Characterization of Equivariant Maps and Application to Entanglement Detection

Bardet, Ivan; Collins, Benoı̂t; Sapra, Gunjan

doi:10.1007/s00023-020-00941-1

Cited by 9 publications

(9 citation statements)

References 19 publications

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“…Here we illustrate our formalism in the tripartite setting, relating it to U ⊗ U ⊗ U-invariant PPT states [15] and a recently introduced class of positive maps [16]. In particular, we illustrate that the considered maps correspond to elements of C[S n ] and WBA(n, d, k), motivating us to study the maps generated by these algebras further.…”

Section: Tripartite Scenariosmentioning

confidence: 99%

Positive maps from the walled Brauer algebra

Balanzó-Juandó,

Studziński,

Huber

2024

J. Phys. A: Math. Theor.

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We present positive maps and matrix inequalities for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps that are in one-to-one correspondence with elements fromthe walled Brauer algebra. Using our formalism, these maps can be obtained in a systematic and clear way by manipulating partially transposed permutation operators under a partial trace. Additionally, these maps are reasonably easy in construction by combining an algorithmic approach with graphical calculus.

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Section: Tripartite Scenariosmentioning

confidence: 99%

Positive maps from the walled Brauer algebra

Balanzó-Juandó,

Studziński,

Huber

2024

J. Phys. A: Math. Theor.

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“…With this section we want to provide the reader of two examples to illustrate the connection of our formalism with well-known examples of existing tripartite maps, such as the ones appearing in Ref. [7], and the tripartite states of Ref. [6].…”

Section: Tripartite Systemsmentioning

confidence: 99%

“…For the values of α and β satisfying conditions 1 and 2, the operator P (α, β) is block-positive in the bipartition 15) is a witness of entanglement in the bipartition 1|23. That is, these are the points for which the map is positive according to condition 2 of [7], and for which the operator P (α, β) is not positive semidefinite.…”

Section: A the Bardet-collins-sapra Mapmentioning

confidence: 99%

“…on the entanglement of states with partial positive transpose [6] into the domain of matrix inequalities. But also, it allows to translate positive multilinear maps back into entanglement witnesses [7].…”

Section: Introductionmentioning

confidence: 99%

“…(15) is a witness of entanglement in the bipartition 1|23. That is, these are the points for which the map is positive according to condition 2 of[7], and for which the operator P (α, β) is not positive semidefinite.…”

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confidence: 99%

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Positive maps from the Walled Brauer Algebra

Balanzó-Juandó¹,

Studziński²,

Huber³

2021

Preprint

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Starting from the Walled Brauer Algebra, we present matrix inequalities in the Löwner order for variables from the positive cone. These inequalities contain partial transpose and reshuffling operations, and can be understood as positive multilinear maps. In turn, we show that these positive multilinear maps are in one-to-one correspondence with entanglement witnesses showing U ⊗(n−k) ⊗ Ū ⊗k -invariance.

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