Optimal Bounds on the Positivity of a Matrix from a Few Moments

Cuevas, Gemma De las; Fritz, Tobias; Netzer, Tim

doi:10.1007/s00220-020-03720-5

Cited by 5 publications

(4 citation statements)

References 22 publications

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“…We start by defining entanglement distillation on * C. Definition 6 (Entanglement distillation on * C). A hyperquantum state ρ is distillable if its shadow sh(ρ) is distillable 5 .…”

Section: Npt Bound Entangled Hyperquantum Statesmentioning

confidence: 99%

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Halos and undecidability of tensor stable positive maps

Mirte¹,

Netzer²,

Cuevas³

2022

J. Phys. A: Math. Theor.

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A map P is tensor stable positive (tsp) if P⊗n is positive for all n, and essential tsp if it is not completely positive or completely co-positive. Are there essential tsp maps? Here we prove that there exist essential tsp maps on the hypercomplex numbers. It follows that there exist bound entangled states with a negative partial transpose (NPT) on the hypercomplex, that is, there exists NPT bound entanglement in the halo of quantum states. We also prove that tensor stable positivity on the matrix multiplication tensor is undecidable, and conjecture that tensor stable positivity is undecidable. Proving this conjecture would imply existence of essential tsp maps, and hence of NPT bound entangled states.

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“…We start by defining entanglement distillation on * C. Definition 6 (Entanglement distillation on * C). A hyperquantum state ρ is distillable if its shadow sh(ρ) is distillable 5 .…”

Section: Npt Bound Entangled Hyperquantum Statesmentioning

confidence: 99%

“…How does the tensor product interact with the elements of the convex cones mentioned above? This is a very rich problem, as the global positivity interacts with the local positivity in highly nontrivial ways [3][4][5]. Here we consider the cone of positive maps, and study the interaction of its elements with the tensor product ⊗.…”

Section: Introductionmentioning

confidence: 99%

Halos and undecidability of tensor stable positive maps

Mirte¹,

Netzer²,

Cuevas³

2022

J. Phys. A: Math. Theor.

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“…Remarkably, an alternative sum-of-squares method has already been used in Ref. [31] for bounding the negative eigenvalues from moments. Here, instead of using the approximation method for general polynomial optimization, we propose an exact method for solving Eq.…”

Section: Optimal Solution To the Pt-moment Problemmentioning

confidence: 99%

Optimal Entanglement Certification from Moments of the Partial Transpose

Yu,

Imai,

Gühne

2021

Preprint

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“…How does the tensor product interact with the elements of the convex cones mentioned above? This is a very rich problem, as the global positivity interacts with the local positivity in highly nontrivial ways [7,10,9]. Here we consider the cone of positive maps, and study the interaction of its elements with the tensor product ⊗.…”

Section: Introductionmentioning

confidence: 99%

Halos and undecidability of tensor stable positive maps

van der Eyden,

Netzer,

Cuevas

2021

Preprint

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A map P is tensor stable positive (tsp) if P ⊗n is positive for all n, and essential tsp if it is not completely positive or completely co-positive. Are there essential tsp maps? Here we prove that there exist essential tsp maps on the hypercomplex numbers. It follows that there exist bound entangled states with a negative partial transpose (NPT) on the hypercomplex, that is, there exists NPT bound entanglement in the halo of quantum states. We also prove that tensor stable positivity on the matrix multiplication tensor is undecidable, and conjecture that tensor stable positivity is undecidable. Proving this conjecture would imply existence of essential tsp maps, and hence of NPT bound entangled states.

show abstract

Optimal Bounds on the Positivity of a Matrix from a Few Moments

Cited by 5 publications

References 22 publications

Halos and undecidability of tensor stable positive maps

Halos and undecidability of tensor stable positive maps

Optimal Entanglement Certification from Moments of the Partial Transpose

Halos and undecidability of tensor stable positive maps

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