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Causal reasoning is essential to science, yet quantum theory challenges it. Quantum correlations violating Bell inequalities defy satisfactory causal explanations within the framework of classical causal models. What is more, a theory encompassing quantum systems and gravity is expected to allow causally nonseparable processes featuring operations in indefinite causal order, defying that events be causally ordered at all. The first challenge has been addressed through the recent development of intrinsically quantum causal models, allowing causal explanations of quantum processes -- provided they admit a definite causal order, i.e. have an acyclic causal structure. This work addresses causally nonseparable processes and offers a causal perspective on them through extending quantum causal models to cyclic causal structures. Among other applications of the approach, it is shown that all unitarily extendible bipartite processes are causally separable and that for unitary processes, causal nonseparability and cyclicity of their causal structure are equivalent., 21 pages, 10 figures. Close to published version

One of the most important algorithmic applications of quantum walks is to solve spatial search problems. A widely used quantum algorithm for this problem, introduced by Childs and Goldstone [Phys. Rev. A 70, 022314 (2004)], finds a marked node on a graph of $n$ nodes via a continuous-time quantum walk. This algorithm is said to be optimal if it can find any of the nodes in $O(\sqrt{n})$ time. However, given a graph, no general conditions for the optimality of the algorithm are known and previous works demonstrating optimal quantum search for certain graphs required an instance-specific analysis. In fact, the demonstration of the necessary and sufficient conditions that a graph must fulfill for quantum search to be optimal has been a long-standing open problem. In this work we make significant progress towards solving this problem. We derive general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled. Our predictions are valid, for example, for (normalized) Hamiltonians whose spectral gap is considerably larger than ${n}^{\ensuremath{-}1/2}$. This allows us to derive necessary and sufficient conditions for optimal quantum search in this regime, as well as provide examples of graphs where quantum search is suboptimal. In addition, by extending this analysis, we are also able to show the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitioned into clusters. Our results imply that, to the best of our knowledge, all prior results analytically demonstrating the optimality of this algorithm for specific graphs can be recovered from our general results.

The problem of sampling from the stationary distribution of a Markov chain finds widespread applications in a variety of fields. The time required for a Markov chain to converge to its stationary distribution is known as the classical mixing time. In this article, we deal with analog quantum algorithms for mixing. First, we provide an analog quantum algorithm that given a Markov chain, allows us to sample from its stationary distribution in a time that scales as the sum of the square root of the classical mixing time and the square root of the classical hitting time. Our algorithm makes use of the framework of interpolated quantum walks and relies on Hamiltonian evolution in conjunction with von Neumann measurements. There also exists a different notion for quantum mixing: the problem of sampling from the limiting distribution of quantum walks, defined in a time-averaged sense. In this scenario, the quantum mixing time is defined as the time required to sample from a distribution that is close to this limiting distribution. Recently we provided an upper bound on the quantum mixing time for Erd\"os-Renyi random graphs [Phys. Rev. Lett. 124, 050501 (2020)]. Here, we also extend and expand upon our findings therein. Namely, we provide an intuitive understanding of the state-of-the-art random matrix theory tools used to derive our results. In particular, for our analysis we require information about macroscopic, mesoscopic and microscopic statistics of eigenvalues of random matrices which we highlight here. Furthermore, we provide numerical simulations that corroborate our analytical findings and extend this notion of mixing from simple graphs to any ergodic, reversible, Markov chain., Comment: Fixes some errors present in the previous versions

One of the most important algorithmic applications of quantum walks is to solve spatial search problems. A widely used quantum algorithm for this problem, introduced by Childs and Goldstone [Phys. Rev. A 70, 022314 (2004)], finds a marked node on a graph of $n$ nodes via a continuous-time quantum walk. This algorithm is said to be optimal if it can find any of the nodes in $O(\sqrt{n})$ time. However, given a graph, no general conditions for the optimality of the algorithm are known and previous works demonstrating optimal quantum search for certain graphs required an instance-specific analysis. In fact, the demonstration of necessary and sufficient conditions a graph must fulfill for quantum search to be optimal has been a long-standing open problem. In this work, we make significant progress towards solving this problem. We derive general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled. Our predictions are valid, for example, for (normalized) Hamiltonians whose spectral gap is considerably larger than $n^{-1/2}$. This allows us to derive necessary and sufficient conditions for optimal quantum search in this regime, as well as provide new examples of graphs where quantum search is sub-optimal. In addition, by extending this analysis, we are also able to show the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitioned into clusters. Our results imply that, to the best of our knowledge, all prior results analytically demonstrating the optimality of this algorithm for specific graphs can be recovered from our general results., This manuscript is a significantly expanded version of the sections on the performance of the Childs and Goldstone algorithm for spatial search presented in arXiv:1807.05957v1 and arXiv:1807.05957v2, along with several additional results. Comments are welcome

Contextuality is one way of capturing the non-classicality of quantum theory. The contextual nature of a theory is often witnessed via the violation of non-contextuality inequalities---certain linear inequalities involving probabilities of measurement events. Using the exclusivity graph approach (one of the two main graph theoretic approaches for studying contextuality), it was shown [PRA 88, 032104 (2013); Annals of mathematics, 51-299 (2006)] that a necessary and sufficient condition for witnessing contextuality is the presence of an odd number of events (greater than three) which are either cyclically or anti-cyclically exclusive. Thus, the non-contextuality inequalities whose underlying exclusivity structure is as stated, either cyclic or anti-cyclic, are fundamental to quantum theory. We show that there is a unique non-contextuality inequality for each non-trivial cycle and anti-cycle. In addition to the foundational interest, we expect this to aid the understanding of contextuality as a resource to quantum computing and its applications to local self-testing. Ministry of Education (MOE) National Research Foundation (NRF) Published version A.S.A. and J.R. acknowledge financial support from the Belgian Fonds de la Recherche Scientifique (FNRS) under Grants No. F.4515.16 (QUICTIME) and No. R.50.05.18.F (QuantAlgo). A.S.A. further acknowledges the FNRS for support through Grant No. F3/5/5–MCF/XH/FC– 16749 FRIA. K.B. acknowledges the Centre for Quantum Technologies (CQT) Graduate Scholarship. K.B. and L.C.K. are grateful to the National Research Foundation and the Ministry of Education, Singapore for financial support.

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In the context of multiparameter quantum estimation theory, we investigate the construction of linear schemes in order to infer two classical parameters that are encoded in the quadratures of two quantum coherent states. The optimality of the scheme built on two phase-conjugate coherent states is proven with the saturation of the quantum Cram&eacute, r&ndash, Rao bound under some global energy constraint. In a more general setting, we consider and analyze a variety of n-mode schemes that can be used to encode n classical parameters into n quantum coherent states and then estimate all parameters optimally and simultaneously.

We investigate weak coin flipping, a fundamental cryptographic primitive where two distrustful parties need to remotely establish a shared random bit. A cheating player can try to bias the output bit towards a preferred value. For weak coin flipping the players have known opposite preferred values. A weak coin-flipping protocol has a bias ϵ if neither player can force the outcome towards their preferred value with probability more than 12 + ϵ. While it is known that all classical protocols have ϵ = 12 ,Mochon showed in 2007 that quantumly weak coin flipping can be achieved with arbitrarily small bias (near perfect) but the former best known explicit protocol has bias 1/6 (also due to Mochon, 2005). We propose a framework to construct new explicit protocols achieving biases below 1/6. In particular, we construct explicit unitaries for protocols with bias down to 1/10. To go lower, we introduce what we call the Elliptic Monotone Align (EMA) algorithm which, together with the framework, allows us to construct protocols with arbitrarily small biases., SCOPUS: cp.p, info:eu-repo/semantics/published

info:eu-repo/semantics/published

Hall et al. [Phys. Rev. X 4 (2014) 041013] recently proposed that quantum theory can be understood as the continuum limit of a deterministic theory in which there is a large, but finite, number of classical “worlds.” A resulting Gaussian limit theorem for particle positions in the ground state, agreeing with quantum theory, was conjectured in Hall et al. [Phys. Rev. X 4 (2014) 041013] and proven by McKeague and Levin [Ann. Appl. Probab. 26 (2016) 2540–2555] using Stein’s method. In this article we show how quantum position probability densities for higher energy levels beyond the ground state may arise as distributional fixed points in a new generalization of Stein’s method. These are then used to obtain a rate of distributional convergence for conjectured particle positions in the first energy level above the ground state to the (two-sided) Maxwell distribution; new techniques must be developed for this setting where the usual “density approach” Stein solution (see Chatterjee and Shao [Ann. Appl. Probab. 21 (2011) 464–483] has a singularity.

Weak coin flipping (WCF) is a fundamental cryptographic primitive for two-party secure computation, where two distrustful parties need to remotely establish a shared random bit whilst having opposite preferred outcomes. It is the strongest known primitive with arbitrarily close to perfect security quantumly while classically, its security is completely compromised (unless one makes further assumptions, such as computational hardness). A WCF protocol is said to have bias $\epsilon$ if neither party can force their preferred outcome with probability greater than $1/2+\epsilon$. Classical WCF protocols are shown to have bias $1/2$, i.e., a cheating party can always force their preferred outcome. On the other hand, there exist quantum WCF protocols with arbitrarily small bias, as Mochon showed in his seminal work in 2007 [arXiv:0711.4114]. In particular, he proved the existence of a family of WCF protocols approaching bias $\epsilon (k)=1/(4k+2)$ for arbitrarily large $k$ and proposed a protocol with bias $1/6$. Last year, Arora, Roland and Weis presented a protocol with bias $1/10$ and to go below this bias, they designed an algorithm that numerically constructs unitary matrices corresponding to WCF protocols with arbitrarily small bias [STOC'19, p.205-216]. In this work, we present new techniques which yield a fully analytical construction of WCF protocols with bias arbitrarily close to zero, thus achieving a solution that has been missing for more than a decade. Furthermore, our new techniques lead to a simplified proof of existence of WCF protocols by circumventing the non-constructive part of Mochon's proof. As an example, we illustrate the construction of a WCF protocol with bias $1/14$., Comment: 13 + 14 pages, 3 figures; In v2, we give a new, simpler and shorter solution. For further details and updates see https://atulsingharora.github.io/WCF2

We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set $P$ and are asked to identify a linear extension of $P$ using pairwise comparisons. For the standard sorting problem, in which $P$ is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04).

Does the information complexity of a function equal its communication complexity? We examine whether any currently known techniques might be used to show a separation between the two notions. Ganor et al. [2014] recently provided such a separation in the distributional case for a specific input distribution. We show that in the non-distributional setting, the relative discrepancy bound is smaller than the information complexity; hence, it cannot separate information and communication complexity. In addition, in the distributional case, we provide a linear program formulation for relative discrepancy and relate it to variants of the partition bound, resolving also an open question regarding the relation of the partition bound and information complexity. Last, we prove the equivalence between the adaptive relative discrepancy and the public-coin partition, implying that the logarithm of the adaptive relative discrepancy bound is quadratically tight with respect to communication., SCOPUS: ar.j, info:eu-repo/semantics/published

Spatial search by discrete-time quantum walk can find a marked node on any ergodic, reversible Markov chain $P$ quadratically faster than its classical counterpart, i.e.\ in a time that is in the square root of the hitting time of $P$. However, in the framework of continuous-time quantum walks, it was previously unknown whether such general speed-up is possible. In fact, in this framework, the widely used quantum algorithm by Childs and Goldstone fails to achieve such a speedup. Furthermore, it is not clear how to apply this algorithm for searching any Markov chain $P$. In this article, we aim to reconcile the apparent differences between the running times of spatial search algorithms in these two frameworks. We first present a modified version of the Childs and Goldstone algorithm which can search for a marked element for any ergodic, reversible $P$ by performing a quantum walk on its edges. Although this approach improves the algorithmic running time for several instances, it cannot provide a generic quadratic speedup for any $P$. Secondly, using the framework of interpolated Markov chains, we provide a new spatial search algorithm by continuous-time quantum walk which can find a marked node on any $P$ in the square root of the classical hitting time. In the scenario where multiple nodes are marked, the algorithmic running time scales as the square root of a quantity known as the extended hitting time. Our results establish a novel connection between discrete-time and continuous-time quantum walks and can be used to develop a number of Markov chain-based quantum algorithms., This version deals only with new algorithms for spatial search by continuous-time quantum walk (CTQW) on ergodic, reversible Markov chains. Please see arXiv:2004.12686 for results on the necessary and sufficient conditions for the optimality of the Childs and Goldstone algorithm for spatial search by CTQW

The symplectic formalism applied to the phase-space representation of bosonic quantum systems provides us with a powerful mathematical tool for the characterisation of Gaussian states and transformations. As a consequence, quantum information protocols involving the latter are very well understood from a theoretical point of view. Nevertheless, it has become clear in recent years that the use of non-Gaussian resources is necessary in order to perform various crucial information-processing tasks. An illustration of this fact can for instance be found in situations where a Gaussian no-go theorem precludes the use of Gaussian transformations in order to achieve a task involving Gaussian states, such as quantum entanglement distillation, quantum error correction, or universal quantum computation. In the first part of this thesis, we develop a new method based on the generating function of a sequence, which gives rise to an elegant description of intrinsically non-Gaussian objects. Building on the generating function of the matrix elements of Gaussian unitaries in Fock basis, our approach gives access to the multi-photon transition probabilities via unexpectedly simple recurrence equations. The method is developed for Gaussian unitaries effecting both passive and active linear coupling between two bosonic modes. It predicts an interferometric suppression term which generalises the Hong-Ou-Mandel effect for more than two indistinguishable photons impinging on a balanced beam splitter. Furthermore, it exhibits an unsuspected 2-photon suppression effect in optical parametric amplification of gain 2, which originates from the indistinguishability between the input and output photon pairs. Finally, we extend our method to Bogoliubov transformations acting on an arbitrary number of modes. In the second part of this thesis, we introduce a class of Gaussian-dilatable bosonic quantum channels (characterised by a Gaussian unitary in their Stinespring dilation) called passive-environment channels. These channels are interesting from a quantum thermodynamical viewpoint because they correspond to the coupling of a bosonic system with a bosonic environment that is passive in the Fock-basis (that is, no energy can be extracted from it by using unitary transformations) followed by discarding the environment. Making use of the generating function, we provide a description of these channels in terms of Gaussian bosonic channels. We then introduce a new preorder relation called Fock-majorization, which coincides with regular majorization for passive states but also induces another relation in terms of mean boson number, thereby connecting the concepts of energy and disorder of a quantum state. We prove various properties of Fock-majorization, showing in particular that the latter can be interpreted as a relation indicating the existence of a heating or amplifying map between two quantum states. This new preorder relation happens to be relevant in the context of passive-environment bosonic channels. Indeed, we show that these channels are Fock-majorization-preserving, so that any two input states that obey a Fock-majorization relation are transformed into output states respecting a similar relation. As a consequence, it also implies that passive-environment channels are majorization-preserving over the set of passive states of the harmonic oscillator. The consequences of majorization preservation are discussed in the context of the so-called entropy photon-number inequality. Most of our results being independent of the specific nature of the system under investigation, they could be generalised to other quantum systems and Hamiltonians, providing new tools that may prove useful in quantum information theory. In the last part of our thesis, we lay out a resource theory of local activity for bosonic systems. We introduce a notion of local-activity distance, and compare it with the work that can be extracted from a quantum state under local unitaries assisted by passive global unitaries. With this framework, we hope to connect the area of continuous-variable bosonic channels together with quantum thermodynamics., Le formalisme symplectique appliqué à la représentation des systèmes bosoniques dans l'espace des phases donne accès à un outil mathématique puissant pour la caractérisation des états gau-ssiens et transformations gaussiennes. Les protocoles d'information quantique impliquant ces derniers sont d'ailleurs très bien compris d'un point de vue théorique. Toutefois, il s'est avéré clair durant ces dernières années que l'utilisation de ressources non-gaussiennes est nécessaire afin d'effectuer des tâches cruciales de traitement de l'information. En effet, certaines tâches — telles que la distillation d’intrication quantique, le codage quantique ou encore le calcul quantique — impliquant des états gaussiens ne peuvent être effectuées avec des transformations gaussiennes. Dans la première partie de cette thèse, nous développons une nouvelle méthode basée sur la fonction génératrice d'une suite qui donne lieu à une description élégante d'objets intrinsèquement non-gaussiens. Se basant sur la fonction génératrice des éléments de matrice d'unitaires gaussiens dans la base de Fock, notre approche donne accès aux probabilités de transition multi-photon via des équations de récurrence étonnamment simples. La méthode est développée pour des unitaires gaussiens produisant des couplages linéaires passifs et actifs entres deux modes bosoniques. Elle prédit un terme d'interférence destructive qui généralise l'effet Hong-Ou-Mandel pour plus de deux photons indistinguables pénétrant dans un diviseur de faisceau équilibré. De plus, elle met en évidence un effet inattendu de suppression de deux photons dans un amplificateur paramétrique optique de gain 2. Cette suppression résulte de l’indistinguabilité entre les paires de photons d’entrée et de sortie. Finalement, nous étendons notre méthode à des transformations de Bogoliubov agissant sur un nombre de modes arbitraire. Dans la seconde partie de cette thèse, nous introduisons une classe de canaux quantiques bosoniques gaussiens-dilatables (caractérisés par un unitaire gaussien dans leur ``Stinespring dilation") appelés canaux à environnement passif. Ces canaux sont intéressants du point de vue de la thermodynamique quantique puisqu’ils correspondent au couplage d’un système bosonique avec un environnement bosonique qui est passif dans la base de Fock (en d’autres termes, il est impossible d’en extraire de l’énergie avec des transformations unitaires), suivi du rejet de l’environnement. Grâce à la fonction génératrice, nous fournissons une description de ces transformations en termes de canaux quantiques bosoniques gaussiens limités par le bruit du vide. Nous introduisons ensuite une nouvelle relation de pré-ordre appelé ``majorization" de Fock, qui coïncide avec la ``majorization" usuelle pour les états passifs mais induit une autre relation en terme du nombre moyen de bosons, connectant ainsi les concepts d’énergie et de désordre d’un état quantique. Dans ce contexte, nous prouvons des propriétés variées de la ``majorization" de Fock et montrons en particulier que cette dernière peut être interprétée comme une relation indiquant l’existence d’une transformation d’amplification entre deux états quantiques. Cette nouvelle relation de pré-ordre s’avère appropriée dans le contexte des canaux bosonique à environnement passif. En effet, nous montrons que ces canaux conservent la ``majorization" de Fock, de sorte que n’importe quels deux états d’entrée obéissant une relation de ``majorization" de Fock sont transformés en états de sortie vérifiant une relation similaire. En particulier, cela implique que les canaux à environnement passif préservent la ``majorization" pour l'ensemble des états passifs de l’oscillateur harmonique. Les conséquences de la préservation de la ``majorization" sont examinées dans le contexte de la ``entropy photon-number inequality". Étant indépendants de la nature spécifique du système étudié, la plupart de nos résultats peuvent être généralisés à d’autres systèmes et hamiltoniens quantiques, donnant lieu à de nouveaux outils qui pourraient s’avérer utiles en théorie de l’information quantique. Dans la dernière partie de notre thèse, nous mettons en place une théorie de l’activité locale pour les système bosoniques. Nous introduisons une notion de distance en terme d'activité locale et la comparons avec le travail qui peut être extrait d'un état quantique avec des unitaires locaux assistés par des unitaires globaux passifs. Le but à long terme est de se baser sur cette théorie afin de connecter les domaines des canaux bosoniques à variables continues et de la thermodynamique quantique., Doctorat en Sciences de l'ingénieur et technologie, info:eu-repo/semantics/nonPublished

The device-independent (DI) framework is a novel approach to quantum information science which exploits the nonlocality of quantum physics to certify the correct functioning of a quantum information processing task without relying on any assumption on the inner workings of the devices performing the task. This thesis focuses on the device-independent certification and generation of true randomness for cryptographic applications. The existence of such true randomness relies on a fundamental relation between the random character of quantum theory and its nonlocality, which arises in the context of Bell tests. Device-independent randomness generation (DIRG) and quantum key distribution (DIQKD) protocols usually evaluate the produced randomness (as measured by the conditional min-entropy) as a function of the violation of a given Bell inequality. However, the probabilities characterising the measurement outcomes of a Bell test are richer than the degree of violation of a single Bell inequality. In this work we show that a more accurate assessment of the randomness present in nonlocal correlations can be obtained if the value of several Bell expressions is simultaneously taken into account, or if the full set of probabilities characterising the behaviour of the device is considered. As a side result, we show that to every behaviour there corresponds an optimal Bell expression allowing to certify the maximal amount of DI randomness present in the correlations. Based on these results, we introduce a family of protocols for DIRG secure against classical side information that relies on the estimation of an arbitrary number of Bell expressions, or even directly on the experimental frequencies of the measurement outcomes. The family of protocols we propose also allows for the evaluation of randomness from a subset of measurement settings, which can be advantageous when considering correlations for which some measurement settings result in more randomness than others. We provide numerical examples illustrating the advantage of this method for finite data, and show that asymptotically it results in an optimal generation of randomness from experimental data without having to assume beforehand that the devices violate a specific Bell inequality., L'approche indépendante des appareils ("device-independent" en anglais) est une nouvelle approche en informatique quantique. Cette nouvelle approche exploite la non-localité de la physique quantique afin de certifier le bon fonctionnement d'une tâche sans faire appel à des suppositions sur les appareils menant à bien cette tâche. Cette thèse traite de la certification et la génération d'aléa indépendante des appareils pour des applications cryptographiques. L'existence de cet aléa repose sur une relation fondamentale entre le caractère aléatoire de la théorie quantique et sa non-localité, mise en lumière dans le cadre des tests de Bell. Les protocoles de génération d'aléa et de distribution quantique de clés indépendants des appareils mesurent en général l'aléa produit en fonction de la violation d'une inégalité de Bell donnée. Cependant les probabilités qui caracterisent les résultats de mesures dans un test de Bell sont plus riches que le degré de violation d'une seule inégalité de Bell. Dans ce travail nous montrons qu'une évaluation plus exacte de l'aléa présent dans les corrélations nonlocales peut être faite si l'on tient compte de plusieurs expressions de Bell à la fois ou de l'ensemble des probabilités (ou comportement) caractérisant l'appareil testé. De plus nous montrons qu'à chaque comportement correspond une expression de Bell optimale permettant de certifier la quantité maximale d'aléa présente dans ces corrélations. À partir de ces resultats, nous introduisons une famille de protocoles de génération d'aléa indépendants des appareils, sécurisés contre des adversaires classiques, et reposant sur l'évaluation de l'aléa à partir d'un nombre arbitraire d'expressions de Bell, ou même à partir des fréquences expérimentales des résultats de mesure. Les protocoles proposés permettent aussi d'évaluer l'aléa à partir d'un sous-ensemble de choix de mesure, ce qui peut être avantageux lorsque l'on considère des corrélations pour lesquelles certains choix de mesure produisent plus d'aléa que d'autres. Nous fournissons des exemples numériques illustrant l'avantage de cette méthode pour des données finies et montrons qu'asymptotiquement cette méthode résulte en un taux de génération d'aléa optimal à partir des données expérimentales, sans devoir supposer à priori que l'expérience viole une inégalité de Bell spécifique., Doctorat en Sciences, info:eu-repo/semantics/nonPublished

The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities., Final version for publication in Quantum

Contextuality is one way of capturing the non-classicality of quantum theory. The contextual nature of a theory is often witnessed via the violation of non-contextuality inequalities---certain linear inequalities involving probabilities of measurement events. Using the exclusivity graph approach (one of the two main graph theoretic approaches for studying contextuality), it was shown [PRA 88, 032104 (2013); Annals of mathematics, 51-299 (2006)] that a necessary and sufficient condition for witnessing contextuality is the presence of an odd number of events (greater than three) which are either cyclically or anti-cyclically exclusive. Thus, the non-contextuality inequalities whose underlying exclusivity structure is as stated, either cyclic or anti-cyclic, are fundamental to quantum theory. We show that there is a unique non-contextuality inequality for each non-trivial cycle and anti-cycle. In addition to the foundational interest, we expect this to aid the understanding of contextuality as a resource to quantum computing and its applications to local self-testing., Comment: 17 pages (5 main, 12 appendix), 4 figures; In v2, generalised the result, upgraded the title, clarified the relation to prior work and rewrote the narrative

The hypothesis (Sparenberg et al. in EPJ Web Conf 58:01016, [1]. https://doi.org/10.1051/epjconf/20135801016) that the particular linear tracks appearing in the measurement of a spherically-emitting radioactive source in a cloud chamber are determined by the (random) positions of atoms or molecules inside the chamber is further explored in the framework of a recently established one-dimensional model (Carlone et al. Comm Comput Phys 18:247, [2]. https://doi.org/10.4208/cicp.270814.311214a). In this model, meshes of localized spins 1/2 play the role of the cloud-chamber atoms and the spherical wave is replaced by a linear superposition of two wave packets moving from the origin to the left and to the right, evolving deterministically according to the Schrödinger equation. We first revisit these results using a time-dependent approach, where the wave packets impinge on a symmetric two-sided detector. We discuss the evolution of the wave function in the configuration space and stress the interest of a non-symmetric detector in a quantum-measurement perspective. Next we use a time-independent approach to study the scattering of a plane wave on a single-sided detector. Preliminary results are obtained, analytically for the single-spin case and numerically for up to 8 spins. They show that the spin-excitation probabilities are sometimes very sensitive to the parameters of the model, which corroborates the idea that the measurement result could be determined by the atom positions. The possible origin of decoherence and entropy increase in future models is finally discussed., SCOPUS: ar.j, info:eu-repo/semantics/published

Different versions of the effective-range function method for charged particle collisions are studied and compared. In addition, a novel derivation of the standard effective-range function is presented from the analysis of Coulomb wave functions in the complex plane of the energy. The recently proposed effective-range function denoted as Δℓ [Ramírez Suárez and Sparenberg, Phys. Rev. C 96, 034601 (2017)] and an earlier variant [Hamilton et al. Nucl. Phys. B 60, 443 (1973)] are related to the standard function. The potential interest of Δℓ for the study of low-energy cross sections and weakly bound states is discussed in the framework of the proton-proton 1S0 collision. The resonant state of the proton-proton collision is successfully computed from the extrapolation of Δℓ instead of the standard function. It is shown that interpolating Δℓ can lead to useful extrapolation to negative energies, provided scattering data are known below one nuclear Rydberg energy (12.5 keV for the proton-proton system). This property is due to the connection between Δℓ and the effective-range function by Hamilton et al. that is discussed in detail. Nevertheless, such extrapolations to negative energies should be used with caution because Δℓ is not analytic at zero energy. The expected analytic properties of the main functions are verified in the complex energy plane by graphical color-based representations., SCOPUS: ar.j, info:eu-repo/semantics/published

info:eu-repo/semantics/published

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set $M$ consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time $HT(P,M)$ of any reversible random walk $P$ on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity $HT^+(backslash mathitP,M)$ which we call extended hitting time. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk $P$ and the absorbing walk $P'$, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk $P$ is not state-transitive. We also provide algorithms in the cases when only approximations or bounds on parameters $p_M$ (the probability of picking a marked vertex from the stationary distribution) and $HT^+(backslash mathitP,M)$ are known., SCOPUS: ar.j, info:eu-repo/semantics/published

In spite of their simple description in terms of rotations or symplectic transformations in phase space, quadratic Hamiltonians such as those modelling the most common Gaussian operations on bosonic modes remain poorly understood in terms of entropy production. For instance, determining the quantum entropy generated by a Bogoliubov transformation is notably a hard problem, with generally no known analytical solution, while it is vital to the characterisation of quantum communication via bosonic channels. Here we overcome this difficulty by adapting the replica method, a tool borrowed from statistical physics and quantum field theory. We exhibit a first application of this method to continuous-variable quantum information theory, where it enables accessing entropies in an optical parametric amplifier. As an illustration, we determine the entropy generated by amplifying a binary superposition of the vacuum and a Fock state, which yields a surprisingly simple, yet unknown analytical expression., SCOPUS: ar.j, info:eu-repo/semantics/published

Here, we present complex resonance states (or Siegert states) that describe the tunnelling decay of a trapped quantum particle from an intuitive point of view that naturally leads to the easily applicable Siegert approximation method. This can be used for analytical and numerical calculations of complex resonances of both the linear and nonlinear Schrödinger equations. This approach thus complements other treatments of the subject that mostly focus on methods based on continuation in the complex plane or on semiclassical approximations., SCOPUS: ar.j, info:eu-repo/semantics/published

The mathematical relations between the regular Coulomb function Fηℓ(ρ) and the irregular Coulomb functions H±ηℓ(ρ) and Gηℓ(ρ) are obtained in the complex plane of the variables η and ρ for integer or half-integer values of ℓ. These relations, referred to as “connection formulas,” form the basis of the theory of Coulomb wave functions and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function Fηℓ(ρ) are studied, in particular, under the transformation ℓ ↦ −ℓ − 1, by means of the modified Coulomb function Φηℓ(ρ), which is entire in the dimensionless energy η−2 and the angular momentum ℓ. Then, it is shown that, for integer or half-integer ℓ, the irregular functions H±ηℓ(ρ) and Gηℓ(ρ) can be expressed in terms of the derivatives of Φη,ℓ(ρ) and Φη,−ℓ−1(ρ) with respect to ℓ. As a consequence, the connection formulas directly lead to the description of the singular structures of H±ηℓ(ρ)and Gηℓ(ρ) at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of η−1., SCOPUS: ar.j, info:eu-repo/semantics/published

Quantum query complexity is known to be characterized by the so-called quantum adversary bound. While this result has been proved in the standard discrete-time model of quantum computation, it also holds for continuous-time (or Hamiltonian-based) quantum computation, due to a known equivalence between these two query complexity models. In this work, we revisit this result by providing a direct proof in the continuous-time model. One originality of our proof is that it draws new connections between the adversary bound, a modern technique of theoretical computer science, and early theorems of quantum mechanics. Indeed, the proof of the lower bound is based on Ehrenfest's theorem, while the upper bound relies on the adiabatic theorem, as it goes by constructing a universal adiabatic quantum query algorithm. Another originality is that we use for the first time in the context of quantum computation a version of the adiabatic theorem that does not require a spectral gap., SCOPUS: cp.p, info:eu-repo/semantics/published

The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid thanks to the use of a Gauss-quadrature approximation. The variational basis related to this Gauss quadrature is composed of Lagrange functions which are infinitely differentiable functions vanishing at all mesh points but one. This method is quite simple to use and, more importantly, can be very accurate with small number of mesh points for a number of problems. The accuracy may however be destroyed by singularities of the potential term. This difficulty can often be overcome by a regularization of the Lagrange functions which does not affect the simplicity and accuracy of the method.The principles of the Lagrange-mesh method are described, as well as various generalizations of the Lagrange functions and their regularization. The main existing meshes are reviewed and extensive formulas are provided which make the numerical calculations simple. They are in general based on classical orthogonal polynomials. The extensions to non-classical orthogonal polynomials and periodic functions are also presented.Applications start with the calculations of energies, wave functions and some observables for bound states in simple solvable models which can rather easily be used as exercises by the reader. The Dirac equation is also considered. Various problems in the continuum can also simply and accurately be solved with the Lagrange-mesh technique including multichannel scattering or scattering by non-local potentials. The method can be applied to three-body systems in appropriate systems of coordinates. Simple atomic, molecular and nuclear systems are taken as examples. The applications to the time-dependent Schrödingerequation, to the Gross-Pitaevskii equation and to Hartree-Fock calculations are also discussed as well as translations and rotations on a Lagrange mesh., SCOPUS: re.j, info:eu-repo/semantics/published

In the context of relativistic quantum mechanics, we obtain a nonlinear first order differential equation for the energy as a function of the coupling constant of a central potential. This differential equation is only exact for power law and logarithmic potentials in the massless limit. For other potentials, we discuss under which conditions the differential equation yields rigorous upper and lower limits on the value of energy levels. These results are applied to the Cornell potential used in meson spectroscopy. We also show that the method applies to noncentral potentials.

The present status of the three-dimensional inverse-scattering method withsupersymmetric transformations is reviewed for the coupled-channel case.We first revisit in a pedagogical way the single-channel case, where thesupersymmetric approach is shown to provide a complete, efficient andelegant solution to the inverse-scattering problem for the radial Schrödingerequation with short-range interactions. A special emphasis is put on thedifferences between conservative and non-conservative transformations, i.e.transformations that do or do not conserve the behaviour of solutions of theradial Schrödinger equation at the origin. In particular, we show that for thezero initial potential, a non-conservative transformation is always equivalentto a pair of conservative transformations. These single-channel results areillustrated on the inversion of the neutron–proton triplet eigenphase shifts forthe S- and D-waves. We then summarize and extend our previous works on thecoupled-channel case, i.e. on systems of coupled radial Schrödinger equations,and stress remaining difficulties and open questions of this problem by puttingit in perspective with the single-channel case. We mostly concentrate on two-channel examples to illustrate general principles while keeping mathematics assimple as possible. In particular, we discuss the important difference betweenthe equal-threshold and different-threshold problems. For equal thresholds,conservative transformations can provide non-diagonal Jost and scatteringmatrices. Iterations of such transformations in the two-channel case are studiedand shown to lead to practical algorithms for inversion. A convenient particulartechnique where the mixing parameter can be fitted without modifying theeigenphases is developed with iterations of pairs of conjugate transformations.This technique is applied to the neutron–proton triplet S–D scattering matrix, forwhich exactly-solvable matrix potential models are constructed. For differentthresholds, conservative transformations do not seem to be able to provide anon-trivial coupling between channels. In contrast, a single non-conservativetransformation can generate coupled-channel potentials starting from the zeropotential and is a promising first step towards a full solution to the coupled-channel inverse problem with threshold differences., SCOPUS: re.j, info:eu-repo/semantics/published

We propose a new approach to calculate perturbatively the effects of a particular deformed Heisenberg algebra on energy spectrum. We use this method to calculate the harmonic oscillator spectrum and find that corrections are in agreement with a previous calculation. Then, we apply this approach to obtain the hydrogen atom spectrum and we find that splittings of degenerate energy levels appear. Comparison with experimental data yields an interesting upper bound for the deformation parameter of the Heisenberg algebra., 7 pages, REVTeX

In this paper, we construct an analytical solution of the one-dimensional spinless Salpeter equation with a Coulomb potential supplemented by a hard core interaction, which keeps the particle in the x positive region., Comment: 10 pages, no figures, Revtex 3.0

We discuss the spectral property of unstable dynamical systems in both classical and quantum mechanics. An important class of unstable dynamical systems corresponds to the Large Poincare Systems (LPS). Conventional perturbation technique leads then to divergences. We introduce methods for the elimination of Poincare divergences to obtain a solution of the spectral problem analytic in the coupling constant. To do so, we have to enlarge the class of permissible transformations, to include non-unitary transformations as well as to extend the Hilbert space. A simple example refers to the Friedrichs model, which was studied independently by George Sudarshan and his co-workers. However, our main interest is the irreducible representations in the Liouville space. In these representations the central quantity is the density matrix, and the eigenfunctions of the Liouville operator cannot be expressed in terms of the wave functions. We suggest that this situation corresponds to quantum chaos. Indeed, classical chaos does not mean that Newton's equation becomes “wrong” but that trajectories loose their operational meaning. Similarly, whenever we have an irreducible representation in the Liouville space this means that the wave function description looses its operational meaning. Additional statistical features appear. A simple example corresponds to persistent interactions in the scattering problem which cannot be treated in the frame of usual S-matrix theory. © 1997, Verlag der Zeitschrift für Naturforschung. All rights reserved., SCOPUS: ar.j, info:eu-repo/semantics/published

The majority of recent works investigating the link between non-locality and randomness, e.g. in the context of device-independent cryptography, do so with respect to some specific Bell inequality, usually the CHSH inequality. However, the joint probabilities characterizing the measurement outcomes of a Bell test are richer than just the degree of violation of a single Bell inequality. In this work we show how to take this extra information into account in a systematic manner in order to optimally evaluate the randomness that can be certified from non-local correlations. We further show that taking into account the complete set of outcome probabilities is equivalent to optimizing over all possible Bell inequalities, thereby allowing us to determine the optimal Bell inequality for certifying the maximal amount of randomness from a given set of non-local correlations., info:eu-repo/semantics/published, Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

We calculate the index of refraction of sodium matter waves propagating through a gas of nitrogen molecules. We use a recent ab initio potential for the ground state of the NaN_2 Van der Waals complex to perform quantal close-coupling calculations and compute the index of refraction as a function of the projectile velocity. We obtain good agreement with the available experimental data. We show that the refractive index contains glory oscillations, but that they are damped by the averaging over the thermal motion of the N_2 molecules. These oscillations appear at lower temperatures and projectile velocity. We also investigate the behavior of the refractive index at low temperature and low projectile velocity to show its dependence on the rotational state of N_2, and discuss the advantage of using diatomic molecules as projectiles., Comment: 6 pages, 4 figures

Measurements on entangled quantum systems necessarily yield outcomes that are intrinsically unpredictable if they violate a Bell inequality. This property can be used to generate certified randomness in a device-independent way, i.e. without making detailed assumptions about the internal working of the quantum devices used to generate the random numbers. Furthermore these numbers are also private; i.e. they appear random not only to the user but also to any adversary that might possess a perfect description of the devices. Since this process requires a small initial random seed to sample the behavior of the quantum devices and to extract uniform randomness from the raw outputs of the devices, one usually speaks of device-independent randomness expansion. The purpose of this paper is twofold. First, we point out that in most real, practical situations, where the concept of device independence is used as a protection against unintentional flaws or failures of the quantum apparatuses, it is sufficient to show that the generated string is random with respect to an adversary that holds only classical side information; i.e. proving randomness against quantum side information is not necessary. Furthermore, the initial random seed does not need to be private with respect to the adversary, provided that it is generated in a way that is independent from the measured systems. The devices, however, will generate cryptographically secure randomness that cannot be predicted by the adversary, and thus one can, given access to free public randomness, talk about private randomness generation. The theoretical tools to quantify the generated randomness according to these criteria were already introduced in S. Pironio, but the final results were improperly formulated. The second aim of this paper is to correct this inaccurate formulation and therefore lay out a precise theoretical framework for practical device-independent randomness generation. © 2013 American Physical Society., SCOPUS: ar.j, info:eu-repo/semantics/published

The classical capacity of quantum channels is the tight upper bound for the transmission rate of classical information. This is a quantum counterpart of the foundational notion of the channel capacity introduced by Shannon. Bosonic Gaussian quantum channels provide a good model for optical communications. In order to properly define the classical capacity for these quantum systems, an energy constraint at the channel input is necessary, as in the classical case. A further restriction to Gaussian input ensembles defines the Gaussian (classical) capacity, which can be studied analytically. It also provides a lower bound on the classical capacity and moreover, it is conjectured to coincide with the classical capacity. Therefore, the Gaussian capacity is a useful and important notion in quantum information theory. Recently, we have shown that the study of both the classical and Gaussian capacity of an arbitrary single mode Gaussian quantum channel can be reduced to the study of a particular fiducial channel. In this work we consider the Gaussian capacity of the fiducial channel, discuss its additivity and analyze its dependence on the channel parameters. In addition, we extend previously obtained results on the optimal channel environment to the single mode fiducial channel. In particular, we show that the optimal channel environment for the lossy, amplification, and phase conjugating channels is given by a pure quantum state if its energy is constrained., info:eu-repo/semantics/published

Rejection sampling is a well-known method to sample from a target distribution, given the ability to sample from a given distribution. The method has been first formalized by von Neumann [1951] and has many applications in classical computing. We define a quantum analogue of rejection sampling: given a black box producing a coherent superposition of (possibly unknown) quantum states with some amplitudes, the problem is to prepare a coherent superposition of the same states, albeit with different target amplitudes. The main result of this article is a tight characterization of the query complexity of this quantum state generation problem. We exhibit an algorithm, which we call quantum rejection sampling, and analyze its cost using semidefinite programming. Our proof of a matching lower bound is based on the automorphism principle that allows to symmetrize any algorithm over the automorphism group of the problem. Our main technical innovation is an extension of the automorphism principle to continuous groups that arise for quantum state generation problems where the oracle encodes unknown quantum states, instead of just classical data. Furthermore, we illustrate how quantum rejection sampling may be used as a primitive in designing quantum algorithms, by providing three different applications.We first show that it was implicitly used in the quantum algorithm for linear systems of equations by Harrow et al. [2009]. Second we show that it can be used to speed up the main step in the quantum Metropolis sampling algorithm by Temme et al. [2011]. Finally, we derive a new quantum algorithm for the hidden shift problem of an arbitrary Boolean function and relate its query complexity to "water-filling" of the Fourier spectrum.©2013 ACM 1942-3454/2013/08-ART12 15.00., SCOPUS: cp.j, info:eu-repo/semantics/published

The apparent random outcome of a quantum measurement is conjectured to be fundamentally determined by the microscopic state of the macroscopic measurement apparatus. The apparatus state thus plays the role of a hidden variable which, in contrast with variables characterizing the measured microscopic system, is shown to lead to a violation of Bell’s inequalities and to agree with standard quantum mechanics. An explicit realization of this interpretation is explored (for details, see [1]) for a primitive model of cloud chamber inspired by Mott [2]. Being highly non local, this interpretation of quantum mechanics is argued to open the way to faster-than-light information transfer., SCOPUS: cp.p, info:eu-repo/semantics/published

The resonance spectrum of a tilted periodic quantum system for a bichromatic periodic potential is investigated. For such a bichromatic Wannier–Stark system, exceptional points, degeneracies of the spectrum, can be localized in parameter space by means of an efficient method for computing resonances. Berry phases and Petermann factors are analysed. Finally, the influence of a nonlinearity of the Gross–Pitaevskii type on the resonance crossing scenario is briefly discussed., SCOPUS: ar.j, info:eu-repo/semantics/published

We study the tunnelling decay of a Bose-Einstein condensate from tilted optical lattices within the mean-field approximation. We introduce a method to calculate ground and excited resonance eigenstates of the Gross-Pitaevskii equation, based on a grid relaxation procedure with complex absorbing potentials. This algorithm works efficiently in a wide range of parameters where established methods fail. It allows us to study the effects of the nonlinearity in detail in the regime of resonant tunnelling, where the decay rate is enhanced by resonant coupling to excited unstable states., SCOPUS: ar.j, info:eu-repo/semantics/published

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. The number of steps of the quantum walk is quadratically smaller than the classical hitting time of any reversible random walk P on the graph. Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the walk P and the absorbing walk P′, whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of the interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not state-transitive, and in the presence of multiple marked vertices. As a consequence we make a progress on an open problem related to the spatial search on the 2D-grid., info:eu-repo/semantics/published

Approximate analytical energy formulas for N-body relativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. This method has already been proved to be a powerful technique in the case of two-body problems. A general procedure is given and applied to various Hamiltonians of interest, in atomic and hadronic physics in particular. A test of formulas is performed for baryons described as a three-quark system., Comment: References added

Few photon systems are best described by their wave function rather than by the usual quantum field formalism. Until recently, the existence of a wave function for photons has been an object of controversy. Important breakthrough happened during the last ten years, sharpening the wave function concept and giving it a precise definition. We show how the photon wave function proposed by Bialynicki-Birula and Sipe connects to experimentally measurable quantities such as multi-photon quantum correlation functions and use it to investigate space-time entanglement of photon pairs produced in parametric down-conversion., info:eu-repo/semantics/published

We address the classical capacity of a quantum bosonic memory channel with additive noise, subject to an input energy constraint. The memory is modeled by correlated noise emerging from a Gauss-Markov process. Under reasonable assumptions, we show that the optimal modulation results from a “quantum water-filling” solution above a certain input energy threshold, similar to the optimal modulation for parallel classical Gaussian channels. We also derive analytically the optimal multimode input state above this threshold, which enables us to compute the capacity of this memory channel in the limit of an infinite number of modes. The method can also be applied to a more general noise environment which is constructed by a stationary Gauss process. The extension of our results to the case of broadband bosonic channels with colored Gaussian noise should also be straightforward., info:eu-repo/semantics/published

We study a pure gluon plasma in the context of quasiparticle models, where the plasma is considered as an ideal gas of massive bosons. In order to reproduce SU(3) gauge field lattice data within such a framework, we review briefly the necessity to use a temperature-dependent gluon mass which accounts for color interactions between the gluons near Tc and agrees with perturbative QCD at large temperatures. Consequently, we discuss the thermodynamics of systems with temperature-dependent Hamiltonians and clarify the situation about the possible solutions proposed in the literature to treat those systems consistently. We then focus our attention on two possible formulations which are thermodynamically consistent, and we extract the gluon mass from the equation of state obtained in SU(3) lattice QCD. We find that the thermal gluon mass is similar in both statistical formalisms. Finally, an interpretation of the gluon plasma as an ideal gas made of glueballs and gluons is also presented. The glueball mass is consistently computed within a relativistic formalism using a potential obtained from lattice QCD. We find that the gluon plasma might be a glueball-rich medium for T 1.13Tc and suggest that glueballs could be detected in future experiments dedicated to quark-gluon plasma. © 2009 The American Physical Society., SCOPUS: ar.j, info:eu-repo/semantics/published

Four-photon scattering in nonlinear waveguides is an important physical process that allows photon-pair generation in well defined guided modes, with high rate and reasonably low noise. Most of the experiments to date used the scalar four-photon scattering process in which the pump photons and the scattered photons have the same polarization. In birefringent waveguides, vectorial four-photon scattering is also allowed: these vectorial scattering processes involve photons with different polarizations. In this article, the theory of four-photon scattering in nonlinear, birefringent, and dispersive fibers is developed in the framework of the quantum theory of light. The work focusses on the spectral properties and quantum correlations (including entanglement) of photon-pairs generated in high-birefringence and low-birefringence fibers., Comment: 12 pages, 5 figures, submitted to Phys. Rev. A

We investigate the possibility for a quark-antiquark pair to form a bound state at temperatures higher than the critical one (T>Tc), thus after deconfinement. Our main goal is to find analytical criteria constraining the existence of such mesons. Our formalism relies on a Schrödinger equation for which we study the physical consequences of using both the free energy and the internal energy as potential term, assuming a widely accepted temperature-dependent Yukawa form for the free energy and a recently proposed nonperturbative form for the screening mass. We show that using the free energy only allows for the 1S bottomonium to be bound above Tc, with a dissociation temperature around 1.5×Tc. The situation is very different with the internal energy, where we show that no bound states at all can exist in the deconfined phase. But, in this last case, quasibound states could be present at higher temperatures because of a positive barrier appearing in the potential. © 2007 The American Physical Society., SCOPUS: ar.j, info:eu-repo/semantics/published

The dynamics of a particle in a gravitational quantum well is studied in the context of nonrelativistic quantum mechanics with a particular deformation of a two-dimensional Heisenberg algebra. This deformation yields a new short-distance structure characterized by a finite minimal uncertainty in position measurements, a feature it shares with noncommutative theories. We show that an analytical solution can be found in perturbation and we compare our results to those published recently, where noncommutative geometry at the quantum mechanical level was considered. We find that the perturbations of the gravitational quantum well spectrum in these two approaches have different signatures. We also compare our modified energy spectrum to the results obtained with the GRANIT experiment, where the effects of the Earth's gravitational field on quantum states of ultracold neutrons moving above a mirror are studied. This comparison leads to an upper bound on the minimal length scale induced by the deformed algebra we use. This upper bound is weaker than the one obtained in the context of the hydrogen atom but could still be useful if the deformation parameter of the Heisenberg algebra is not a universal constant but a quantity that depends on the energetic content of the system. © 2006 The American Physical Society., SCOPUS: ar.j, info:eu-repo/semantics/published

Assuming that the Casimir scaling hypothesis is well verified in QCD, masses of glueballs and hybrid gluelumps (gluon attached to a point-like c̄ pair) are computed within the framework of the rotating string formalism. In our model, two gluons are attached by an adjoint string in a glueball, while the gluon and the colour octet c̄ pair are attached by two fundamental strings in a hybrid gluelump. Masses for such exotic hadrons are computed with very few free parameters. These predictions can serve as a guide for experimental searches. In particular, the ground-state glueballs lie on a Regge trajectory and the lightest 2++ state has a mass compatible with some experimental candidates., SCOPUS: ar.j, info:eu-repo/semantics/published

In a recent article, W.-S. Hou and G.-G. Wong [Phys. Rev. D {\bf 67}, 034003 (2003)] have investigated the spectrum of two-gluon glueballs below 3 GeV in a potential model with a dynamical gluon mass. We point out that, among the 18 states calculated by the authors, only three are physical. The other states either are spurious or possess a finite mass only due to an arbitrary restriction of the variational parameter., Comment: Comment on paper