qubits) dimension case. The criterion is applied on a variety of physical systems including bipartite and multipartite mixed states and reveals itself to be stronger than the Bell inequalities and other criteria. The criterion also work on continuous variable cat states and angular momentum states of the radiation field.PACS numbers: 03.65. Ud, 03.67.Mn In the past few years, many criteria detecting entanglement in bipartite and multipartite systems have been developed [1]. The Peres-Horodecki positive partial transpose (PPT) criterion [2] has played a crucial role in the field and provides, in some cases, necessary and sufficient conditions to entanglement. That criteria is formulated in terms of the density operator and any practical application involves state tomography. Other criteria have been proposed so they could be tested experimentally in a direct manner, as the Bell inequalities [3,4] or the entanglement witnesses [5]. More recently, criteria based on variance measurements have been studied for continuous and discrete variable systems [6,7,8,9,10,11,12,13,14].In [11] the Heisenberg relation has been used along with the partial transpose operation to obtain a criterion detecting entanglement condition in bipartite nongaussian states. That idea was generalized in [13,14] with use of the Schrödinger-Robertson relation instead of the Heisenberg inequality. In this paper, we generalize completely those concepts and prove that the Schrödinger-Robertson type inequality is able to detect entanglement in any pure state of bipartite and tripartite systems. Experimentally, it can be realized by measuring mean values and variances of different observables; however we show that all observables are not suitable and we yield the general condition they must satisfy to be eligible. For 2 × 2 systems, we explicitly give their general form. The inequality has a wide application range : qubits, angular momentum states of harmonic oscillators, cat states, etc. For the mixed state case, the inequality detects entanglement of bipartite Werner states better than the Bell inequalities [3] and also leads to a good characterization of multipartite Werner states.For any observables A, B and any density operator ρ, the Schrödinger-Robertson uncertainty relation The Heisenberg uncertainty relation is obtained if the last term is not considered, which gives a weaker inequality.The PPT criterion [2] is a sufficient condition for entanglement, saying that if a bipartite state ρ is separable it can be written as ρ = i p i ρ 1 i ⊗ ρ 2 i with usual notations and its partial transpose ρ pt ≡ i p i ρ 1 ...

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Testing nonclassicality in multimode fields: A unified derivation of classical inequalities

et al. 2010

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We consider a way to generate operational inequalities to test nonclassicality (or quantumness) of multimode bosonic fields (or multiparty bosonic systems) that unifies the derivation of many known inequalities and allows to propose new ones. The nonclassicality criteria are based on Vogel's criterion corresponding to analyzing the positivity of multimode P functions or, equivalently, the positivity of matrices of expectation values of, e.g., creation and annihilation operators. We analyze not only monomials but also polynomial functions of such moments, which can sometimes enable simpler derivations of physically relevant inequalities. As an example, we derive various classical inequalities which can be violated only by nonclassical fields. In particular, we show how the criteria introduced here easily reduce to the well-known inequalities describing (a) multimode quadrature squeezing and its generalizations, including sum, difference, and principal squeezing; (b) two-mode one-time photon-number correlations, including sub-Poisson photon-number correlations and effects corresponding to violations of the Cauchy-Schwarz and Muirhead inequalities; (c) two-time single-mode photon-number correlations, including photon antibunching and hyperbunching; and (d) two-and three-mode quantum entanglement. Other simple inequalities for testing nonclassicality are also proposed. We have found some general relations between the nonclassicality and entanglement criteria, in particular those resulting from the Cauchy-Schwarz inequality. It is shown that some known entanglement inequalities can be derived as nonclassicality inequalities within our formalism, while some other known entanglement inequalities can be seen as sums of more than one inequality derived from the nonclassicality criterion. This approach enables a deeper analysis of the entanglement for a given nonclassicality.

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Entanglement conditions for tripartite systems via indeterminacy relations

Cited by 6 publications

References 27 publications

Entanglement criteria and nonlocality for multimode continuous-variable systems

Entanglement criteria and nonlocality for multimode continuous-variable systems

Multipartite entanglement criterion from uncertainty relations

Testing nonclassicality in multimode fields: A unified derivation of classical inequalities

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