Sufficient and necessary condition of separability for generalized Werner states

Deng, Dan; Chen, Jing-Ling

doi:10.1016/j.aop.2008.12.006

Cited by 6 publications

(5 citation statements)

References 40 publications

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“…This generalizes previous results in the literature [32,33]. However, notice that we can only conclude about the genuine entanglement of multipartite quantum states and not about their separability, since in this case it is necessary to distinguish between full separability and m-order separability [34].…”

Section: Results 4 Letsupporting

confidence: 86%

Section: Full-rank Quantum Statessupporting

confidence: 54%

“…Thus states likeρ are relevant per se; they are indeed Werner states in arbitrary dimensions [32]. Secondly, this result will allow us to generalize the sufficient and necessary condition of separability already proved for some generalized Werner states [32,33] in the multipartite realm (see below).Merging both results we have a first sufficient and necessary criterion of separability for bipartite states of arbitrary dimensions of the form ρ = (1 − λ)M 1 ⊗ M 2 + λ|z z|:3.2 Full-rank states with r E ρ > 1Following along the same line as above, we focus on those cases in which the decomposition (4) yields a matrix E ρ such that r E ρ > 1. We arrive at the first incomplete result of this approach.…”

supporting

confidence: 55%

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An analytic approach to the problem of separability of quantum states based upon the theory of cones

Salgado¹,

Sánchez-Gómez

Ferrero

2010

Quantum Inf Process

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Exploiting the cone structure of the set of unnormalized mixed quantum states, we offer an approach to detect separability independently of the dimensions of the subsystems. We show that any mixed quantum state can be decomposed as ρ = (1 − λ)C ρ + λE ρ , where C ρ is a separable matrix whose rank equals that of ρ and the rank of E ρ is strictly lower than that of ρ. With the simple choice C ρ = M 1 ⊗ M 2 we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E ρ equals 1. We give a first extension of this result to detect genuine entanglement in multipartite states and show a natural connection between the multipartite separability problem and the classification of pure states under stochastic local operations and classical communication. We argue that this approach is not exhausted with the first simple choices included herein. Entanglement is […] the characteristic trait of quantum mechanics" [1]. This statement remains valid, even stronger, more than 70 years after its formulation. Nowadays entanglement is believed to lie behind most quantum phenomena and is still considered the source of conceptual challenges [2]. In a few words, entanglement stands for the presence of quantum correlations, i.e. statistical correlations which cannot be reproduced using classical probability theory. Despite its importance, we are still far from a full comprehension of this fundamental feature of Nature, not to mention the fact that the problem of discerning whether a given composite quantum system portrays entanglement has only a partial solution. By and large, a complete understanding will follow only after apprehending the notion of quantum correlation both qualitatively and quantitatively for all sort of composite systems, either bipartite or multipartite and either with infinite or with finite dimensions.Today a big gap exists between our understanding of bipartite and multipartite entanglement, probably as big as radical the difference is between having two subsystems and having more than two. This gap is also present between finite-and infinite-dimensional systems. These latter have been analyzed mainly in the Gaussian domain (see e.g. [3]). In the following we will concentrate upon finite-dimensional systems. From a broad mathematical point of view two approaches can be followed to discern whether a given quantum state is entangled or not. On the one hand we can adopt an analytical point of view and try to find operational criteria to settle the question upon the quantum state itself. As prominent instances we can cite the Schmidt decomposition (only valid for pure states of any dimensions) [4], the PPT criterion (valid as a necessary and sufficient condition for 2 ⊗ 2 or 2 ⊗ 3 and a necessary condition for the rest of bipartite systems) [5][6][7], the range criterion (a necessary condition for all bipartite systems) [8] and the computable cross norm or realignment criterion (a necessary condition for all bipartite systems) [9,10], to name a few (see [11][12][13][14] for a...

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Section: Results 4 Letsupporting

confidence: 86%

Section: Full-rank Quantum Statessupporting

confidence: 54%

supporting

confidence: 55%

See 1 more Smart Citation

An analytic approach to the problem of separability of quantum states based upon the theory of cones

Salgado¹,

Sánchez-Gómez

Ferrero

2010

Quantum Inf Process

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“…We conjecture that this incompatibility holds for general linear Bell inequalities even though there are unbounded violations [36]. The possible reason is that almost all critical points require exponential violations [42] (Appendices E, F) going beyond polynomial violations [36]. Unfortunately, our algorithmic proof (Appendix F) can only provide evidences for small m due to high computational complexity.…”

Section: Discussionmentioning

confidence: 94%

Undetectable Werner states using linear Bell inequalities

Luo

2017

Preprint

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Bell inequality serves as an important method to detect quantum entanglement, a problem which is generally known to be NP-hard. Our goal in this work is to detect Werner states using linear Bell inequality. Surprisingly, we show that Werner states of almost all generalized multipartite Greenberger-Horne-Zeilinger (GHZ) states cannot be detected by homogeneous linear Bell inequalities with dichotomic inputs and outputs of each local observer. The main idea is to estimate the largest violations of Werner states in the case of general linear Bell inequalities. The presented method is then applied to Werner states of all pure states to show a similar undetectable result. Moreover, we provide an accessible method to determine undetectable Werner states for general linear Bell inequality including sub-correlations. The numeric algorithm shows that there are nonzero measures of Werner states with small number of particles that cannot be detected by general linear Bell inequalities.

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“…Instead, we exhibit some characteristics, in Fig. 3, by a family of generalized Werner states [37], ρ W (θ) = 1 − x 4 I 2 ⊗ I 2 + x|Φ(θ) Φ(θ)|,…”

Section: Discussionmentioning

confidence: 99%

Bounds of concurrence and their relation with fidelity and frontier states

Zhang

Deng

et al. 2009

Physics Letters A

Self Cite

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The bounds of concurrence in [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98 (2007) 140505] and [C. Zhang et. al., Phys. Rev. A 78 (2008) 042308] are proved by using two properties of the fidelity. In two-qubit systems, for a given value of concurrence, the states achieving the maximal upper bound, the minimal lower bound or the maximal difference upper-lower bound are determined analytically.

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Sufficient and necessary condition of separability for generalized Werner states

Cited by 6 publications

References 40 publications

An analytic approach to the problem of separability of quantum states based upon the theory of cones

An analytic approach to the problem of separability of quantum states based upon the theory of cones

Undetectable Werner states using linear Bell inequalities

Bounds of concurrence and their relation with fidelity and frontier states

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