One of the main foci of modern applied quantum information theory\nis the production of large-scale quantum entanglement involving many\nparticles, to achieve the next-generation quantum technologies. Although a full-scale quantum computer is still far away in time, in the next\nfew years we can expect to have quantum devices composed of up to a\nhundred controllable qubits. The primary challenge in moving towards\nsuch devices lies in the development of reliable and resource-efficient\ndetection techniques to prove genuine quantum advantage. The main\nobjective of this thesis is to investigate novel methods to detect the presence of quantum correlations in such systems, in particular, quantum\nentanglement and quantum nonlocality.\nThe first part of this thesis is dedicated to entanglement detection\nin large-scale quantum systems. Our main goal is to develop a novel\nprobabilistic method in which entanglement is seen as an ability of a\nquantum system to accomplish certain information-processing tasks.\nWe show that for certain classes of large (e.g. few tens of qubits) quantum states, even a single copy of a quantum state suffices to detect\nentanglement with a high confidence. Compared to the standard detection methods, this makes our method exceptionally resource-efficient.\nThe developed probabilistic scheme applies to multiple classes of states, vital for quantum computation, such as cluster states or ground\nstates of local Hamiltonians.\nIn the second part of this thesis, we develop a generic framework\nfor translating any entanglement witness into a resource-efficient probabilistic scheme. We show that the confidence level of entanglement\ndetection grows exponentially fast with the number of detection events,\nwhich makes this ansatz very efficient and reliable. Furthermore, we\npresent the first experimental performance of our method by verifying\nthe presence of entanglement in a photonic six-qubit cluster state.\nThe third part of this thesis describes the extension of our entanglement verification method to quantum nonlocality. Concrete examples\nare presented where we convert Bell’s inequalities into the probabilistic\nprocedure and incorporate our method into self-testing schemes. We\nshow that the performance of our method significantly exceeds the ability of the existing methods for detection of quantum nonlocality and\nself-testing.\nThe last part of this thesis is a study of the weak-convergence properties of random variables generated by quantum measurements. Beginning with a sequence of random variables generated by repeated\nunsharp quantum measurements, we study the limit distribution of\nmeasured relative frequency. We provide the de Finetti-type of representation theorem for all separable states, showing that the measured\ndistribution can be well approximated by a mixture of normal distributions. Additionally, we investigate the convergence rates and show\nthat the relative frequency converges to some constant at the rate of\norder 1=pN for all separable inputs. Finally, we provide an example of\na strictly unsharp quantum measurement where we obtain better scaling by using entangled inputs. We find such behaviour of entangled\nstates relevant for quantum information processing.
One of the main foci of modern applied quantum information theory\nis the production of large-scale quantum entanglement involving many\nparticles, to achieve the next-generation quantum technologies. Although a full-scale quantum computer is still far away in time, in the next\nfew years we can expect to have quantum devices composed of up to a\nhundred controllable qubits. The primary challenge in moving towards\nsuch devices lies in the development of reliable and resource-efficient\ndetection techniques to prove genuine quantum advantage. The main\nobjective of this thesis is to investigate novel methods to detect the presence of quantum correlations in such systems, in particular, quantum\nentanglement and quantum nonlocality.\nThe first part of this thesis is dedicated to entanglement detection\nin large-scale quantum systems. Our main goal is to develop a novel\nprobabilistic method in which entanglement is seen as an ability of a\nquantum system to accomplish certain information-processing tasks.\nWe show that for certain classes of large (e.g. few tens of qubits) quantum states, even a single copy of a quantum state suffices to detect\nentanglement with a high confidence. Compared to the standard detection methods, this makes our method exceptionally resource-efficient.\nThe developed probabilistic scheme applies to multiple classes of states, vital for quantum computation, such as cluster states or ground\nstates of local Hamiltonians.\nIn the second part of this thesis, we develop a generic framework\nfor translating any entanglement witness into a resource-efficient probabilistic scheme. We show that the confidence level of entanglement\ndetection grows exponentially fast with the number of detection events,\nwhich makes this ansatz very efficient and reliable. Furthermore, we\npresent the first experimental performance of our method by verifying\nthe presence of entanglement in a photonic six-qubit cluster state.\nThe third part of this thesis describes the extension of our entanglement verification method to quantum nonlocality. Concrete examples\nare presented where we convert Bell’s inequalities into the probabilistic\nprocedure and incorporate our method into self-testing schemes. We\nshow that the performance of our method significantly exceeds the ability of the existing methods for detection of quantum nonlocality and\nself-testing.\nThe last part of this thesis is a study of the weak-convergence properties of random variables generated by quantum measurements. Beginning with a sequence of random variables generated by repeated\nunsharp quantum measurements, we study the limit distribution of\nmeasured relative frequency. We provide the de Finetti-type of representation theorem for all separable states, showing that the measured\ndistribution can be well approximated by a mixture of normal distributions. Additionally, we investigate the convergence rates and show\nthat the relative frequency converges to some constant at the rate of\norder 1=pN for all separable inputs. Finally, we provide an example of\na strictly unsharp quantum measurement where we obtain better scaling by using entangled inputs. We find such behaviour of entangled\nstates relevant for quantum information processing.
Abstract One of the main challenges of quantum information is the reliable verification of quantum entanglement. The conventional detection schemes require repeated measurement on a large number of identically prepared systems. This is hard to achieve in practice when dealing with large-scale entangled quantum systems. In this letter we formulate verification as a decision procedure, i.e., entanglement is seen as the ability of quantum system to answer certain “yes-no questions”. We show that for a variety of large quantum states even a single copy suffices to detect entanglement with a high probability by using local measurements. For example, a single copy of a 16-qubit k-producible state or one copy of 24-qubit linear cluster state suffices to verify entanglement with more than 95% confidence. Our method is applicable to many important classes of states, such as cluster states or ground states of local Hamiltonians in general.
Quantum information has the potential to disrupt the present computational landscape. Much of this potential rests on the existence of efficient quantum algorithms for classically intractable problems and of quantum cryptographic protocols for tasks that are provably impossible to realize classically. At the heart of many quantum advantages is one of the most counterintuitive features of quantum mechanics, known as entanglement. The central motivating question of this thesis is the following: if quantum devices will perform tasks that are beyond the reach of classical devices, can we hope to certify that they are performing these tasks correctly? Bell's theorem, a landmark result in physics, provides a partial answer to this question: it asserts that measurements on spatially isolated, but entangled, particles can result in outcomes that are correlated in a way that cannot be explained by any local hidden variable theory (such as Newtonian physics). A direct operational consequence of this theorem is that one can devise a statistical test to certify the presence of entanglement (and hence of genuine quantumness). Remarkably, nature allows us to take this certification one step further: in some cases, the correlation of measurement outcomes is sufficient to single out a unique quantum setup compatible with this correlation. This phenomenon is often referred to as self-testing, and is the central topic of this thesis. In the first part of this thesis, we review the basic terminology and results in the theory of self-testing. We then explore a concrete application to the problem of verifiably delegating a quantum computation. Our main technical contribution is a test that robustly certifies products of single-qubit Clifford measurements on many EPR pairs. We employ this test to obtain a protocol which allows a classical user to verifiably delegate her quantum computation to two spatially isolated quantum servers. The overall complexity of our protocol is near-optimal, requiring resources that scale almost linearly in the size of the circuit being delegated. In the second part of this thesis, the driving question is the following: what is the class of quantum states and measurements that can be certified through self-testing? Does self-testing only apply to a few special cases, like EPR pairs or copies of EPR pairs, or are these instances of a more general phenomenon? One of the main results of this thesis is that we settle this question for the case of bipartite states. We show the existence of a self-testing correlation for any pure bipartite entangled state of any finite local dimension. We then move on to explore the multipartite case, and we show that a significantly larger class of states can be self-tested than was previously known. This includes all multipartite partially entangled GHZ states, and more generally all multipartite qudit states which admit a Schmidt decomposition. In the final part of this thesis, we explore connections of the theory of self-testing to basic questions about entanglement and quantum correlation sets. In particular, we set out to understand the expressive power of infinite-dimensional quantum systems. We consider two questions: can spatially isolated quantum systems of infinite dimension produce correlations that are unattainable by finite-dimensional systems? Does there exist a correlation that cannot be attained exactly by spatially isolated quantum systems (not even infinite-dimensional ones), but can be approximated arbitrarily well by a sequence of finite or infinite-dimensional systems? The first question was posed by Tsirelson in 1993, and its answer has been elusive. One of the main results of this thesis is a resolution of this question. The second question is better known as the "non-closure of the set of quantum correlations", and was answered affirmatively in a breakthrough of Slofstra. We give a new elementary proof of this result which leverages one of our self-testing results and a phenomenon known as embezzlement.
Although the realization of useful quantum computers poses significant challenges, swift progress in emerging quantum technologies is making this goal realistically approachable. In this context, one of the essential resources is quantum entanglement, which allows for quantum computations outperforming their classical counterparts. However, the task of entanglement detection is not always straightforward for several reasons. One of the main challenges is that standardly-used methods rapidly become unfeasible when dealing with quantum states containing more than a few qubits. Typically, this is due to the fact that a vast amount of measurements is needed on many copies of the state. Generally, it is not unusual to deal with a very limited number of state copies in experimental settings - in fact, this may be the case for many large quantum systems. In this article, an overview is provided of a probabilistic approach that enables high-confidence genuine multipartite entanglement detection using an exceptionally low number of state copies. Additionally, a study is presented that shows that this protocol remains efficient also in the presence of noise, thus confirming the practicality of the method for near-term quantum devices and its suitability for complex experimental settings.
In the past two decades, the union of quantum mechanics and information science has led to significant progress in our understanding of the quantum world and in our ability to coherently control individual quantum systems. New technologies have appeared, and some of them, such as quantum key distribution (QKD), are now commercially available. We are now envisioning quantum networks that could allow us to communicate over intercontinental scales, as the internet does, but with a provable security. An open question in this framework is how to certify that a given quantum network performs well, i.e. how to detect entanglement between 2, 5, 10 or 100 nodes located far away from each other. Bell tests are appealing for such a task as they allow one to certify the presence of entanglement deviceindependently, i.e. without assumptions on the Hilbert space dimension or on the proper alignment of the measurement devices. In the first chapter, I start by examining in depth the most standard approach to perform a Bell test in which a source based on spontaneous parametric down conversion (SPDC) is used to create photon pairs entangled in polarization. I show, in particular, that in this case the Bell inequality violation is fundamentally limited by vacuum and multi-photon emissions. In the second part of this thesis, I show how to overcome this limitation by using the SPDC source to create single photon entanglement, i.e. entanglement between two paths sharing a single photon, or two-mode squeezed states. Here, the non local content of the states is revealed using photon counting techniques preceded by small displacements in phase space. In the third part, I show how these measurements can be used for an optomechanical Bell test. This opens the way for device-independent quantum information processing with optomechanical systems and might be relevant to test collapse models in a theory independent manner. In the fourth chapter, the same measurements are used to build up a witness that can reveal genuine path entanglement. This witness has triggered an experiment which highlights its potential for network certification. I conclude the thesis by showing how path entanglement can be revealed with the human eye by exploiting these displacement operations.
Quantum entanglement, has been acknowledged as a precious resource due to its inherent nonclassical correlations between subsystems. These quantum correlations have the potential for many quantum processes, including canonical ones: quantum cryptography, quantum teleportation, and dense coding. To exploit the advantages of quantum entanglement, two essential premises are required, i.e., to prepare high-quality entanglement and characterize quality level of prepared entanglement. Thus far, quantum entanglement can be produced in various quantum systems; however, it appears that this new resource is complex and difficult to characterize. The standard methods to characterize multipartite entanglement, e.g., entanglement witness, state tomography, or quantum state verification, require full knowledge of the Hilbert space dimension and precise calibration of measurement devices, which are usually difficult to acquire in experiment. The most radical way to overcome these problems is to detect entanglement solely based on the Bell-like correlations of measurement outcomes collected in the experiment, namely, device-independent characterization of entanglement. This article reviews the recently developed device-independent methods to characterize entanglement, including self-testing and device-independent certification of entanglement. These approaches can be widely applied in kinds of quantum information processing, especially for those with security demands.
Entanglement is one of the most studied properties of quantum mechanics for its application in quantum information protocols. Nevertheless, detecting the presence of entanglement in large multipartite sates keeps being a great challenge both from the theoretical and the experimental point of view. Most of the known methods either have computational costs that scale inefficiently with the number of parties or require more information on the state than what is attainable in every-day experiments. We introduce a new technique for entanglement detection that provides several important advantages in these respects. First, its scales efficiently with the number of parties, thus allowing for application to systems composed by up to few tens of parties. Second, it needs only the knowledge of a subset of all possible measurements on the state, therefore being apt for experimental implementation. Moreover, since it is based on the detection of nonlocality, our method is device-independent. We report several examples of its implementation for well-known multipartite states, showing that the introduced technique has a promising range of applications.