Entanglement Properties of Composite Quantum Systems

Eckert, K.; Gühne, Otfried; Hulpke, Florian; Hyllus, Philipp; Korbicz, J. K.; Mompart, J.; Bruß, Dagmar; Lewenstein, Maciej; Sanpera, Anna

doi:10.1002/3527603549.ch7

Cited by 10 publications

(11 citation statements)

References 37 publications

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“…The second relation of Eq. (2) shows that the time reversal transformation ϑ is unitarily equivalent to the transposition map T . Hence, the PPT criterion is equivalent to the condition that the partial time reversal ϑ 2 is positive:…”

Section: B Nondecomposable Positive Maps and Optimal Entanglement Wimentioning

confidence: 99%

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Separability criteria and bounds for entanglement measures

Breuer

2006

J. Phys. A: Math. Gen.

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Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N ≥ 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.

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Section: B Nondecomposable Positive Maps and Optimal Entanglement Wimentioning

confidence: 99%

“…A central problem in quantum information theory [1,2] is the formulation of appropriate measures that quantify the degree of entanglement in composite systems. Particularly important entanglement measures are the concurrence [3,4,5] and the entanglement of formation [6,7].…”

Section: Introductionmentioning

confidence: 99%

Separability criteria and bounds for entanglement measures

Breuer

2006

J. Phys. A: Math. Gen.

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“…Subsequently, the study of separability criteria and their relation to positive maps attracted a great deal of attention and several new criteria were formulated [3]. By now there exists a sophisticated theory based on so-called entanglement witnesses [3,4,5]. However, for a long time the PPT criterion remained the most powerful and versatile operational separability criterion.…”

Section: Introductionmentioning

confidence: 99%

Some properties of the computable cross-norm criterion for separability

2003

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The computable cross norm (CCN) criterion is a new powerful analytical and computable separability criterion for bipartite quantum states, that is also known to systematically detect bound entanglement. In certain aspects this criterion complements the well-known Peres positive partial transpose (PPT) criterion. In the present paper we study important analytical properties of the CCN criterion. We show that in contrast to the PPT criterion it is not sufficient in dimension 2 × 2. In higher dimensions we prove theorems connecting the fidelity of a quantum state with the CCN criterion. We also analyze the behaviour of the CCN criterion under local operations and identify the operations that leave it invariant. It turns out that the CCN criterion is in general not invariant under local operations.

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“…States which cannot be written in this way are called inseparable or entangled. Much effort in quantum information theory has been devoted to the problems of the characterization, the classification and the quantification of mixed state entanglement [2,3]. Although considerable progress has been made in recent years (see, e. g., Refs.…”

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confidence: 99%

Optimal Entanglement Criterion for Mixed Quantum States

2006

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We develop a strong and computationally simple entanglement criterion. The criterion is based on an elementary positive map Φ which operates on state spaces with even dimension N ≥ 4. It is shown that Φ detects many entangled states with positive partial transposition (PPT) and that it leads to a class of optimal entanglement witnesses. This implies that there are no other witnesses which can detect more entangled PPT states. The map Φ yields a systematic method for the explicit construction of high-dimensional manifolds of bound entangled states.PACS numbers: 03.67. Mn,03.65.Ud,03.65.Yz Entanglement and quantum inseparability are key features of quantum mechanics which are connected to the tensor product structure of the state spaces of composite systems. A mixed state ρ of a bipartite system, for instance, is defined to be separable or classically correlated if it can be written as a convex linear combination of uncorrelated product states, i. e., if it can be represented in the form ρ = i p i ρ i 1 ⊗ρ i 2 , where {p i } is a probability distribution and the ρ i 1 , ρ i 2 are density matrices describing states of the first and the second subsystem, respectively [1]. States which cannot be written in this way are called inseparable or entangled. Much effort in quantum information theory has been devoted to the problems of the characterization, the classification and the quantification of mixed state entanglement [2,3]. Although considerable progress has been made in recent years (see, e. g., Refs.[4]), we are still far away from a true understanding of many aspects of these problems.A problem of central importance in entanglement theory is the development of computationally efficient criteria which allow us to decide whether or not a given state is entangled. Peres [5] has developed a very strong entanglement criterion which is known as criterion of positive partial transposition (PPT). It states that a necessary condition for a given state ρ to be separable is that its partial transpose is a positive operator. Usually, one writes this condition as (I ⊗ T )ρ ≥ 0, where T denotes the transposition of operators in a chosen basis and I is the identity map, indicating that the transposition is carried out only on the second part of the composite system. The PPT condition represents a necessary and sufficient separability criterion for certain low-dimensional systems [6], but it is only necessary in higher dimensions. Hence, there are generally entangled PPT states which belong to the class of bound entangled states [7].The transposition T is a distinguished example of a positive but not completely positive map. This means that T maps all positive operators on the subsystems to positive operators, while there exist states ρ of the combined system for which (I ⊗T )ρ has negative eigenvalues. There are many other maps with this property. The significance of positive maps in entanglement theory is provided by a fundamental theorem established in Ref. [6]. This theorem states that a necessary and sufficient conditio...

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Entanglement Properties of Composite Quantum Systems

Cited by 10 publications

References 37 publications

Separability criteria and bounds for entanglement measures

Separability criteria and bounds for entanglement measures

Some properties of the computable cross-norm criterion for separability

Optimal Entanglement Criterion for Mixed Quantum States

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