Various notions of positivity for bi-linear maps and applications to tri-partite entanglement

Han, Kyung Hoon; Kye, Seung-Hyeok

doi:10.1063/1.4931059

Cited by 20 publications

(26 citation statements)

References 30 publications

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“…If ̺ is GHZ diagonal, then we have ∆ ̺ = min{a 1 , a 2 , a 3 , a 4 }. The number C(z) above turns out to coincide essentially with the number in the characterization of three qubit X-shaped entanglement witnesses [21]. In this way, we got a simpler expression for C(z).…”

Section: Ghz Diagonal Statesmentioning

confidence: 81%

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Separability of three qubit Greenberger–Horne–Zeilinger diagonal states

Han¹,

Kye²

2017

J. Phys. A: Math. Theor.

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Abstract. We characterize the separability of three qubit GHZ diagonal states in terms of entries. This enables us to check separability of GHZ diagonal states without decomposition into the sum of pure product states. In the course of discussion, we show that the necessary criterion of Gühne [1] for (full) separability of three qubit GHZ diagonal states is sufficient with a simpler formula. The main tool is to use entanglement witnesses which are tri-partite Choi matrices of positive bi-linear maps. IntroductionEntanglement is now considered as one of the most important resources in quantum information theory, and it is crucial to detect entanglement from separability. Positivity of partial transposes is a simple but powerful criterion for separability [2,3]. In fact, it is known [4,5] that PPT property is equivalent to separability for 2⊗2 or 2⊗3 systems. But, it is very difficult in general to distinguish separability from entanglement, as it is known to be an NP -hard problem [6].The purpose of this note is to give a complete characterization of separability for three qubit Greenberger-Horne-Zeilinger diagonal states, which are diagonal in the GHZ basis. Those include mixtures of GHZ states and identity, as they were considered in [7,8] for examples. In multi-qubit systems, GHZ states [9,10] are key examples of maximally entangled states, and they are known to have many applications in quantum information theory. They also play central roles in the classification of multi-qubit entanglement [7,11,12]. See survey articles [13,14] for general theory of entanglement as well as various aspects of GHZ states.Our main tool is to use the notion of entanglement witnesses. In the bi-partite cases, positive linear maps are very useful to detect entanglement through the duality between tensor products and linear maps [5,15]. This duality has been formulated as the notion of entanglement witnesses [16], which is still valid in multi-partite cases. In the tri-partite cases, the second author [17] has interpreted entanglement witnesses as positive bi-linear maps. With this interpretation, we carefully choose useful entanglement witnesses for our purpose.1991 Mathematics Subject Classification. 81P15, 15A30, 46L05, 46L07.

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Section: Ghz Diagonal Statesmentioning

confidence: 81%

“…For a three qubit X-shaped self-adjoint matrix W = X(s, t, u), the authors [21] have shown that W = X(s, t, u) is an entanglement witness if and only if it is non-positive and satisfies the inequality…”

Section: Entanglement Witnessesmentioning

confidence: 99%

Separability of three qubit Greenberger–Horne–Zeilinger diagonal states

Han¹,

Kye²

2017

J. Phys. A: Math. Theor.

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“…In other words, detecting entanglement with the PPT property depends on the anti-diagonal phases. It was shown in [17,19] that the anti-diagonal phases also play a role to characterize three-qubit X-shaped entanglement witnesses.…”

Section: Resultsmentioning

confidence: 99%

“…Note that the number in the right side of (7) appear in the characterization of X-shaped three-qubit entanglement witnesses [17], which correspond to positive bi-linear maps between 2 × 2 matrices [25]. In order to get a preliminary criterion, we introduce the number…”

Section: Separability Criterion With Anti-diagonal Phasesmentioning

confidence: 99%

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The role of phases in detecting three-qubit entanglement

Han

Kye

2017

Journal of Mathematical Physics

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Abstract. We propose separability criteria for three-qubit states in terms of diagonal and anti-diagonal entries to detect entanglement with positive partial transposes. We report that the phases, that is, the angular parts of anti-diagonal entries, play a crucial role in determining whether a given three-qubit state is separable or entangled, and they must obey even an identity for separability in some cases. These criteria are strong enough to detect PPT (positive partial transpose) entanglement with nonzero volume. In several cases when all the entries are zero except for diagonal and anti-diagonal entries, we characterize separability using phases. These include the cases when anti-diagonal entries of such states share a common magnitude, and when ranks are less than or equal to six. We also compute the lengths of rank six cases, and find three-qubit separable states with lengths 8 whose maximum ranks of partial transposes are 7.

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Schmidt number of bipartite and multipartite states under local projections

Chen

Tang

2017

Quantum Inf Process

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The Schmidt number is a fundamental parameter characterizing the properties of quantum states, and the local projections are a fundamental operation in quantum physics. We investigate the relation between the Schmidt numbers of bipartite states and their projected states. We show that there exist bipartite positive-partial-transpose (PPT) entangled states of any given Schmidt number. We further construct the notion of joint Schmidt number for multipartite states, and its relation with the Schmidt number of bipartite reduced density operators.

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Various notions of positivity for bi-linear maps and applications to tri-partite entanglement

Cited by 20 publications

References 30 publications

Separability of three qubit Greenberger–Horne–Zeilinger diagonal states

Separability of three qubit Greenberger–Horne–Zeilinger diagonal states

The role of phases in detecting three-qubit entanglement

Schmidt number of bipartite and multipartite states under local projections

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