A class of exposed indecomposable positive maps

Sarbicki, Gniewomir; Chruściński, Dariusz

doi:10.1088/1751-8113/46/1/015306

Cited by 19 publications

(21 citation statements)

References 45 publications

(111 reference statements)

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“…For N = 2 it reproduces the Robertson map (8.2). Recently, it was shown [111], that Φ 2N is exposed and hence extremal. The second generalization uses well known representation of R 2…”

Section: Generalizations Of the Robertson Mapmentioning

confidence: 99%

Entanglement witnesses: construction, analysis and classification

Chruściński¹,

Sarbicki²

2014

J. Phys. A: Math. Theor.

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185

182

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From the physical point of view entanglement witnesses define a universal tool for analysis and classification of quantum entangled states. From the mathematical point of view they provide highly nontrivial generalization of positive operators and they find elegant correspondence with the theory of positive maps in matrix algebras. We concentrate on theoretical analysis of various important notions like (in)decomposability, atomicity, optimality, extremality and exposedness. Several methods of construction are provided as well. Our discussion is illustrated by many examples enabling the reader to see the intricate structure of these objects. It is shown that the theory of entanglement witnesses finds elegant geometric formulation in terms of convex cones and related geometric structures.

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“…For N = 2 it reproduces the Robertson map (8.2). Recently, it was shown [111], that Φ 2N is exposed and hence extremal. The second generalization uses well known representation of R 2…”

Section: Generalizations Of the Robertson Mapmentioning

confidence: 99%

Entanglement witnesses: construction, analysis and classification

Chruściński¹,

Sarbicki²

2014

J. Phys. A: Math. Theor.

Self Cite

185

182

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“…See Section 8 for a discussion of exposed faces. As a related example we have looked at a witness Ω Z belonging to a new class of witnesses in dimension N × (2K) introduced by Zwolak and Chruściński [28,55,56]. We take N = 2, K = 1, and |z 12 | = 1 in their notation.…”

Section: The Robertson Witnessmentioning

confidence: 99%

“…Positive linear maps, and the extremal positive maps, were studied already in 1963 by Størmer in the context of partially ordered vector spaces and C * algebras [14,15]. Since then the extremal positive maps and extremal witnesses have only received scattered attention [21][22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Extremal entanglement witnesses

Hansen

Hauge

Myrheim

et al. 2015

Int. J. Quantum Inform.

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We present a study of extremal entanglement witnesses on a bipartite composite quantum system. We define the cone of witnesses as the dual of the set of separable density matrices, thus Tr Ωρ ≥ 0 when Ω is a witness and ρ is a pure product state, ρ = ψψ † with ψ = φ ⊗ χ. The set of witnesses of unit trace is a compact convex set, uniquely defined by its extremal points. The expectation value f (φ, χ) = Tr Ωρ as a function of vectors φ and χ is a positive semidefinite biquadratic form. Every zero of f (φ, χ) imposes strong real-linear constraints on f and Ω. The real and symmetric Hessian matrix at the zero must be positive semidefinite. Its eigenvectors with zero eigenvalue, if such exist, we call Hessian zeros. A zero of f (φ, χ) is quadratic if it has no Hessian zeros, otherwise it is quartic. We call a witness quadratic if it has only quadratic zeros, and quartic if it has at least one quartic zero. A main result we prove is that a witness is extremal if and only if no other witness has the same, or a larger, set of zeros and Hessian zeros. A quadratic extremal witness has a minimum number of isolated zeros depending on dimensions. If a witness is not extremal, then the constraints defined by its zeros and Hessian zeros determine all directions in which we may search for witnesses having more zeros or Hessian zeros. A finite number of iterated searches in random directions, by numerical methods, lead to an extremal witness which is nearly always quadratic and has the minimum number of zeros. We discuss briefly some topics related to extremal witnesses, in particular the relation between the facial structures of the dual sets of witnesses and separable states. We discuss the relation between extremality and optimality of witnesses, and a conjecture of separability of the so called structural physical approximation (SPA) of an optimal witness. Finally, we discuss how to treat the entanglement witnesses on a complex Hilbert space as a subset of the witnesses on a real Hilbert space.

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“…An older construction of Terpstra [36] yields only special isolated examples which are in most cases not extremal. Most of the examples that can be found in the literature are obtained as generalizations of Choi's example, see [6,10,15,19,24,28,30,32,35].…”

Section: Introductionmentioning

confidence: 99%

New examples of extremal positive linear maps

Buckley

Šivic

2020

Linear Algebra and its Applications

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New families of nonnegative biquadratic forms that have 8, 9 or 10 real zeros in P 2 × P 2 are constructed. These are the first examples with 8, 9 or 10 real zeros. It is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros; our examples show that the upper bound is obtained. Such biquadratic forms define positive linear maps that are not completely positive. Our constructions are explicit, and moreover we are able to determine which of the examples are extremal.

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A class of exposed indecomposable positive maps

Cited by 19 publications

References 45 publications

Entanglement witnesses: construction, analysis and classification

Entanglement witnesses: construction, analysis and classification

Extremal entanglement witnesses

New examples of extremal positive linear maps

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