Relaxations of separability in multipartite systems: Semidefinite programs, witnesses and volumes

Lancien, Cécilia; Gühne, Otfried; Sengupta, Ritabrata; Huber, Marcus

doi:10.1088/1751-8113/48/50/505302

Cited by 32 publications

(35 citation statements)

References 28 publications

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“…This insight has led to the concept of PPT mixers [264,265], yielding effective numerical tools for low dimensions. At the same time, this connection can be used to effectively 'lift' bipartite witnesses for multipartite usage [266,267], and to obtain generalisations to maps which are positive on biseparable states [268]. GME witnesses.…”

Section: Contemporary Challenges: Multipartite Entanglementmentioning

confidence: 99%

Entanglement certification from theory to experiment

et al. 2018

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Entanglement is an important resource that allows quantum technologies to go beyond the classically possible. There are many ways quantum systems can be entangled, ranging from the archetypal two-qubit case to more exotic scenarios of entanglement in high dimensions or between many parties. Consequently, a plethora of entanglement quantifiers and classifiers exist, corresponding to different operational paradigms and mathematical techniques.However, for most quantum systems, exactly quantifying the amount of entanglement is extremely demanding, if at all possible. This is further exacerbated by the difficulty of experimentally controlling and measuring complex quantum states. Consequently, there are various approaches for experimentally detecting and certifying entanglement when exact quantification is not an option, with a particular focus on practically implementable methods and resource efficiency. The applicability and performance of these methods strongly depends on the assumptions one is willing to make regarding the involved quantum states and measurements, in short, on the available prior information about the quantum system. In this review we discuss the most commonly used paradigmatic quantifiers of entanglement. For these, we survey state-of-the-art detection and certification methods, including their respective underlying assumptions, from both a theoretical and experimental point of view.In the early twentieth century, the phenomenon of quantum entanglement rose to prominence as a central feature of the famous thought experiment by Einstein, Podolsky, and Rosen [1]. Initially disregarded as a mathematical artefact that showcases the incompleteness of quantum theory, the properties of entanglement were largely ignored until 1964, when John Bell famously proposed an experimentally testable inequality able to distinguish between the predictions of quantum mechanics and those of any local-realistic theory [2]. With the advent of the first experimental tests [3], spearheaded by , emerged the realisation that entanglement constitutes a resource for information processing *

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Section: Contemporary Challenges: Multipartite Entanglementmentioning

confidence: 99%

Entanglement certification from theory to experiment

et al. 2018

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“…Finally, let us note that in the relaxation one could use optimization over states, which remain positive under a different positive map, not necessarily the transposition; this could for example be the Breuer-Hall map [47,48]. To make a better approximation of the set of (bi)separable states, this could also be performed over states which stay positive under a set of positive maps [49]. All these relaxations are SDPs.…”

Section: Lower Bounds On the Entanglement Of A Subspace In Terms Omentioning

confidence: 99%

Entanglement of genuinely entangled subspaces and states: Exact, approximate, and numerical results

Demianowicz

Augusiak

2019

Phys. Rev. A

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Genuinely entangled subspaces (GESs) are those subspaces of multipartite Hilbert spaces that consist only of genuinely multiparty entangled (GME) pure states. They are natural generalizations of the well-known notion of completely entangled subspaces (CESs), which by definition are void of fully product vectors. Entangled subspaces are an important tool of quantum information theory as they directly lead to construtions of entangled states, since any state supported on such a subspace is automatically entangled. Moreover, they have also proven useful in the area of quantum error correction (QEC). In our recent contribution [M. Demianowicz and R. Augusiak, Phys. Rev. A 98, 012313 (2018)], we have studied the notion of a GES qualitatively in relation to socalled nonorthogonal unextendible product bases and provided a few novel constructions of such subspaces. The main aim of the present work is to perform a quantitative study of the entanglement properties of GESs. First, we show how one can attempt to compute analytically the subspace entanglement, defined as the entanglement of the least entangled vector from the subspace, of a GES and illustrate our method applying it to a new class of GESs. Second, we show that certain semi-definite programming (SDP) relaxations can be exploited to estimate the entanglement of a GES and apply this observation to few classes of GESs revealing that in many cases the method provides the exact results. Finally, we study the entanglement of certain states supported on GESs, which is compared to the obtained values of the entanglement of the corresponing subspaces, and find the white noise robustness of several GESs. In our study we use the (generalized) geometric measure as the quantifier of entanglement.

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“…It should be noticed that, as anticipated, in such an average-state case E N cannot be interpreted as a quantifier of genuine multipartite entanglement. Indeed, the revelation of multipartite entanglement in general multiparty mixed states requires a more refined approach (see [16] for a recent assessment of this point and the provision of useful criteria). Nevertheless, this figure of merit is still very useful for our analysis, as it provides valuable information on the average amount of bipartite entanglement within the statistically average stage of the network, and we will thus make further use of E N in the remainder of this work.…”

Section: Analysis Of the Entanglement Structure In A Random Four-qubimentioning

confidence: 99%

Structure of Multipartite Entanglement in Random Cluster-Like Photonic Systems

Ciampini

Mataloni

Paternostro

2017

Entropy

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Quantum networks are natural scenarios for the communication of information among distributed parties, and the arena of promising schemes for distributed quantum computation. Measurement-based quantum computing is a prominent example of how quantum networking, embodied by the generation of a special class of multipartite states called cluster states, can be used to achieve a powerful paradigm for quantum information processing. Here we analyze randomly generated cluster states in order to address the emergence of correlations as a function of the density of edges in a given underlying graph. We find that the most widespread multipartite entanglement does not correspond to the highest amount of edges in the cluster. We extend the analysis to higher dimensions, finding similar results, which suggest the establishment of small world structures in the entanglement sharing of randomised cluster states, which can be exploited in engineering more efficient quantum information carriers.

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Relaxations of separability in multipartite systems: Semidefinite programs, witnesses and volumes

Cited by 32 publications

References 28 publications

Entanglement certification from theory to experiment

Entanglement certification from theory to experiment

Entanglement of genuinely entangled subspaces and states: Exact, approximate, and numerical results

Structure of Multipartite Entanglement in Random Cluster-Like Photonic Systems

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