Structural approximations to positive maps and entanglement-breaking channels

Any two-qubit state can be represented geometrically by a steering ellipsoid inside the Bloch sphere. We extend this approach to represent any block positive two-qubit operator B. We derive a classification scheme based on the positivity of det B and det B T B ; this shows that any ellipsoid inside the Bloch sphere must represent either a two-qubit state or a two-qubit entanglement witness. We focus on such witnesses and their corresponding ellipsoids, finding that properties such as witness optimality are naturally manifest in this geometric representation.

show abstract

“…We consider a family of weakly optimal witnesses originally presented in Ref. [30] and studied further in Refs. [31,32].…”

Section: Examples Of Entanglement Witnessesmentioning

confidence: 99%

Geometric representation of two-qubit entanglement witnesses

2015

View full text Add to dashboard Cite

show abstract

“…The Choi map is an example of this kind [5,9,10]. Variations of the Choi map given by the second author [11] also give rise to such maps.…”

mentioning

confidence: 99%

“…See the recent paper [13] for related topics. Although there are some examples of optimal EW [9,[14][15][16][17], to the best of the author's knowledge, only known examples of indecomposable optimal EWs are ones which come from extremal indecomposable positive linear maps.We consider another variations of the Choi map given in [18]. …”

mentioning

confidence: 99%

“…This kinds of witness is said to be an indecomposable optimal EW. It is known [4,9] that W is an indecomposable optimal entanglement witness if and only if both W and W Γ are optimal entanglement witnesses.Typical examples of indecomposable optimal EW come from indecomposable positive linear maps which generate an extremal ray of the convex cone consisting of all positive linear maps. The Choi map is an example of this kind [5,9,10].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

One-parameter family of indecomposable optimal entanglement witnesses arising from generalized Choi maps

Kye

2011

Phys. Rev. A

View full text Add to dashboard Cite

In the recent paper [Chruściński and Wudarski, arXiv:1105.4821], it was conjectured that the entanglement witnesses arising from some generalized Choi maps are optimal. We show that this conjecture is true. Furthermore, we show that they provide a one parameter family of indecomposable optimal entanglement witnesses.PACS numbers: 03.65. Ud, Quantum entanglement is a basic resource in quantum information processing and communication [1]. Therefore, so much effort naturally has been put into developing theoretical and experimental methods of entanglement detection. Among them, one of the most general approach is based on the notion of entanglement witness [2,3]. Recall that an observable W = W † is said to be an entanglement witness (EW) if tr(W σ) ≥ 0 for all separable states σ, and there exists an entangled state ρ for which tr(W ρ) < 0. In this case we say that W detects ρ. Following [4], an EW W is said to be optimal if the set of entanglement detected by W is maximal with respect to the set inclusion.Note that the Choi-Jamio lwski isomorphism [5,6] gives rise to an entanglement witnesswhich is acting on M M ⊗ M N , for every positive linear map Λ : M M → M N which is not completely positive, where M K denotes the C * -algebra of all K × K matrices over the complex field C, and P + denotes the projector onto the maximally entangled state in C M ⊗C M . It is well known that decomposable positive maps give decomposable entanglement witnesses which take the general form W = P + Q Γ , where P, Q ≥ 0 and Γ refers to partial transposition with respect to the second subsystem, that is, Q Γ = (1 1 ⊗ T )Q. If a given witness can not be written in this form, we call it indecomposable. Of course, indecomposable EWs arise from indecomposable positive maps [4,7,8] Note that an EW is indecomposable if and only if it detects entangled states with positive partial transposes [4]. Thus, so far indecomposable EWs are concerned, it is natural to consider the optimality by requiring the witness to be finer with respect to entangled states with positive partial transposes only. This kinds of witness is said to be an indecomposable optimal EW. It is known [4,9] that W is an indecomposable optimal entanglement witness if and only if both W and W Γ are optimal entanglement witnesses.Typical examples of indecomposable optimal EW come from indecomposable positive linear maps which generate an extremal ray of the convex cone consisting of all positive linear maps. The Choi map is an example of this kind [5,9,10]. Variations of the Choi map given by the second author [11] also give rise to such maps. Some of them, parameterized by three real variables, were shown to be extremal in [12]. See the recent paper [13] for related topics. Although there are some examples of optimal EW [9,[14][15][16][17], to the best of the author's knowledge, only known examples of indecomposable optimal EWs are ones which come from extremal indecomposable positive linear maps.We consider another variations of the Choi map given in [18].

show abstract

Entanglement Detection for Multipartite States Based on Structural Physical Approximation of Partial Transposition

Zhang,

Shen,

2024

Int J Theor Phys

View full text Add to dashboard Cite

Structural approximations to positive maps and entanglement-breaking channels

Cited by 76 publications

References 59 publications

Geometric representation of two-qubit entanglement witnesses

Geometric representation of two-qubit entanglement witnesses

One-parameter family of indecomposable optimal entanglement witnesses arising from generalized Choi maps

Entanglement Detection for Multipartite States Based on Structural Physical Approximation of Partial Transposition

Contact Info

Product

Resources

About