Sufficient separability criteria and linear maps

Lewenstein, Maciej; Augusiak, Remigiusz; Chruściński, Dariusz; Rana, Swapan; Samsonowicz, Jan

doi:10.1103/physreva.93.042335

Cited by 15 publications

(18 citation statements)

References 41 publications

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“…Another interesting research program may be devoted to further analysis of more general construction of maps in terms of MUBs in the spirit of [60].…”

Section: Discussionmentioning

confidence: 99%

Entanglement witnesses from mutually unbiased bases

2018

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We provide a class of entanglement witnesses constructed in terms of Mutually Unbiased Bases (MUBs). This construction reproduces many well-known examples like the celebrated reduction map and Choi map together with its generalizations. We illustrated our construction by the detailed analysis of the 3-dimensional case: in this case, one obtains a family of entanglement witnesses parameterized by an L-dimensional torus (L = 2, 3, 4 being a number of MUBs used in the construction).

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“…Another interesting research program may be devoted to further analysis of more general construction of maps in terms of MUBs in the spirit of [60].…”

Section: Discussionmentioning

confidence: 99%

Entanglement witnesses from mutually unbiased bases

2018

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“…Our method to work with matrix contractions and positive maps rests on generalizing the swap identities from the previous section [Eq. (10)]. We formalize the translation of permutations into matrix products.…”

Section: How To Translate a Permutation Into A Matrix Multiplicationmentioning

confidence: 99%

“…Linear maps from the positive cone to itself are known as positive maps. These have applications in the study of quantum dynamics and entanglement [9][10][11][12][13][14][15][16], and can be understood as matrix inequalities for the positive cone.…”

Section: Introductionmentioning

confidence: 99%

Positive maps and trace polynomials from the symmetric group

Huber

2020

Preprint

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This article develops a systematic method to obtain operator inequalities in several matrix variables. These take the form of polynomial-like expressions that involve matrix monomials X α 1 • • • X α r and their traces tr(X α 1 • • • X α r ). Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps, and construct swap polynomials and other matrix identities on tensor product spaces. Further connections to concepts from invariant theory, polynomial identity rings, and multipartite entanglement are given.

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“…The key ingredient in our investigation is the operator obtained by applying the so-called universal state inversion [21,22], see also Refs. [23,24,25,26]. Consider a d-dimensional Hilbert space H and B(H), the set of bounded positive semidefinite operators on H. Horodecki et al [21] defined the operator…”

Section: Universal State Inversion 1definitionmentioning

confidence: 99%

Distribution of entanglement and correlations in all finite dimensions

Eltschka

Siewert

2018

Quantum

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The physics of a many-particle system is determined by the correlations in its quantum state. Therefore, analyzing these correlations is the foremost task of manybody physics. Any 'a priori' constraint for the properties of the global vs. the local states-the so-called marginals-would help in order to narrow down the wealth of possible solutions for a given many-body problem, however, little is known about such constraints. We derive an equality for correlation-related quantities of any multipartite quantum system composed of finite-dimensional local parties. This relation defines a necessary condition for the compatibility of the marginal properties with those of the joint state. While the equality holds both for pure and mixed states, the pure-state version containing only entanglement measures represents a fully general monogamy relation for entanglement. These findings have interesting implications in terms of conservation laws for correlations, and also with respect to topology.Correlations between different parts of manyparticle quantum systems are manifestly different from their classical counterparts. The hallmark of the latter is that, if the global state is completely known, so are the local states, that is, the individual states of each particle. For quantum states this may be different: Even if the global state is known with certainty, there may be no information whatsoever regarding the local states such as, for example, in maximally entangled states of two d-dimensional systems [1,2].Often, different degrees of departure from the classical behavior are distinguished, namely entanglement, steerability, and nonlocality [3]. While all these correlation types are based on en-

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Sufficient separability criteria and linear maps

Cited by 15 publications

References 41 publications

Entanglement witnesses from mutually unbiased bases

Entanglement witnesses from mutually unbiased bases

Positive maps and trace polynomials from the symmetric group

Distribution of entanglement and correlations in all finite dimensions

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