Entanglement plays a pervasive role nowadays throughout quantum information science, and at the same time provides a bridging notion between quantum information science and fields as diverse as condensed-matter theory, quantum gravity, and quantum foundations. In recent years, a notion of ''Generalized Entanglement'' (GE) has emerged, based on the idea that entanglement may be directly defined through expectation values of preferred observables -- without reference to a preferred subsystem decomposition. Preferred observables capture the physically relevant point of view, as defined by dynamical, operational, or fundamental constraints. While reducing to the standard entanglement notion when preferred observables are restricted to arbitrary local observables acting on individual subsystems, GE substantially expands subsystem-based entanglement theories, in terms of both conceptual foundations and range of applicability. Remarkably, the GE framework allows for non-trivial entanglement to exist within a single, indecomposable quantum system, demands in general a distinction between quantum separability and absence of entanglement, and naturally extends to situations where existing approaches may not be directly useful -- such as entanglement in arbitrary convex-cones settings and entanglement for indistinguishable quantum particles. In this paper, we revisit the main motivations leading to GE, and summarize the accomplishments and prospects of the GE program to date, with an eye toward conceptual developments and implications. In particular, we explain how the GE approach both shares strong points of contact with abstract operational quantum theories and, ultimately, calls for an observer-dependent redefinition of concepts like locality, completeness, and reality in quantum theory.
Entanglement plays a pervasive role nowadays throughout quantum information science, and at the same time provides a bridging notion between quantum information science and fields as diverse as condensed-matter theory, quantum gravity, and quantum foundations. In recent years, a notion of ''Generalized Entanglement'' (GE) has emerged, based on the idea that entanglement may be directly defined through expectation values of preferred observables -- without reference to a preferred subsystem decomposition. Preferred observables capture the physically relevant point of view, as defined by dynamical, operational, or fundamental constraints. While reducing to the standard entanglement notion when preferred observables are restricted to arbitrary local observables acting on individual subsystems, GE substantially expands subsystem-based entanglement theories, in terms of both conceptual foundations and range of applicability. Remarkably, the GE framework allows for non-trivial entanglement to exist within a single, indecomposable quantum system, demands in general a distinction between quantum separability and absence of entanglement, and naturally extends to situations where existing approaches may not be directly useful -- such as entanglement in arbitrary convex-cones settings and entanglement for indistinguishable quantum particles. In this paper, we revisit the main motivations leading to GE, and summarize the accomplishments and prospects of the GE program to date, with an eye toward conceptual developments and implications. In particular, we explain how the GE approach both shares strong points of contact with abstract operational quantum theories and, ultimately, calls for an observer-dependent redefinition of concepts like locality, completeness, and reality in quantum theory.
This paper addresses the following main question: Do we have a theoretical understanding of entanglement applicable to a full variety of physical settings? It is clear that not only the assumption of distinguishability, but also the few-subsystem scenario, are too narrow to embrace all possible physical settings. In particular, the need to go beyond the traditional subsystem-based framework becomes manifest when one tries to apply the conventional concept of entanglement to the physics of matter, since the constituents of a quantum many-body system are indistinguishable particles. We shall discuss here a notion of generalized entanglement, which can be applied to any operator language (fermions, bosons, spins, etc.) used to describe a physical system and which includes the conventional entanglement settings introduced to date in a unified fashion. This is realized by noticing that entanglement is an observer-dependent concept, whose properties are determined by the expectations of a distinguished set of observables without reference to a preferred subsystem decomposition, i.e., it depends on the physically relevant point of view. This viewpoint depends in turn upon the relationship between different sets of observables that determine our ability to control the system of interest. Indeed, the extent to which entanglement is present depends on the observables used to measure a system and describe its states. This represents a most conspicuous advantage as will be highlighted by the condensed-matter application we will discuss.
We present a notion of generalized entanglement which goes beyond the conventional definition based on quantum subsystems. This is accomplished by directly defining entanglement as a property of quantum states relative to a distinguished set of observables singled out by Physics. While recovering standard entanglement as a special case, our notion allows for substantially broader generality and flexibility, being applicable, in particular, to situations where existing tools are not directly useful.
We present a generalization of entanglement based on the idea that entanglement is relative to a distinguished subspace of observables rather than a distinguished subsystem decomposition. A pure quantum state is entangled relative to such a subspace if its expectations are a proper mixture of those of other states. Many information-theoretic aspects of entanglement can be extended to this observable-based setting, suggesting new ways of measuring and classifying multipartite entanglement. By going beyond the distinguishable-subsystem framework, generalized entanglement also provides novel tools for probing quantum correlations in interacting many-body systems.
Parastatistics is an exotic type of exchange statistics beyond fermions and bosons. Paraparticles transform in higher dimensional representations of the exchange symmetry group, analogous to non-Abelian anyons, yet consistently defined in any dimension. Although paraparticles have long been proposed, they were widely believed to be physically equivalent to fermions or bosons. Nevertheless, a recent paper proposed a different theory, called R-parastatistics, and demonstrated that nontrivial R-paraparticles can emerge as quasiparticles in condensed matter systems, and are observably distinct from both fermions and bosons. This paper develops the theoretical foundation and several extensions of R-parastatistics, with particular emphasis on its observable consequences. Central to this paper is a general theory of local observables extending the basic family introduced before. First, we define local observables that distinguish particle types. Second, we formulate local observables at special point defects that probe the internal indices of R-paraparticles, crucial for observing R-parastatistics and for the proposed applications in quantum information. Third, we introduce local observables that create or annihilate particle-antiparticle pairs, important for building a relativistic quantum field theory for R-paraparticles. We further introduce generalized hidden symmetries that act on internal indices of R-paraparticles while preserving the local observable algebra, providing a basis for proving local indistinguishability and for connecting to a categorical description of R-paraparticles. This work sets a solid theoretical foundation for understanding the fundamental physical properties of R-paraparticles and pave the way for finding them in nature.
Unentangled pure states on a bipartite system are exactly the coherent states with respect to the group of local transformations. What aspects of the study of entanglement are applicable to generalized coherent states? Conversely, what can be learned about entanglement from the well-studied theory of coherent states? With these questions in mind, we characterize unentangled pure states as extremal states when considered as linear functionals on the local Lie algebra. As a result, a relativized notion of purity emerges, showing that there is a close relationship between purity, coherence, and (non)entanglement. To a large extent, these concepts can be defined and studied in the even more general setting of convex cones of states. Based on the idea that entanglement is relative, we suggest considering these notions in the context of partially ordered families of Lie algebras or convex cones, such as those that arise naturally for multipartite systems. The study of entanglement includes notions of local operations and, for information-theoretic purposes, entanglement measures and ways of scaling systems to enable asymptotic developments. We propose ways in which these may be generalized to the Lie-algebraic setting and, to a lesser extent, to the convex-cones setting. One of our motivations for this program is to understand the role of entanglementlike concepts in condensed matter. We discuss how our work provides tools for analyzing the correlations involved in quantum phase transitions and other aspects of condensed-matter systems.
Quantum theory is considered to be the most fundamental and most accurate physical theory of today. Although quantum theory is conceptually difficult to understand, its mathematical structure is quite simple. What determines this particularly simple and elegant mathematical structure? In short: Why is quantum theory as it is? Addressing such questions is the aim of investigating the foundations of quantum theory. In the past this field of research was sometimes considered as an academic subject without much practical impact. However, with the emergence of quantum information theory this perception has changed significantly and both fields started to fruitfully influence each other. Today fundamental aspects of quantum theory attract increasing attention and the field belongs to the most exciting subjects of theoretical physics. This thesis is concerned with a particular branch in this field, namely, with so-called Generalized Probabilistic Theories (GPTs), which provide a unified theoretical framework in which classical and quantum theory emerge as special cases. This is used to examine nonlocal features that help to distinguish quantum theory from alternative toy theories. In order to extend the scope of theories that can be examined with the framework, we also introduce several generalizations to the framework itself. We start in Chapter 1 with introducing the standard GPT framework and summarize previous results, based on a review paper of the author [New J. Phys. 13, 063024 (2011)]. To keep the introduction accessible to a broad readership, we follow a constructive approach. Starting from few basic physically motivated assumptions we show how a given set of observations can be manifested in an operational theory. Furthermore, we characterize consistency conditions limiting the range of possible extensions. We point out that non-classical features of single systems can equivalently result from higher dimensional classical theories that have been restricted. Entanglement and non-locality, however, are shown to be genuine non-classical features. We review features that have been found to be specific for quantum theory separably or single and joint systems. Chapter 2 incorporates results published in [J. Phys. A 47(32), pp. 1-32 (2014)] and [Proc. QPL 2011 via EPTCS vol. 95, pp. 183–192 (2012)]. The GPT framework is applied to show how the structure of local state spaces indirectly affects possible nonlocal correlations, which are global properties of a theory. These correlations are stronger than those possible in a classical theory, but happen to show different restrictions that can be linked to the structure of subsystems. We first illustrate the phenomenon with toy theories with particular local state spaces. We than show that a particular class of joint states (inner product states), whose existence depends on geometrical properties of the local subsystems, can only have correlations for a known limited set called Q1. All bipartite correlations of both, quantum and classical correlations, can be mapped to measurement statistics from such joint states. Chapter 3 shows unpublished results on entanglement swapping in GPTs. This protocol, which is well known in quantum information theory, allows to nonlocally transfer entanglement to initially unentangled parties with the help of a third party that shares entanglement with each. We review our approach published in [Proc. QPL 2011 via EPTCS vol. 95, pp. 183–192 (2012)], which mimics the joint systems' structure of quantum theory by modifying a popular toy theory known as boxworld. However, it is illustrated that this approach fails for bigger multipartite systems due to inconsistencies evoked by entanglement swapping. It turns out that the GPT framework does not allow entanglement swapping for general subsystems with two-dimensional state spaces with transitive pure states. Altering the GPT framework to allow completely globally degrees of freedom, however, enables us to construct consistent entanglement swapping for these subsystems. This construction resembles the situation in quantum theory on a real Hilbert space. A questionable assumption usually taken in the standard GPT framework is the so-called no-restriction hypothesis. It states that the measurement that are possible in a theory can be derived from the state space. In fact, this assumption seems to exist for reasons of mathematical convenience, but it seems to lack physical motivation. We generalize the GPT framework to also account for systems that do not obey the no-restriction hypothesis in Chapter 4, which presents results published in [Phys. Rev. A 87, 052131 (2013)] and [Proc. QPL 2013, to be published in EPTCS]. The extended framework includes new classes of probabilistic theories. As an example, we show how to construct theories that include intrinsic noise. We also provide a "self-dualization" procedure that requires the violation of the no-restriction hypothesis. This procedure restricts the measurement of arbitrary theories such that the theories act as if they were self-dual. Self-duality has recently gathered lots of interest, since such theories share many features of quantum theory. For example Tsirelson’s bound holds for correlations on the maximally entangled state in these theories. Finally, we characterize the maximal set of joint states that can be consistently defined for given subsystems. This generalizes the maximal tensor product of the standard GPT framework.