Since the 1980s, there has been a physical problem relating to the detection of multipartite entangled states that helps us to investigate fundamental point of view on interpretations of quantum mechanics. On the other hand, information theories based on the mathematical structure of quantum mechanics, for example, quantum cryptography and quantum computation, have been investigated by researchers whose number has been increasing day by day in the world. This field became to be called quantum information theories. A practical experimental realization of quantum information processing needs that we detect experimentally multipartite entangled states. The number of researchers for multipartite entangled states is increasing rapidly. Especially, Greenberger-Horne-Zeilinger (GHZ) states are useful for quantum information processing information processing using quantum mechanics, which improves the conventional information processing ability which had been based on classical theories. However, practical experimental realization of the theories and controlling quantum states have inevitable difficulties due to the interaction with environments. The requirement in maintaining multipartite entangled states in relation to the time that takes to make a practical experiment is also a problem. Many experimentalists are tackling this problem, and recently, several groups have reported that multipartite entangled states have been physically detected in their laboratories. The main difficulty is how to judge whether or not the observed quantum state is indeed a multipartite entangled state. For this purpose, it is necessary to construct theoretical models to analyze such experimental data. In this thesis, we apply a mathematical structure as to density operators described in a high-dimensional Hilbert space, and we employ mathematical theories to prove if the corresponding density operator represents a multipartite entangled state or not. The main aim for constructing such models is to analyze the raw experimental data. Moreover, this model helps experimentalists in the sense that it allows them to see how to confirm that observed states are indeed the desired state. With mathematical theories developed before this thesis, only limited applications for this purpose were allowed. Some theories cover two-dimensional Hilbert space only. It has not been even discussed how to analyze the fidelity to GHZ states of the observed state. This has led to some doubt of that experimental data might not always confirm three-partite entangled states. However, by using the method of this thesis, statistical check based on such experimental data of the existence of three particle-entangled states is possible under appropriate assumptions. This example symbolizes that our analytical knowledge has improved very much in the last few years. This thesis gives us an advanced mathematical model for analyzing experimental data.
Since the 1980s, there has been a physical problem relating to the detection of multipartite entangled states that helps us to investigate fundamental point of view on interpretations of quantum mechanics. On the other hand, information theories based on the mathematical structure of quantum mechanics, for example, quantum cryptography and quantum computation, have been investigated by researchers whose number has been increasing day by day in the world. This field became to be called quantum information theories. A practical experimental realization of quantum information processing needs that we detect experimentally multipartite entangled states. The number of researchers for multipartite entangled states is increasing rapidly. Especially, Greenberger-Horne-Zeilinger (GHZ) states are useful for quantum information processing information processing using quantum mechanics, which improves the conventional information processing ability which had been based on classical theories. However, practical experimental realization of the theories and controlling quantum states have inevitable difficulties due to the interaction with environments. The requirement in maintaining multipartite entangled states in relation to the time that takes to make a practical experiment is also a problem. Many experimentalists are tackling this problem, and recently, several groups have reported that multipartite entangled states have been physically detected in their laboratories. The main difficulty is how to judge whether or not the observed quantum state is indeed a multipartite entangled state. For this purpose, it is necessary to construct theoretical models to analyze such experimental data. In this thesis, we apply a mathematical structure as to density operators described in a high-dimensional Hilbert space, and we employ mathematical theories to prove if the corresponding density operator represents a multipartite entangled state or not. The main aim for constructing such models is to analyze the raw experimental data. Moreover, this model helps experimentalists in the sense that it allows them to see how to confirm that observed states are indeed the desired state. With mathematical theories developed before this thesis, only limited applications for this purpose were allowed. Some theories cover two-dimensional Hilbert space only. It has not been even discussed how to analyze the fidelity to GHZ states of the observed state. This has led to some doubt of that experimental data might not always confirm three-partite entangled states. However, by using the method of this thesis, statistical check based on such experimental data of the existence of three particle-entangled states is possible under appropriate assumptions. This example symbolizes that our analytical knowledge has improved very much in the last few years. This thesis gives us an advanced mathematical model for analyzing experimental data.
Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.
Despite the success of quantum mechanics in predicting the outcomes of experiments in many branches of physics, the foundations of the theory have remained subject of research and dispute. At the basis of this struggle with the theory lie the phenomena of nonlocality and entanglement. Since it was first predicted by Bell in 1964, nonlocality was not only verified in numerous experiments, but also identified as a useful resource for quantum information processing. Thus, the study of nonlocality is important both from a fundamental point of view and with respect to new applications in quantum information theory, such as secure cryptography and randomness generation. The identification of entanglement as a resource for quantum information led to a strong theoretical effort devoted to its characterisation and detection. Many of the resulting mathematical tools find application in several domains of physics. Although the only known way to create nonlocal correlations is to measure entangled quantum systems, it has been shown that entanglement and nonlocality constitute two inequivalent properties. Therefore, in the light of the success of entanglement theory, it is of interest to also devise a resource theory of nonlocality. The task of this thesis is to develop such a theoretical framework for the characterisation of nonlocality as a resource. To gain a better understanding of nonlocal correlations it will be necessary to investigate correlation scenarios that go beyond the situation originally considered by Bell. In doing so, this thesis provides new description of nonlocality that also have implications for the characterisation of quantum correlations and the detection of new forms of nonlocality. The first question we address is how nonlocality can consistently be defined in a scenario of arbitrarily many parties that may collaborate among each other. To this end we recognise which are the allowed physical operations in this situation and then define nonlocality as the resource that cannot be created by these operations. Our approach shows that the conventional definition of multipartite nonlocality, adopted by the community so far, is inconsistent with this operational definition; we further propose and analyse new models that do not suffer from these inconsistencies. Furthermore, we show that our findings have implications for the characterisation of quantum correlations. A recent approach to describe the set of quantum correlation consists in using principles inspired from information theory. By using a special instance of the models we defined earlier we show a fundamental limitation of this approach: no bipartite information principle is sufficient to single out the set of quantum correlations from the set of nonsignalling correlations. We then developed a description of nonlocality in an even more generalised scenario of several parties. Motivated by a result of Popescu we study scenarios where the parties are allowed to perform not only a single but sequences of measurements on their systems. Characterising nonlocality also in this scenario in operational terms and defining local models compatible with this definition, we show that a new form of nonlocality can be detected Lastly, we examine the problem of detecting the presence of nonlocality in a multipartite scenario when one is given only partial access to the global system. We find that one can verify that the total system must display nonlocality, even though the accessible subsystems only exhibit local correlations.
The focal point of this thesis is strongly based on quantum entanglement within Quantum Information Science. We provide an indepth analysis of entanglement quantification in both bipartite and multipartite quantum systems, with an emphasis on many-body entanglement. We examine the definitions of entanglement, introduce several entanglement measures, and examine the challenges in expanding these measures to multipartite scenarios. For bipartite systems, we detail entropy-based measures such as von Neumann and Rényi entropies, the entanglement spectrum and important algebraic metrics such as negativity and concurrence. We then elaborate on multipartite entanglement, introducing modern measures and detection techniques including geometric and global entanglement measurements, generalised multipartite concurrence, multipartite negativity, fidelity-based entanglement benchmarks, and entanglement witnesses. Some examples include Greenberger-Horne-Zeilinger (GHZ), and the W states to illustrate the distinction between biseparable and truly multi-partite-entangled states, as well as the SLOCC classification of families of entangled states. We also briefly introduce practical ways of measuring entanglement, such as the SWAP test, quantum state tomography for experimental estimation of entanglement and consequently randomised measurements. We apply these measures to specific quantum states (GHZ, W , graph states, and random states), analysing how entanglement evolves under operations and how different metrics capture distinct features of multi-qubit entanglement. Finally, we conclude with a summary of findings, a discussion of current limitations such as scalability of measures. The thesis is organised into five chapters: Introduction, Entanglement, Multipartite Entanglement Measures, Negativity vs. Three-Tangle, and Conclusion.
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.
Judging by the compelling number of innovations based on taming quantum mechanical effects, such as the development of transistors and lasers, further research in this field promises to tackle further technological challenges in the years to come. This statement gains even more importance in the information processing scenario. Here, the growing data generation and the correspondingly higher need for more efficient computational resources and secure high bandwidth networks are central problems which need to be tackled. In this sense, the required CPU minituarization makes the design of structures at atomic levels inevitable, as foreseen by Moore's law.
From these perspectives, it is necessary to concentrate further research efforts into controlling and manipulating quantum mechanical systems. This enables for example to encode quantum superposition states to tackle problems which are computationally NP hard and which therefore cannot be solved efficiently by classical computers. The only limitation affecting these solutions is the low scalability of existing quantum systems. Similarly, quantum communication schemes are devised to certify the secure transmission of quantum information, but are still limited by a low transmission bandwidth.
This thesis follows the guideline defined by these research projects and aims to further increase the scalability of the quantum mechanical systems required to perform these tasks. The method used here is to encode quantum states into photons generated by spontaneous parametric down-conversion (SPDC). An intrinsic limitation of photons is that the scalability of quantum information schemes employing them is limited by the low detection efficiency of commercial single photon detectors. This is addressed by encoding higher dimensional quantum states into two photons, increasing the scalability of the scheme in comparison to multi-photon states. Further on, the encoding of quantum information into the emission-time degree of freedom improves its applicability to long distance quantum communication schemes. By doing that, the intrinsic limitations of other schemes based on the encoding into the momentum and polarization degree of freedom are overcome.
This work presents results on a scalable experimental implementation of time-energy encoded higher dimensional states, demonstrating the feasibility of the scheme. Further tools are defined and used to characterize the properties of the prepared quantum states, such as their entanglement, their dimension and their preparation fidelity. Finally, the method of quantum state tomography is used to fully determine the underlying quantum states at the cost of an increased measurement effort and thus operation time. It is at this point that results obtained from the research field of compressed sensing help to decrease the necessary number of measurements. This scheme is compared with an adaptive tomography scheme designed to offer an additional reconstruction speedup. These results display the scalability of the scheme to bipartite dimensions higher than 2x8, equivalent to the encoding of quantum information into more than 6 qubits.
The research field of quantum entanglement theory is comparatively new. While a basic understanding of the most simple systems in question (i.e. bipartite systems) has been established over the past few decades, multipartite entanglement still holds many unsolved questions and intriguing riddles. In particular, it is completely unclear how several concepts from the bipartite case can be generalised in a meaningful way to multipartite scenarios. In this work, the main issues of multipartite entanglement detection, characterisation and classification are discussed. The differences and similarities between the bipartite and the multipartite situation are reviewed, various possible generalisations are presented and results are obtained in several areas. The focus of this work particularly lies on a formalism - the so called HMGH framework, which has been developed and expanded for the past several years - which allows for construction of very specific separability criteria, capable of discriminating between different kinds of multipartite entanglement. By means of these criteria, the questions of partial separability, genuine multipartite entanglement and - ultimately - multipartite entanglement classification (which appear to contain the most striking differences to bipartite entanglement) are adressed and discussed. In order to illustrate the theoretical conclusions in these respects, several examples are given from different (and differently closely related) fields, showing the capabilities, strengths and weaknesses of the HMGH framework as well as giving insights into the current status of research in multipartite entanglement theory as a whole.