In the past decades, rapid advances in the experimental control of quantum systems have opened up unparalleled capabilities of engineering exotic quantum states. Nowadays, we may consider ourselves witnesses of a second quantum revolution as strongly correlated quantum matter is regularly created, on a daily basis, within the most diverse platforms, e.g. arrays of Rydberg atoms, ultra-cold atoms in optical lattices, superconducting qubits, trapped ions, and quantum dots. In this era, dubbed Noisy-Intermediate Scale Quantum (NISQ) era, the effort in pursuing research for realizing quantum technologies with practical purposes, such as quantum computing, simulations, communication and metrology, has greatly accelerated, and we are just starting to experience the immense progress that could be achieved. The remarkable breakthrough we are facing builds upon seminal theoretical and experimental advances in quantum physics. The key achievements in atomic, molecular, and optical (AMO) physics have opened up the possibility of controlling, trapping, and measuring single atoms, one by one, with high accuracy and reliability. In parallel, from a theoretical point of view, the study of quantum entanglement and correlations has bridged AMO physics and quantum information with crucial proposals for the realization of universal quantum computers. In this context, entanglement has emerged as one of the key tools to characterize and to exploit quantum many-body systems for quantum information purposes. We will study quantum entanglement in several scenarios to investigate and probe complex quantum many-body systems. We will consider examples ranging from generic mixed states in equilibrium to out-of-equilibrium dynamics, with and without dissipation, and topologically non-trivial systems. The leitmotif of this work will be how entanglement and correlations can be exploited to characterize the many-body quantum state describing a physical system and how entanglement can be detected in an experimentally efficient manner.
In the past decades, rapid advances in the experimental control of quantum systems have opened up unparalleled capabilities of engineering exotic quantum states. Nowadays, we may consider ourselves witnesses of a second quantum revolution as strongly correlated quantum matter is regularly created, on a daily basis, within the most diverse platforms, e.g. arrays of Rydberg atoms, ultra-cold atoms in optical lattices, superconducting qubits, trapped ions, and quantum dots. In this era, dubbed Noisy-Intermediate Scale Quantum (NISQ) era, the effort in pursuing research for realizing quantum technologies with practical purposes, such as quantum computing, simulations, communication and metrology, has greatly accelerated, and we are just starting to experience the immense progress that could be achieved. The remarkable breakthrough we are facing builds upon seminal theoretical and experimental advances in quantum physics. The key achievements in atomic, molecular, and optical (AMO) physics have opened up the possibility of controlling, trapping, and measuring single atoms, one by one, with high accuracy and reliability. In parallel, from a theoretical point of view, the study of quantum entanglement and correlations has bridged AMO physics and quantum information with crucial proposals for the realization of universal quantum computers. In this context, entanglement has emerged as one of the key tools to characterize and to exploit quantum many-body systems for quantum information purposes. We will study quantum entanglement in several scenarios to investigate and probe complex quantum many-body systems. We will consider examples ranging from generic mixed states in equilibrium to out-of-equilibrium dynamics, with and without dissipation, and topologically non-trivial systems. The leitmotif of this work will be how entanglement and correlations can be exploited to characterize the many-body quantum state describing a physical system and how entanglement can be detected in an experimentally efficient manner.
Quantum many-body systems, as they appear in different branches of physics and chemistry, are very hard to simulate. The number of parameters defining their quantum states grows exponentially with the number of constituents. This prevents us from exploring and investigating relevant problems involving relatively small numbers of subsystems even with the help of the most powerful supercomputers. Richard Feynman already noticed this difficulty back in 1981 when he wrote his visionary paper “Simulating Physics with Computers” [Int. J. Theor. Phys. 21:6/7, 467 (1982)]. There, he proposed to use quantum systems (as opposed to classical computers) to perform the simulation and in this way to circumvent the need to store and compute a colossal number of parameters, necessary to describe the pertaining superpositions. What was a visionary idea at that time has become a very active field of research in the last few years. The extraordinary progress of both theoretical and experimental research in quantum physics allows us now to tame, control, and manipulate various quantum systems with unprecedented precision. Laser, magnetic, electronic and other technologies enable us to arrange quantum subsystems in different geometries, modify their interactions, and detect them. This allows us to engineer Hamiltonians and emulate systems displaying some of the most intriguing quantum phenomena. This tremendous progress is, to a large extent, due to the pioneering work of many researchers during the last twenty to thirty years. Here, the investigations of basic phenomena with one or few atoms or photons were of particular impact. The degree of control achieved in the experiments has enabled the observation of mind-boggling quantum effects, related to the superposition principle or to entangled states, and their disappearance due to decoherence. In fact, the 2012 Nobel Prize has been awarded to two scientists, S. Haroche and D. Wineland, for such ground-breaking experiments. Such experiments, often in combination with laser cooling of atoms or ions and Bose-Einstein condensation, have formed one of the pillars on which the field of quantum simulations is being established. Quantum simulations with cold atoms, either in magnetic traps or in optical lattices, with trapped ions, with photons, with superconducting devices, with quantum dots, etc., have been theoretically proposed and experimentally considered by many researchers. Most of the proposals so far deal with problems in condensed matter physics that are either difficult to tackle with modern supercomputers or are difficult to observe in solid state systems. This has established a very close link between different disciplines, like atomic, molecular, and optical (AMO) physics, and condensed matter physics. Although some of the experiments are still in their infancy, and a lot of research is still needed, the extraordinary progress made in all these research areas during the last few years makes us feel confident that in the not too distant future quantum simulators will provide a key tool to investigate many-body quantum systems. Furthermore, we also expect that quantum simulators will help quantum physicists to establish new and fruitful links to other scientific communities, like high-energy physics or quantum chemistry. The present special issue contains various review papers as well as research papers on quantum simulations. It starts out with two papers written by last year's Nobel Prize winners describing their foundational work (Wineland, p. 739 and Haroche, p. 763). The issue continues with three thorough, invited review papers covering different topics on quantum simulations. The first one reviews recent theoretical proposals to use cold atoms in optical lattices to simulate lattice gauge theories of the sort that appear in high-energy physics (U.-J. Wiese, p. 783). The second covers both theoretical proposals and experimental demonstrations of cold-atom systems to simulate the physics of matter in the presence of gauge fields, displaying so-called topological phenomena, which so far have only been observed in solid-state systems (I. Spielman, p. 794). The third one contains an extensive overview and new proposals to perform a broad range of quantum simulations using quantum dot systems (P. Barthelmy et al., p. 813). The issue also includes six original papers with a variety of theoretical proposals as well as the introduction of novel theoretical techniques related to the field of quantum simulation. The field of quantum simulations is just starting, but we are confident that it will soon generate an even larger plethora of new ideas and discoveries, trigger interdisciplinary research, and lead us to the adventure of describing and understanding the quantum world. We hope that this special issue will help future researchers in such an exciting adventure. Rainer Blatt received his PhD degree in physics from the University of Mainz in 1981. As a postdoctoral fellow he worked on atomic beam cooling in Boulder, CO and later as a research assistant in Hamburg with single trapped ions. In 1995 he accepted a chair in experimental physics at the University of Innsbruck working on quantum information processing with trapped ions. Since 2003 he is also scientific director at the Institute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences. Immanuel Bloch received his PhD degree in physics from the Ludwig Maximilians University of Munich in 2000. As a junior group leader, he continued in Munich starting his work on ultracold quantum gases in optical lattices. In 2008 he was appointed scientific director of the newly founded division on Quantum Many-Body Systems at the Max Planck Institute of Quantum Optics. Since 2009 he also holds a chair in quantum optics at the Ludwig-Maximilians University of Munich. Ignacio Cirac received his PhD degree in physics from the Universidad Complutense de Madrid in 1991. He was professor at the Universities of Castilla-La Mancha and Innsbruck from 1991–1996 and 1996–2001, respectively. In 2001 he joined the Max-Planck Institute of Quantum Optics, where he leads the Theory Division. His current research interests include quantum information theory and quantum optics. Peter Zoller obtained his doctorate from the University of Innsbruck in 1977. He was Max Kade Fellow at the University of Southern California from 1978/79, and was JILA visiting Fellow 1981/82 and 1988. In 1990 he became JILA Fellow and professor of physics at the University of Colorado. At the end of 1994, he accepted a chair at the University of Innsbruck, where he has worked ever since. Since 2003, he has also held the position of Scientific Director at the Institute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences.
Microscopic systems such as elementary particles, atoms, and molecules, can be comprehensively described only by quantum mechanics, which thus plays a crucial role for a better understanding of fundamental physics. A very important phenomenon that only occurs in quantum mechanics is entanglement, which describes non-classical correlations between microscopic systems. These correlations exist independently of the physical distance between them, as well as independently of the basis in which these are measured. Specifically, this is the aspect that makes entanglement a purely quantum phenomenon. The importance of quantum mechanics however goes much beyond fundamental physics, since quantum phenomena potentially allow us to solve complex issues associated with currently available technologies. For example, quantum mechanics and entanglement can allow for faster and more powerful computing and information processing, as well as guarantee fully secure communications. Complex entangled states are specifically required for these tasks, and many efforts have been spent towards their experimental realization. However, a full exploitation of entanglement necessitates a deep understanding of this phenomenon, as well as a complete characterization of entangled states. This still represents an open issue, especially in the case of quantum systems which are characterized by a large number of parties (i.e., the physical components - or units - forming a quantum system) and/or a high dimensionality. To date, a technique that enables the entanglement characterization of any arbitrarily complex quantum state in an experimentally feasible manner has not been developed yet. Moreover, realizing complex states in practical, scalable, and low-cost platforms, as demanded by modern technology, is still a big milestone to achieve. Motivated by the importance as well as by the crucial role of entanglement, we oriented our research interests towards the realization of complex quantum systems, as well as towards the derivation of a universal approach capable of characterizing their entanglement with a minimal experimental effort. Experimental feasibility is indeed at the basis of entanglement application for modern technology. Towards this goal, we first explored quantum systems that we could use as a basis for the realization of complex entangled states, i.e. consisting of multiple parties and levels (or dimensions). This investigation led us to the first ever generation of high dimensional frequency-entangled photon pairs on-chip. We further coherently manipulated these states in order to characterize their dimensionality through Schmidt mode decomposition, as well as their entanglement via quantum interference measurements and Bell inequality violations. The quantum state analysis pursued here provided us the basis for developing a universal approach for entanglement characterization. In this research work, we theoretically derived a universal entanglement witness that allows us to detect the presence of any arbitrarily complex pure quantum state, as well as of its eventual entanglement. We provided a technique that permits us to customize the witness (still preserving its mathematical properties) based on specific experimental restrictions and available measurement settings, in such a way to make the witness measurement experimentally feasible. Such a feature bridges the theoretical work pursued in this thesis and the experiments presented here. We explicitly derived a witness which is capable of detecting the presence of high-dimensional multipartite cluster states, and used this operator to test the robustness of these states toward noise. In connection to this, we present the first ever realization of a four-partite three level cluster state by making use of two photons that are simultaneously entangled (i.e. hyper-entangled) in time and in frequency. Cluster states are a class of multipartite systems with unique entanglement properties which make them crucial to realize so-called ‘one-way’ quantum computers. We confirmed the genuine multipartite entanglement of the generated cluster state by making use of the developed witness. We finally implemented two-partite projection measurements on the cluster state both in time and in frequency, thus demonstrating proof-of-principle high-dimensional one-way quantum processing. While cluster states have been already realized in quantum optical platforms for two-level systems, their extension to multiple levels (i.e. more dimensions) has never been explored prior to this work. The fundamental importance as well as the potential applications of cluster states in quantum technology, motivated us towards the investigation and the exploitation of these states in higher dimensions. The realizations demonstrated here were achieved in an integrated photonic platform, while quantum state manipulations were implemented by making use of fiber-based components that are compatible with off-the-shelf telecommunication infrastructures. In view of this, the results presented here are extremely timely with the latest technology advancements, which aim at reducing the size and the cost of platforms as much as possible. The work that has been specifically accomplished in the context of this research line and that is the main subject of this thesis has a theoretical footprint. In particular, the pursued studies have been focused on deriving a theoretical, yet experimentally oriented, formalism enabling the entanglement characterization of complex (i.e. multipartite and/or high-dimensional) photon states. More specifically, such a theoretical work was crucial for the experimental realization of entangled photon states, and to perform the quantum measurements necessary for their entanglement characterization. One of the main tasks described in this thesis was indeed to theoretically provide all the measurements needed for quantum state analysis in such a way that these could be reproduced and implemented in an experimentally feasible manner. In order to fully understand the formalism and the tools that we exploited in this research work, this thesis is organized as follows. Chapter 1 is an introduction to quantum mechanics. It contains the formalism that we used to describe quantum states, as well as the operations implemented on them. Chapter 2 is about the entanglement phenomenon. Here, we present classes of entangled states that are crucial for this research work. These are the so-called Bell states and hyper-entangled states. In Chapter 3 we introduce cluster states and witness operators. From the literature, we provide a description of how witnesses can be used to detect entanglement of two-level cluster states. Finally, Chapters 4, 5, and 6 report the investigation and results accomplished in this research work, with a main focus on the theoretical framework, in line with the theory work that has been accomplished during the PhD studies and that is the main subject of this thesis. La meccanica quantistica è la sola branca della fisica in grado di descrivere le proprietà e la dinamica dei sistemi microscopici, quali particelle elementari, atomi e molecole, ragion per cui essa gioca un ruolo cruciale per una maggiore comprensione della fisica fondamentale. Un concetto basilare della meccanica quantistica e che è privo di qualsiasi analogo classico è il fenomeno dell’entanglement, il quale descrive correlazioni tra due o più sistemi microscopici. Queste correlazioni persistono indipendentemente sia dalla distanza fisica che li separa sia dalla base rispetto alla quale tali sistemi vengono misurati. È proprio quest’ultimo aspetto a rendere l’entanglement un fenomeno di natura esclusivamente quantistica. L’importanza della meccanica quantistica va tuttavia ben al di là della comprensione della fisica fondamentale. In particolare, i fenomeni quantistici possono svolgere un ruolo chiave sia per lo sviluppo di nuove tecnologie sia per superare le capacità e i limiti delle tecnologie che sono attualmente disponibili. Ad esempio, certi fenomeni quantistici permettono di realizzare computer (chiamati appunto computer quantistici) con potenza ed efficacia di calcolo di gran lunga superiori al loro analogo classico e che sono inoltre capaci di immagazzinare e processare una maggiore quantità di dati. Oppure, l’entanglement fra due o più sistemi permette di comunicare in totale sicurezza. L’introduzione dei principi e dei fenomeni della meccanica quantistica nell’ambito delle tecnologie richiede l’utilizzo di stati entangled complessi. Per questo motivo, la comunità scientifica si è impegnata parecchio negli ultimi decenni nella realizzazione sperimentale di questi ultimi. Tuttavia, un appropriato utilizzo dell’entanglement in tutti i suoi aspetti richiede una piena comprensione di tale fenomeno, nonché la capacità di caratterizzare in toto qualsiasi stato entangled. Ciò rimane tuttora una questione aperta, soprattutto nel caso di stati quantistici complessi caratterizzati da un numero elevato di parti e/o di livelli (cioè multipartiti e/o multidimensionali). Ad oggi, non esiste un approccio universale in grado di studiare e caratterizzare l’entanglement di un sistema quantistico qualsivoglia complesso. In più, la realizzazione di stati entangled complessi su piattaforme pratiche, riducibili in scala e a basso costo, come richiesto dalle tecnologie moderne, rimane ancora un grande obiettivo da raggiungere. Motivati dall’importanza e dal ruolo cruciale dell’entanglement, abbiamo rivolto la nostra attività di ricerca alla realizzazione di stati entangled complessi, così come alla teorizzazione di un approccio universalmente valido e in grado di caratterizzare in pieno l’entanglement di questi. Un obiettivo importante che ci siamo prefissati nella derivazione di tale approccio era che questo permettesse di caratterizzare l’entanglement attraverso misure da effettuare con il minore sforzo sperimental
The ability to precisely and coherently control atomic systems has improved dramatically in the last two decades, driving remarkable advancements in quantum computation and simulation. In recent years, atomic and atom-like systems have also been served as a platform to study topological phases of matter and non-equilibrium many-body physics. Integrated with rapid theoretical progress, the employment of these systems is expanding the realm of our understanding on a range of physical phenomena. In this dissertation, I draw on state-of-the-art experimental technology to develop several new ideas for controlling and applying atomic systems. In the first part of this dissertation, we propose several novel schemes to realize, detect, and probe topological phases in atomic and atom-like systems. We first theoretically study the intriguing properties of Hopf insulators, a peculiar type of topological insulators beyond the standard classification paradigm of topological phases. Using a solid-state quantum simulator, we report the first experimental observation of Hopf insulators. We demonstrate the Hopf fibration with fascinating topological links in the experiment, showing clear signals of topological phase transitions for the underlying Hamiltonian. Next, we propose a feasible experimental scheme to realize the chiral topological insulator in three dimensions. They are a type of topological insulators protected by the chiral symmetry and have thus far remained unobserved in experiment. We then introduce a method to directly measure topological invariants in cold-atom experiments. This detection scheme is general and applicable to probe of different topological insulators in any spatial dimension. In another study, we theoretically discover a new type of topological gapless rings, dubbed a Weyl exceptional ring, in three-dimensional dissipative cold atomic systems. In the second part of this dissertation, we focus on the application of atomic systems in quantum computation and simulation. Trapped atomic ions are one of the leading platforms to build a scalable, universal quantum computer. The common one-dimensional setup, however, greatly limits the system's scalability. By solving the critical problem of micromotion, we propose a two-dimensional architecture for scalable trapped-ion quantum computation. Hamiltonian tomography for many-body quantum systems is essential for benchmarking quantum computation and simulation. By employing dynamical decoupling, we propose a scalable scheme for full Hamiltonian tomography. The required number of measurements increases only polynomially with the system size, in contrast to an exponential scaling in common methods. Finally, we work toward the goal of demonstrating quantum supremacy. A number of sampling tasks, such as the boson sampling problem, have been proposed to be classically intractable under mild assumptions. An intermediate quantum computer can efficiently solve the sampling problem, but the correct operation of the device is not known to be classically verifiable. Toward practical verification, we present an experimental friendly scheme to extract useful and robust information from the quantum boson samplers based on coarse-grained measurements. In a separate study, we introduce a new model built from translation-invariant Ising-interacting spins. This model possesses several advantageous properties, catalyzing the ultimate experimental demonstration of quantum supremacy.
We review methods that allow one to detect and characterize quantum correlations in many-body systems, with a special focus on approaches which are scalable. Namely, those applicable to systems with many degrees of freedom, without requiring a number of measurements or computational resources to analyze the data that scale exponentially with the system size. We begin with introducing the concepts of quantum entanglement, Einstein-Podolsky-Rosen steering, and Bell nonlocality in the bipartite scenario, to then present their multipartite generalization. We review recent progress on characterizing these quantum correlations from partial information on the system state, such as through data-driven methods or witnesses based on low-order moments of collective observables. We then review state-of-the-art experiments that demonstrate the preparation, manipulation and detection of highly-entangled many-body systems. For each platform (e.g. atoms, ions, photons, superconducting circuits) we illustrate the available toolbox for state preparation and measurement, emphasizing the challenges that each system poses. To conclude, we present a list of timely open problems in the field.
The enormous experimental progress in atomic, molecular and optical (AMO) physics during the last decades allows us nowadays to isolate single, a few or even many-body ensembles of microscopic particles, and to manipulate their quantum properties at a level of precision, which still seemed unthinkable some years ago. This versatile set of tools has enabled the development of the well-established concept of engineering of many-body Hamiltonians in various physical platforms. These available tools, however, can also be harnessed to extend the scenario of Hamiltonian engineering to a more general Liouvillian setting, which in addition to coherent dynamics also includes controlled dissipation in many-body quantum systems. Here, we review recent theoretical and experimental progress in different directions along these lines, with a particular focus on physical realizations with systems of atoms and ions. This comprises digital quantum simulations in a general open system setting, as well as engineering and understanding new classes of systems far away from thermodynamic equilibrium.