[EN] In this Bachelor Thesis, I present some of the fundaments of entanglement detection. I introduce basic entanglement criteria namely the PPT and CCNR criteria. Entanglement witnesses are defined and used in illustrative examples. I also explain Bell inequalities and their relation with other entanglement witnesses. Lastly, in the last section of the theoretical background I introduce some collective observables and show how we can use them to detect entanglement in our experiments. Once the basics have been explained, I present a new criterion that gives us a lower bound in the amount of entanglement particles of a system. This caracterizes the quantum system from a new perspective and can be of great interest in modern multipartite entanglement experiments. Some results will be used to validate our approach, demostrating that it complements previous methods.
[EN] In this Bachelor Thesis, I present some of the fundaments of entanglement detection. I introduce basic entanglement criteria namely the PPT and CCNR criteria. Entanglement witnesses are defined and used in illustrative examples. I also explain Bell inequalities and their relation with other entanglement witnesses. Lastly, in the last section of the theoretical background I introduce some collective observables and show how we can use them to detect entanglement in our experiments. Once the basics have been explained, I present a new criterion that gives us a lower bound in the amount of entanglement particles of a system. This caracterizes the quantum system from a new perspective and can be of great interest in modern multipartite entanglement experiments. Some results will be used to validate our approach, demostrating that it complements previous methods.
This thesis deals with methods of the detection of entanglement.After recalling some facts and definitions concerning entanglement and separability, we investigate two methods of the detection of entanglement.In the first part of this thesis we consider so-called entanglement witnesses, mainly in view of the detection of multipartite entanglement.Entanglement witnesses are observables for which a negative expectation value indicates entanglement.We first present a simple method to construct these witnesses.Since witnesses are nonlocal observables, they are not easy to measure in a real experiment.However, as we will show, one can circumvent this problem by decomposing the witness into several local observables which can be measured separately.We calculate the local decompositions for several interesting witnesses for two, three and four qubits.Local decompositions can be optimized in the number of measurement settings which are needed for an experimental implementation.We present a method to prove that a given local decomposition is optimal and discuss with this the optimality of our decompositions.Then we present another method of designing witnesses which are by construction measurable with local measurements.Finally, we shortly report on experiments where some of the witnesses derived in this part have been used to detect three-and four-partite entanglement of polarized photons.The second part of this thesis deals with separability criteria which are written in terms of uncertainty relations.There are two different formulations of uncertainty relations since one can measure the uncertainty of an observable by its variance as well as by entropic quantities.We show that both formulations are useful tools for the derivation of separability criteria for finite-dimensional systems and investigate the resulting criteria.Our results in this part exhibit also some more fundamental properties of entanglement: We show how known separability criteria for infinitedimensional systems can be translated to finite-dimensional systems.Furthermore, we prove that any entropic uncertainty relation on one part of the system gives rise to a separability criterion on the composite system.
Nonlocality detection in multipartite quantum systems is of great interest. The most popular tool to detect nonlocality in quantum systems are Bell inequalities. Most of the provided constructions of multipartite Bell inequalities involve correlations between all parties which quickly becomes computationally intractable and hard to test experimentally in many-body quantum systems. Recently, J. Tura and collaborators have shown in [Science 344.6189 (2014): 1256-1258] that detection of nonlocality in multipartite systems is possible with Bell inequalities involving only one- and two- body correlation functions. However, it is uncertain how efficient these new inequalities are. One of the objectives of the present work is to address this question by numerical means. The other objective is to show that these inequalities can also serve as as device independent witnesses of different forms of entanglement such as genuine multipartite entanglement.
Despite the success of quantum mechanics in predicting the outcomes of experiments in many branches of physics, the foundations of the theory have remained subject of research and dispute. At the basis of this struggle with the theory lie the phenomena of nonlocality and entanglement. Since it was first predicted by Bell in 1964, nonlocality was not only verified in numerous experiments, but also identified as a useful resource for quantum information processing. Thus, the study of nonlocality is important both from a fundamental point of view and with respect to new applications in quantum information theory, such as secure cryptography and randomness generation. The identification of entanglement as a resource for quantum information led to a strong theoretical effort devoted to its characterisation and detection. Many of the resulting mathematical tools find application in several domains of physics. Although the only known way to create nonlocal correlations is to measure entangled quantum systems, it has been shown that entanglement and nonlocality constitute two inequivalent properties. Therefore, in the light of the success of entanglement theory, it is of interest to also devise a resource theory of nonlocality. The task of this thesis is to develop such a theoretical framework for the characterisation of nonlocality as a resource. To gain a better understanding of nonlocal correlations it will be necessary to investigate correlation scenarios that go beyond the situation originally considered by Bell. In doing so, this thesis provides new description of nonlocality that also have implications for the characterisation of quantum correlations and the detection of new forms of nonlocality. The first question we address is how nonlocality can consistently be defined in a scenario of arbitrarily many parties that may collaborate among each other. To this end we recognise which are the allowed physical operations in this situation and then define nonlocality as the resource that cannot be created by these operations. Our approach shows that the conventional definition of multipartite nonlocality, adopted by the community so far, is inconsistent with this operational definition; we further propose and analyse new models that do not suffer from these inconsistencies. Furthermore, we show that our findings have implications for the characterisation of quantum correlations. A recent approach to describe the set of quantum correlation consists in using principles inspired from information theory. By using a special instance of the models we defined earlier we show a fundamental limitation of this approach: no bipartite information principle is sufficient to single out the set of quantum correlations from the set of nonsignalling correlations. We then developed a description of nonlocality in an even more generalised scenario of several parties. Motivated by a result of Popescu we study scenarios where the parties are allowed to perform not only a single but sequences of measurements on their systems. Characterising nonlocality also in this scenario in operational terms and defining local models compatible with this definition, we show that a new form of nonlocality can be detected Lastly, we examine the problem of detecting the presence of nonlocality in a multipartite scenario when one is given only partial access to the global system. We find that one can verify that the total system must display nonlocality, even though the accessible subsystems only exhibit local correlations.
Various kinds of methods of entanglement detection and definitions of entanglement measurement in quantum systems were reviewed.For entanglement of two or more particles,some separability criterions were summarized.Their relationships with positive map and the relationships between themselves were also analyzed.Particularly,concentrations on entanglement witness are paid which is a special criterion involving its definition and construction methods.Further,its applications are also investigated from the viewpoint of experiment.Based on axiomatic hypothesis of entanglement measurement,the theoretical definitions of two-particle and multipartite entanglement,and their estimations in experiment were discussed.At last, various nonlinear separability criterions were analyzed.
We address the question of assessing the number of particles sharing genuinely nonlocal correlations in a multipartite system. While the interest in multipartite nonlocality has grown in recent years, its existence in large quantum systems is difficult to confirm experimentally. This is mostly due to the inadequacy of standard multipartite Bell inequalities to many-body systems: Such inequalities usually rely on expectation values involving many parties, usually all, and require individual addressing of each party. In addition, known Bell inequalities for genuine nonlocality are composed of a number of expectation values that scales exponentially with the number of observers, which makes such inequalities impractical from the experimental point of view. In a recent work [Tura et al., Science 344, 1256 (2014)], it was shown that it is possible to detect nonlocality in multipartite systems using Bell inequalities with only two-body correlators. This opened the way for the detection of Bell correlations with trusted collective measurements through Bell correlation witnesses [Schmied et al., Science 352, 441 (2016)]. These witnesses were recently tested experimentally in many-body systems such as Bose-Einstein condensate or thermal ensembles, hence demonstrating the presence of Bell correlations with assumptions on the statistics. Here, we address the problem of detecting nonlocality depth, a notion that quantifies the number of particles sharing nonlocal correlation in a multipartite system. We introduce a general framework allowing us to derive Bell-like inequalities for nonlocality depth from symmetric two-body correlators. We characterize all such Bell-like inequalities for a finite number of parties and we show that they reveal Bell correlation depth $k\ensuremath{\le}6$ in arbitrarily large systems. We then show how Bell correlation depth can be estimated using quantities that are within reach in current experiments. On one hand, we use the standard multipartite Bell inequalities such the Mermin and Svetlichny ones to derive Bell correlations witnesses of any depth that involves only two collective measurements, one of which being the parity measurement. On the other hand, we show that our two-body Bell inequalities can be turned into witnesses of depth $k\ensuremath{\le}6$ that require measuring total spin components in certain directions. Interestingly, such a witness is violated by existing data from an ensemble of 480 atoms.
In this thesis, two topics are studied: mathematical inequalities and non-linear quantum entanglement witnesses. First, various inequalities, like the Cauchy-Schwarz inequality (on finite dimensional vector spaces) and Jensen's inequality, along with their extensions and generalisations, are proved and discussed. The intimate relationship between these inequalities is studied. Because this thesis was restricted to finite dimensional vector spaces, the consequences of generalising the results to infinite dimensional vector spaces are finally determined. Secondly, the topic of entanglement detection is discussed - specifically, non-linear entanglement witnesses are considered. A bipartite and multipartite entanglement criterion based on the previously discussed inequalities are introduced and assessed extensively by considering their optimality, how they relate to other criteria as well as their limitations.