LazySets.jl: Scalable Symbolic-Numeric Set Computations$^*$

doi:10.21105/jcon.00097

Cited by 10 publications

(5 citation statements)

References 20 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this work, the implementation of Ref. 57 , was used to check an unknown state for separability based on the convex hull of a given set of separable states. Using known separable states in as vertices, they form a polytope which approximates the convex set of all separable states in .…”

Section: Methodsmentioning

confidence: 99%

Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts

Popp

Hiesmayr

2023

Sci Rep

View full text Add to dashboard Cite

We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension $$d \ge 3$$ d ≥ 3 , a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with special algebraic and geometric structures and applications in quantum information processing tasks in different dimensions. Extending analytical and numerical methods and results for Bell diagonal qutrits ($$d=3$$ d = 3 ), we successfully classify more than $$75\%$$ 75 % of representative Bell diagonal PPT states for $$d=4$$ d = 4 . Via those representative states we are able to estimate the volumes of separable and bound entangled states among PPT ququarts ($$d=4$$ d = 4 ). We find that at least $$75.7\%$$ 75.7 % of all PPT states are separable, $$1.7\%$$ 1.7 % bound entangled and for $$22.6\%$$ 22.6 % it remains unclear whether they are separable or bound entangled. Comparing the structure of bound entangled states and their detectors, we find considerable differences in the detection capabilities for different dimensions and relate those to differences of the Euclidean geometry for qutrits ($$d=3$$ d = 3 ) and ququarts ($$d=4$$ d = 4 ). Finally, using a detailed visual analysis of the set of separable and bound entangled Bell diagonal states in both dimensions, qualitative observations are made that allow to better distinguish bound entangled from separable states.

show abstract

Section: Methodsmentioning

confidence: 99%

Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts

Popp

Hiesmayr

2023

Sci Rep

View full text Add to dashboard Cite

show abstract

“…RangeEnclosures currently only supports functions with univariate range. To represent multivariate ranges as convex and non-convex sets, we plan to use LazySets.jl [6].…”

Section: Future Applicationsmentioning

confidence: 99%

RangeEnclosures.jl: A framework to bound function ranges

2024

JCON

View full text Add to dashboard Cite

Computing the range of a function is needed in several application domains. During the past decades, several algorithms to compute or approximate the range have been developed, each with its own merits and limitations. Motivated by this, we introduce RangeEnclosures.jl, a unified framework to bound the range of univariate and multivariate functions. In addition to its own algorithms, the package allows to easily integrate third-party algorithms, offering a unified interface that can be used across different domains and allows to easily benchmark different approaches.

show abstract

“…Leveraging geometric properties of a certain class of mixed Bell states that are related by Weyl transformations, BellDiagonalQudits combines known analytical results by Baumgartner et al (2007) and numerical methods for quantum state representation and analysis. It leverages and depends on the Julia package QuantumInformation, the convex sets of LazySets (Forets & Schilling, 2021) and the optimization methods of Optim (Mogensen & Riseth, 2018).…”

Section: Statement Of Needmentioning

confidence: 99%

BellDiagonalQudits: A package for entanglement analyses of mixed maximally entangled qudits

Popp¹

2023

JOSS

View full text Add to dashboard Cite

LazySets.jl: Scalable Symbolic-Numeric Set Computations$^*$

Cited by 10 publications

References 20 publications

Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts

Comparing bound entanglement of bell diagonal pairs of qutrits and ququarts

RangeEnclosures.jl: A framework to bound function ranges

BellDiagonalQudits: A package for entanglement analyses of mixed maximally entangled qudits

Contact Info

Product

Resources

About