Abstract Quantum information theory is built upon the realisation that quantum resources like coherence and entanglement can be exploited for novel or enhanced ways of transmitting and manipulating information, such as quantum cryptography, teleportation, and quantum computing. We now know that there is potentially much more than entanglement behind the power of quantum information processing. There exist more general forms of non-classical correlations, stemming from fundamental principles such as the necessary disturbance induced by a local measurement, or the persistence of quantum coherence in all possible local bases. These signatures can be identified and are resilient in almost all quantum states, and have been linked to the enhanced performance of certain quantum protocols over classical ones in noisy conditions. Their presence represents, among other things, one of the most essential manifestations of quantumness in cooperative systems, from the subatomic to the macroscopic domain. In this work we give an overview of the current quest for a proper understanding and characterisation of the frontier between classical and quantum correlations (QCs) in composite states. We focus on various approaches to define and quantify general QCs, based on different yet interlinked physical perspectives, and comment on the operational significance of the ensuing measures for quantum technology tasks such as information encoding, distribution, discrimination and metrology. We then provide a broader outlook of a few applications in which quantumness beyond entanglement looks fit to play a key role.
Abstract Quantum information theory is built upon the realisation that quantum resources like coherence and entanglement can be exploited for novel or enhanced ways of transmitting and manipulating information, such as quantum cryptography, teleportation, and quantum computing. We now know that there is potentially much more than entanglement behind the power of quantum information processing. There exist more general forms of non-classical correlations, stemming from fundamental principles such as the necessary disturbance induced by a local measurement, or the persistence of quantum coherence in all possible local bases. These signatures can be identified and are resilient in almost all quantum states, and have been linked to the enhanced performance of certain quantum protocols over classical ones in noisy conditions. Their presence represents, among other things, one of the most essential manifestations of quantumness in cooperative systems, from the subatomic to the macroscopic domain. In this work we give an overview of the current quest for a proper understanding and characterisation of the frontier between classical and quantum correlations (QCs) in composite states. We focus on various approaches to define and quantify general QCs, based on different yet interlinked physical perspectives, and comment on the operational significance of the ensuing measures for quantum technology tasks such as information encoding, distribution, discrimination and metrology. We then provide a broader outlook of a few applications in which quantumness beyond entanglement looks fit to play a key role.
Quantum coherence and quantum correlations are of fundamental and practical significance for the development of quantum mechanics.They are also cornerstones of quantum computation and quantum communication theory. Searching physically meaningful and mathematically rigorous quantifiers of them are long-standing concerns of the community of quantum information science, and various faithful measures have been introduced so far. We review in this paper the measures of discordlike quantum correlations for bipartite and multipartite systems, the measures of quantum coherence for any single quantum system, and their relationship in different settings. Our aim is to provide a full review about the resource theory of quantum coherence, including its application in many-body systems, and the discordlike quantum correlations which were defined based on the various distance measures of states. We discuss the interrelations between quantum coherence and quantum correlations established in an operational way, and the fundamental characteristics of quantum coherence such as their complementarity under different basis sets, their duality with path information of an interference experiment, their distillation and dilution under different operations, and some new viewpoints of the superiority of the quantum algorithms from the perspective of quantum coherence. Additionally, we review properties of geometric quantum correlations and quantum coherence under noisy quantum channels. Finally, the main progresses for the study of quantum correlations and quantum coherence in the relativistic settings are reviewed. All these results provide an overview for the conceptual implications and basic connections of quantum coherence, quantum correlations, and their potential applications in various related subjects of physics.
The project concerned the study of quantum correlations (QC) in compound systems, i.e. statistical correlations more general than entanglement which are predicted by quantum mechanics but not described in any classical scenario. I aimed to understand the technical and operational properties of the measures of QC, their interplay with entanglement quantifiers and the experimental accessibility.\nIn the first part of my research path, after having acquired the conceptual and technical rudiments of the project, I provided solutions for some computational issues: I developed analytical and numerical algorithms for calculating bipartite QC in finite dimensional systems. Then, I tackled the problem of the experimental detection of QC. There is no Hermitian operator associated with entanglement measures, nor with QC ones. However, the information encoded in a density matrix is redundant to quantify them, thus the full knowledge of the state is not required to accomplish the task. I reported the first protocol to measure the QC of an unknown state by means of a limited number of measurements, without performing the tomography of the state. My proposal has been implemented experimentally in a NMR (Nuclear Magnetic Resonance) setting.\nIn the final stage of the project, I explored the foundational and operational merits of QC. I showed that the QC shared by two subsystems yield a genuinely quantum kind of uncertainty on single local observables. The result is a promising evidence of the potential exploitability of separable (unentangled) states for quantum metrology in noisy conditions.
The most common type of quantum correlations is quantum entanglement, which plays an indisputable vital role in a number of quantum information theory tasks, including quantum teleportation, quantum dense coding, and quantum cryptography. However, current findings indicate that different Entanglement is not necessary for applications in quantum information theory, and their performance can be described by a novel kind of quantum correlations that transcends entanglement. The most well-liked candidate for such widespread quantum connections is quantum discord, which Zurek first proposed more than ten years ago. We provide an overview of this current line of inquiry in this work. Following a brief overview of the fundamental ideas of quantum theory and entanglement, we introduce quantum discord and generic quantum correlations and go over three applications that are predicated on this novel kind of correlation: transmission of correlations, entanglement dispersion, and remote state preparation. We also provide an outlook for future research in this area.
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.
Quantum correlations in a physical system are usually studied with respect to a unique and fixed decomposition of the system into subsystems, without fully exploiting the rich structure of the state space. Here, we show several examples in which the consideration of different ways to decompose a physical system enhances the quantum resources and accounts for a more flexible definition of quantumness measures. Furthermore, we give a different perspective regarding how to reassess the fact that local operations play a key role in general quantumness measures that go beyond entanglement---as discordlike ones. We propose a family of measures to quantify the maximum quantumness of a given state. For the discord-based case, we present some analytical results for $2\ifmmode\times\else\texttimes\fi{}d$-dimensional states. Applying our definition to low-dimensional bipartite states, we show that different behaviors can be reported for separable and entangled states vis-\`a-vis those corresponding to the usual measures of quantum correlations. We show that there is a close link between our proposal and the criterion to witness quantum correlations based on the rank of the correlation matrix, proposed by Daki\ifmmode \acute{c}\else \'{c}\fi{}, Vedral, and Brukner [Phys. Rev. Lett. 105, 190502 (2010)].
Quantum entanglement is the most popular kind of quantum correlations. Its role in several tasks in quantum information theory like quantum cryptography, quantum dense coding and quantum teleportation is indisputable. The role of entanglement in quantum algorithms has also attracted enormous amount of attention in the literature. The research in this direction is motivated by the finding that a quantum computer can efficiently factorize arbitrary integer numbers. Since no classical algorithm is known today that can solve this problem efficiently, this example clearly demonstrates the superiority of a quantum computer compared to its classical counterpart. In this context, entanglement also plays a crucial role: if the quantum computer operates on a pure state without entanglement, then the computational process can be simulated efficiently on a classical computer. In this case entanglement is the key ingredient that makes the difference between a quantum and a classical computer.
However, the situation is more involved if the computational process is not perfect. In this case the quantum computer operates on a mixed state, and the role of entanglement is less obvious. A seminal result in this direction is a quantum algorithm known under the acronym DQC1. This algorithm operates on an almost maximally mixed state with vanishingly little entanglement. However, DQC1 can still perform tasks efficiently for which no efficient classical algorithm is known today. Thus, it is reasonable to assume that entanglement in general is not related to the efficiency of a quantum algorithm.
Triggered by this observation other kinds of quantum correlations have been studied. In this context a particular measure of quantum correlations known as quantum discord is referred to frequently in the literature. Quantum discord is a measure of quantum correlations beyond entanglement, i.e., a quantum state can have no entanglement, but nonvanishing quantum discord. Quantum discord is also regarded as the key resource in the DQC1 algorithm. Although this statement is still controversial, it was shown under very general assumptions that quantum discord is required if a quantum computer is to show an exponential speedup over any classical algorithm.
In this thesis we give a short introduction into the theory of entanglement and general quantum correlations, and further discuss our results. In particular, we show that two different kinds of entanglement measures can coincide. This result is used to build an algorithm for computing entanglement. We also consider the role of quantum correlations for the distribution of entanglement and for the quantum measurement process. We further discuss the behavior of quantum correlations under local noise, and their monogamy properties.