Extremal extensions of entanglement witnesses: Finding new bound entangled states

Sengupta, Ritabrata; Arvind, .

doi:10.1103/physreva.84.032328

Cited by 14 publications

(18 citation statements)

References 36 publications

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“…From the positivity of φ k (X[x 1 , x 2 , x 3 ]), we can correlate the variables b k 's in (6) with the variables a ℓ 's in (5). 9 be the variables in (6), and a 1 , a 2 , a 3 be the variables in (5). Then we have…”

Section: Extremality Of the Choi Mapmentioning

confidence: 99%

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Notes on extremality of the Choi map

2013

Linear Algebra and its Applications

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Abstract. It is widely believed that the Choi map generates an extremal ray in the cone P(M 3 ) of all positive linear maps between C * -algebra M 3 of all n × n matrices over the complex field. But the only proven fact is that the Choi map generates the extremal ray in the cone of all positive linear map preserving all real symmetric 3 × 3 matrices. In this note, we show that the Choi map is indeed extremal in the cone P(M 3 ). We also clarify some misclaims about the correspondence between positive semi-definite biquadratic real forms and postive linear maps, and discuss possible positive linear maps which coincide with the Choi map on symmetric matrices.

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Section: Extremality Of the Choi Mapmentioning

confidence: 99%

“…where X = (x ij ) ∈ M 3 . It is widely believed [8,7,6,9] that the Choi map generates an extremal ray in P(M 3 ). It is not as trivial as one may think.…”

Section: Introductionmentioning

confidence: 99%

Notes on extremality of the Choi map

2013

Linear Algebra and its Applications

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“…Studies of entanglement have led to a well developed mathematical theory of entanglement where positive maps (P) which are not completely positive (CP) [7][8][9][10][11][12][13][14] and unextendable product bases (UPB) [15][16][17] play an important role [18][19][20][21]. These mathematical advances have led to the discovery of bound entangled states [22]: states from which one cannot distill EPR pairs although they are still provably non-separable.…”

Section: Introductionmentioning

confidence: 99%

Measurement-based local quantum filters and their ability to transform quantum entanglement

2017

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We introduce local filters as a means to detect the entanglement of bound entangled states which do not yield to detection by witnesses based on positive (P) maps which are not completely positive (CP). We demonstrate how such non-detectable bound entangled states can be locally filtered into detectable bound entangled states. Specifically, we show that a bound entangled state in the orthogonal complement of the unextendible product bases (UPB), can be locally filtered into a another bound entangled state that is detectable by the Choi map. We reinterpret these filters as local measurements on locally extended Hilbert spaces. We give explicit constructions of a measurementbased implementation of these filters for 2 ⊗ 2 and 3 ⊗ 3 systems. This provides us with a physical mechanism to implement such local filters.

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“…See ( [12], [13], [14], [15]) for more details. One may observe this tendency in [11], [12], [14], [15], [17], [18], [19], [20], [21], [22], [23], [24], [26], [36], [38], [45], [47], [56], [57], [62], [66], [73], [74], [76], [77], [78], [89], for instance. Such maps have been tested for more properties like separability, entanglement and Schmidt numbers and used to construct Entanglement witnesses, Entanglement breaking or partially entanglement breaking channels by some of them.…”

Section: H Separability and Entanglement Of Generalized Choi Maps mentioning

confidence: 99%

Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

Singh

2015

ELA

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Abstract. Power symmetric matrices defined and studied by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studied using Choi-Jamiolkowsky matrix of such maps and the famous Horodecki's criteria (1996). A Trichotomy Theorem has been proved, and examples have been given that depend mainly on generalized Choi maps and clearly distinguish the levels of entanglement breaking.

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Extremal extensions of entanglement witnesses: Finding new bound entangled states

Cited by 14 publications

References 36 publications

Notes on extremality of the Choi map

Notes on extremality of the Choi map

Measurement-based local quantum filters and their ability to transform quantum entanglement

Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

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