Multipartite entanglement in three-mode Gaussian states of continuous variable systems: Quantification, sharing structure and decoherence

We present a complete analysis of multipartite entanglement of three-mode Gaussian states of continuous variable systems. We derive standard forms which characterize the covariance matrix of pure and mixed three-mode Gaussian states up to local unitary operations, showing that the local entropies of pure Gaussian states are bound to fulfill a relationship which is stricter than the general Araki-Lieb inequality. Quantum correlations will be quantified by a proper convex roof extension of the squared logarithmic negativity (the contangle), satisfying a monogamy relation for multimode Gaussian states, whose proof will be reviewed and elucidated. The residual contangle, emerging from the monogamy inequality, is an entanglement monotone under Gaussian local operations and classical communication and defines a measure of genuine tripartite entanglement. We analytically determine the residual contangle for arbitrary pure three-mode Gaussian states and study the distribution of quantum correlations for such states. This will lead us to show that pure, symmetric states allow for a promiscuous entanglement sharing, having both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. We thus name these states GHZ/$W$ states of continuous variable systems because they are simultaneous continuous-variable counterparts of both the GHZ and the $W$ states of three qubits. We finally consider the action of decoherence on tripartite entangled Gaussian states, studying the decay of the residual contangle. The GHZ/$W$ states are shown to be maximally robust under both losses and thermal noise.Comment: 20 pages, 5 figures. (v2) References updated, published versio

Experimental single-setting quantum state tomography

Quantum computers solve ever more complex tasks using steadily growing system sizes. Characterizing these quantum systems is vital, yet becoming increasingly challenging. The gold-standard is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge. Measurement and classical computing costs, however, increase exponentially in the system size - a bottleneck given the scale of existing and near-term quantum devices. Here, we demonstrate a scalable and practical QST approach that uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVM). We implement these nonorthogonal measurements on an ion trap device by utilizing more energy levels in each ion - without ancilla qubits. More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, providing full tomographic information in every shot. Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination. SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods. The latter enables in-depth entanglement studies, which we experimentally showcase on a 5-qubit absolutely maximally entangled (AME) state. Moreover, the fact that the full tomography information is available in every shot enables online QST in real time. We demonstrate this on an 8-qubit entangled state, as well as for fast state identification. All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.Comment: 34 pages, 15 figure

Quantum Mechanical Vistas on the Road to Quantum Gravity

In this thesis, we lay out the goal, and a broad outline, for a program that takes quantum mechanics in its minimal form to be the fundamental ontology of the universe. Everything else, including features like space-time, matter and gravity associated with classical reality, are emergent from these minimal quantum elements. We argue that the Hilbert space of quantum gravity is locally finite-dimensional, in sharp contrast to that of conventional field theory, which could have observable consequences for gravity. We also treat time and space on an equal footing in Hilbert space in a reparametrization invariant setting and show how symmetry transformations, both global and local, can be treated as unitary basis changes. Motivated by the finite-dimensional context, we use Generalized Pauli Operators as finite-dimensional conjugate variables and define a purely Hilbert space notion of locality based on the spread induced by conjugate operators which we call "Operator Collimation." We study deviations in the spectrum of physical theories, particularly the quantum harmonic oscillator, induced by finite-dimensional effects, and show that by including a black hole-based bound in a lattice field theory, the quantum contribution to the vacuum energy can be suppressed by multiple orders of magnitude. We then show how one can recover subsystem structure in Hilbert space which exhibits emergent quasi-classical dynamics. We explicitly connect classical features (such as pointer states of the system being relatively robust to entanglement production under environmental monitoring and the existence of approximately classical trajectories) with features of the Hamiltonian. We develop an in-principle algorithm based on extremization of an entropic quantity that can sift through different factorizations of Hilbert space to pick out the one with manifest classical dynamics. This discussion is then extended to include direct sum decompositions and their compatibility with Hamiltonian evolution. Following this, we study quantum coarse-graining and state-reduction maps in a broad context. In addition to developing a first-principle quantum coarse-graining algorithm based on principle component analysis, we construct more general state-reduction maps specified by a restricted set of observables which do not span the full algebra (as could be the case of limited access in a laboratory or in various situations in quantum gravity). We also present a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. Throughout the thesis, we discuss implications of our work in the broader goal of understanding quantum gravity from minimal elements in quantum mechanics.</p

Algebraic Methods for Dynamical Systems and Optimisation

This thesis develops various aspects of Algebraic Geometry and its applications in different fields of science. In Chapter 2 we characterise the feasible set of an optimisation problem relevant in chemical process engineering. We consider the polynomial dynamical system associated with mass-action kinetics of a chemical reaction network. Given an initial point, the attainable region of that point is the smallest convex and forward closed set that contains the trajectory. We show that this region is a spectrahedral shadow for a class of linear dynamical systems. As a step towards representing attainable regions we develop algorithms to compute the convex hulls of trajectories. We present an implementation of this algorithm which works in dimensions 2,3 and 4. These algorithms are based on a theory that approximates the boundary of the convex hull of curves by a family of polytopes. If the convex hull is represented as the output of our algorithms we can also check whether it is forward closed or not. Chapter 3 has two parts. In this first part, we do a case study of planar curves of degree 6. It is known that there are 64 rigid isotopy types of these curves. We construct explicit polynomial representatives with integer coefficients for each of these types using different techniques in the literature. We present an algorithm, and its implementation in software Mathematica, for determining the isotopy type of a given sextic. Using the representatives various sextics for each type were sampled. On those samples we explored the number of real bitangents, inflection points and eigenvectors. We also computed the tensor rank of the representatives by numerical methods. We show that the locus of all real lines that do not meet a given sextic is a union of up to 46 convex regions that is bounded by its dual curve. In the second part of Chapter 3 we consider a problem arising in molecular biology. In a system where molecules bind to a target molecule with multiple binding sites, cooperativity measures how the already bound molecules affect the chances of other molecules binding. We address an optimisation problem that arises while quantifying cooperativity. We compute cooperativity for the real data of molecules binding to hemoglobin and its variants. In Chapter 4, given a variety X in n-dimensional projective space we look at its image under the map that takes each point in X to its coordinate-wise r-th power. We compute the degree of the image. We also study their defining equations, particularly for hypersurfaces and linear spaces. We exhibit the set-theoretic equations of the coordinate-wise square of a linear space L of dimension k embedded in a high dimensional ambient space. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with degenerate eigenspectrum

Tree tensor networks for high-dimensional quantum systems and beyond

This thesis presents the development of a numerical simulation technique, the Tree Tensor Network, aiming to overcome current limitations in the simulation of two- and higher-dimensional quantum many-body systems. The development and application of methods based on Tensor Networks (TNs) for such systems are one of the most relevant challenges of the current decade with the potential to promote research and technologies in a broad range of fields ranging from condensed matter physics, high-energy physics, and quantum chemistry to quantum computation and quantum simulation. The particular challenge for TNs is the combination of accuracy and scalability which to date are only met for one-dimensional systems by other established TN techniques. This thesis first describes the interdisciplinary field of TN by combining mathematical modelling, computational science, and quantum information before it illustrates the limitations of standard TN techniques in higher-dimensional cases. Following a description of the newly developed Tree Tensor Network (TTN), the thesis then presents its application to study a lattice gauge theory approximating the low-energy behaviour of quantum electrodynamics, demonstrating the successful applicability of TTNs for high-dimensional gauge theories. Subsequently, a novel TN is introduced augmenting the TTN for efficient simulations of high-dimensional systems. Along the way, the TTN is applied to problems from various fields ranging from low-energy to high-energy up to medical physics.In dieser Arbeit wird die Entwicklung einer numerischen Simulationstechnik, dem Tree Tensor Network (TTN), vorgestellt, die darauf abzielt, die derzeitigen Limitationen bei der Simulation von zwei- und höherdimensionalen Quanten-Vielteilchensystemen zu überwinden. Die Weiterentwicklung von auf Tensor-Netzwerken (TN) basierenden Methoden für solche Systeme ist eine der aktuellsten und relevantesten Herausforderungen. Sie birgt das Potential, Forschung und Technologien in einem breiten Spektrum zu fördern, welches sich von der Physik der kondensierten Materie, der Hochenergiephysik und der Quantenchemie bis hin zur Quantenberechnung und Quantensimulation erstreckt. Die besondere Herausforderung für TN ist die Kombination von Genauigkeit und Skalierbarkeit, die bisher nur für eindimensionale Systeme erfüllt wird. Diese Arbeit beschreibt zunächst das interdisziplinäre Gebiet der TN als eine Kombination von mathematischer Modellierung, Computational Science und Quanteninformation, um dann die Grenzen der Standard-TN-Techniken in höherdimensionalen Fällen aufzuzeigen. Nach einer Beschreibung des neu entwickelten TTN stellt die Arbeit dessen Anwendung zur Untersuchung einer Gittereichtheorie vor, die das Niederenergieverhalten der Quantenelektrodynamik approximiert und somit die erfolgreiche Anwendbarkeit von TTNs für hochdimensionale Eichtheorien demonstriert. Anschließend wird ein neuartiges TN eingeführt, welches das TTN für effiziente Simulationen hochdimensionaler Systeme erweitert. Zusätzlich wird das TTN auf diverse Probleme angewandt, die von Niederenergie- über Hochenergie- bis hin zur medizinischen Physik reichen

Redistribuição de correlação devido a horizontes causais

Orientador: Marcos César de OliveiraDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb WataghinResumo: Em 1975 Stephen Hawking deduziu que devido a efeitos da mecânica quântica um buraco negro formado via colapso gravitacional irá emitir um espectro térmico de partículas. Em 1976 William Unruh inspirado por esse resultado, descobriu que um observador de Rindler no espaço-tempo de Minkowski, vivendo em sua região de Rindler, percebe o vácuo de Minkowski como um banho térmico e, por conta da região de Rindler aproximar a geometria próxima do horizonte de eventos de um buraco negro eterno de Schwarzschild, o vácuo para um observador em queda livre seria percebido como um banho térmico pelo observador de Schwarzschild. Trabalhando com um exemplo simples que foi vastamente usado na literatura para esse tipo de investigação e empregando métodos de teoria de informação quântica, calculamos a correlação clássica, quântica e total entre os subsistemas observados por um observador de Minkowski e um observador de Rindler à esquerda ou à direita; e também por um observador em queda livre e um observador de Schwarzschild à esquerda ou à direita na região próxima ao horizonte. Conseguimos calcular o emaranhamento de formação para o estado experimentado pelos observadores de Rindler à esquerda e à direita, e vemos que este sinaliza a redistribuição de correlação devido a presença do horizonte causal. Concluímos que esse exemplo simples mostra que um horizonte causal de Rindler ou Schwarzschild redistribui correlações por intermédio do emaranhamento entre as duas partes causalmente desconexas do sistemaAbstract: In 1975 Stephen Hawking derived that because of quantum mechanical effects a black hole formed by gravitational collapse will emit a thermal spectrum of particles. In 1976 William Unruh inspired by this discovered that one Rindler observer in Minkowski spacetime, living on its Rindler wedge, perceives the Minkowksi vacuum as a thermal bath and, since the Rindler wedge approximates the near-horizon geometry of an eternal Schwarzschild black hole, the vacuum state for a freely-faling observer would be perceived by as a thermal bath for a Schwarzschild observer. Working with a simple example that has been vastly used in the literature for this kind of investigation and employing quantum information methods, we compute classical, quantum and total correlations between the subsystems observed by a Minkowski and either a left or right Rindler observer; and also by a freely-falling and either a left or right Schwarzschild observer. We are able to compute the entanglement of formation for the state probed by the left and right Rindler observers and left and right Schwarzschild observers, and following the methods of we see that it signals the correlation redistribution imparted by the presence of the causal horizon. We conclude that this simple example shows that a causal horizon like the Rindler or Schwarzschild horizons redistribute correlations by means of entanglement between the two causally disconnected parts of the systemMestradoFísicaMestre em Física132437/2017-1CNP

Entanglement in Many-Body Systems

The recent interest in aspects common to quantum information and condensed matter has prompted a prosperous activity at the border of these disciplines that were far distant until few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermionic and bosonic model systems. Both bipartite and multipartite entanglement will be considered. At equilibrium we emphasize on how entanglement is connected to the phase diagram of the underlying model. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.Comment: 61 pages, 29 figure

Entanglement and correlations in composite quantum systems

Verschränkung ist eine der wichtigsten Besonderheiten der Quantentheorie, welche kein klassisches Analogon besitzt. Aufgrund dieser ist es nicht länger angemessen zusammengesetzte Systeme lediglich als eine Kombination von unabhängig voneinander beschreibbaren Subsystemen zu betrachten. Anstatt dessen gibt es Quantenzustände die inseparabel bezüglich der einzelnen Bestandteile des Systems sind. Wohingegen Verschränkung zunächst als eine Art künstlicher Fehler der Theorie angesehen wurde, hat sich deren Existenz mittlerweile durch zahlreiche Experimente bestätigt. Diese Gegebenheit hat weitreichende Konsequenzen. Auf einer fundamentalen Ebene lässt sich hiermit zeigen, daß es keine lokal-realistische Alternative zur Quantentheorie gibt. Zusätzlich stellt Verschränkung eine Ressource für neue Technologien der Informationsverarbeitung dar, wie z.B. Quantenkryptographie, Dense Coding, Quantenteleportation oder der Quantencomputer. Trotz intensiver Forschung in den letzten Jahrzehnten gibt es noch viele offene Fragen bezüglich Verschränkung und deren Manifestationen. Dies gilt insbesondere für Verschränkung in komplexen Vielteilchensystemen. Gegenstand dieser Dissertation ist es Verschränkung und nicht-lokal-realitsche Korrelationen in zusammengesetzten endlich-dimensionalen Quantensystemen (Multipartite Qudits) zu untersuchen und zu verstehen. Um die Analyse solcher Systeme zu erleichtern, werden neue mathematische Hilfsmittel, wie zum Beispiel praktische Parameterisierungen von unitären Gruppen, Dichtematrizen und Unterräumen vorgestellt. Die Struktur von Verschränkung in Vielteilchensystemen wird betrachtet, und eine exakte Charakterisierung von Multilevel-Vielteilchenverschränkung wird eingeführt. Es werden Methoden zur Verschränkungsdetektion angegeben, und deren experimentelle Implementierung diskutiert. Ein weiteres Thema ist die Quantifizierung von Verschränkung. In diesem Zusammenhang wird ein nützliches Maß für Vielteilchenverschränkung vorgestellt. Darüber hinaus wird auch die Klassifizierung von Vielteilchenverschränkung thematisiert, und ein systematischer Ansatz zur Unterscheidung verschiedener Klassen angegeben. Der letzte Teil dieser Arbeit beschäftigt sich mit Relationen zwischen Verschränkung und anderen fundamentalen Aspekten der Quantentheorie. Insbesondere wird eine Verbindung zwischen dem Komplementaritätsprinzip und dem Separabilitätsproblem hergestellt. Besondere Aufmerksamkeit gilt auch dem Zusammenhang zwischen Verschränkung und nicht-lokal-realitischen Korrelationen. Hier wird eine geometrische Struktur, welche diskreten zusammengesetzten Systemen zugrunde liegt, verwendet, um deren Verschiedenartigkeit veranschaulichen zu können.Entanglement is a key feature of quantum theory which has no classical analogue. Due to this feature it is no longer accurate to regard composite systems as a mere combination of independently describable subsystems. Instead there are quantum states which are inseparable with respect to the individual parts of the system. Initially regarded as an artifact of the theory, numerous experiments performed in the last decades have provided evidence of the existence of entanglement in nature. The consequences of entanglement are far-reaching. On a fundamental level, it allows to demonstrate that there is no local-realistic alternative to quantum theory. In addition, entanglement also serves as a resource for novel information processing technologies such as quantum cryptography, dense coding, quantum teleportation and quantum computing. Despite extensive research in recent years, several questions concerning entanglement and its manifestations still remain open --- especially in complex many-body systems. The aim of this dissertation is to investigate entanglement and non-local-realistic correlations in composite finite-dimensional quantum systems (i.e. multipartite qudits). In order to simplify the analysis of those systems, we present new mathematical tools such as convenient parameterizations for unitary groups, density matrices and subspaces. We study the structure of entanglement in multipartite systems and introduce a precise characterization of multilevel-multipartite entanglement. We consider methods for entanglement detection and discuss their implementation in experiments. Another problem treated in this thesis concerns the quantification of entanglement. Here, we introduce a useful measure of multipartite entanglement and derive computable lower bounds. Moreover, the classification of multipartite entanglement is also addressed, where a systematic approach for discriminating between different classes is given. The last part of this work deals with relations between entanglement and other foundational aspects of quantum theory. Specifically, we establish a link between complementarity and the separability problem. Particular attention is also devoted to the connection between entanglement and non-local-realistic correlations. Here, we exploit a geometric structure underlying discrete composite systems to illustrate their dissimilarities