Is entanglement an exclusive feature of quantum systems, or is it common to all non-classical theories? And if this is the case, how strong is quantum mechanical entanglement as compared to that exhibited by other theories? The first part of this thesis deals with these questions by considering quantum theory as part of a wider landscape of physical theories, collectively called general probabilistic theories (GPTs). Among the other things, this manuscript contains a detailed introduction to the abstract state space formalism for GPTs. We start with a comprehensive review of the proof of a famous theorem by Ludwig that constitutes one of its cornerstones (Ch. 1). After explaining the basic rules of the game, we translate our questions into precise conjectures and present our progress toward a full solution (Ch. 2). In Ch. 3 we consider entanglement at the level of measurements instead of states, focusing on one of its main implications, i.e. data hiding. We determine the maximal data hiding strength that a quantum mechanical system can exhibit, and also the maximum value among all GPTs, finding that the former scales as the square root of the latter. In the second part of this manuscript we look into quantum entanglement. In Ch. 4 we discuss the entanglement transformation properties of a class of maps that model white noise acting either locally or globally on a bipartite system. In Ch. 5 we employ matrix analysis tools to develop a unified approach to Gaussian entanglement. The third part of this thesis concerns more general forms of non-classical correlations in bipartite continuous variable systems. In Ch. 6 we devise a general scheme that allows to consistently classify correlations of bipartite Gaussian states into classical and quantum ones. Finally, Ch. 7 explores some problems connected with a certain strong subadditivity matrix inequality.
Is entanglement an exclusive feature of quantum systems, or is it common to all non-classical theories? And if this is the case, how strong is quantum mechanical entanglement as compared to that exhibited by other theories? The first part of this thesis deals with these questions by considering quantum theory as part of a wider landscape of physical theories, collectively called general probabilistic theories (GPTs). Among the other things, this manuscript contains a detailed introduction to the abstract state space formalism for GPTs. We start with a comprehensive review of the proof of a famous theorem by Ludwig that constitutes one of its cornerstones (Ch. 1). After explaining the basic rules of the game, we translate our questions into precise conjectures and present our progress toward a full solution (Ch. 2). In Ch. 3 we consider entanglement at the level of measurements instead of states, focusing on one of its main implications, i.e. data hiding. We determine the maximal data hiding strength that a quantum mechanical system can exhibit, and also the maximum value among all GPTs, finding that the former scales as the square root of the latter. In the second part of this manuscript we look into quantum entanglement. In Ch. 4 we discuss the entanglement transformation properties of a class of maps that model white noise acting either locally or globally on a bipartite system. In Ch. 5 we employ matrix analysis tools to develop a unified approach to Gaussian entanglement. The third part of this thesis concerns more general forms of non-classical correlations in bipartite continuous variable systems. In Ch. 6 we devise a general scheme that allows to consistently classify correlations of bipartite Gaussian states into classical and quantum ones. Finally, Ch. 7 explores some problems connected with a certain strong subadditivity matrix inequality.
Inspired by its fundamental importance in quantum mechanics, we define and study the notion of entanglement for abstract physical theories, investigating its profound connection with the concept of superposition. We adopt the formalism of general probabilistic theories (GPTs), encompassing all physical models whose predictive power obeys minimal requirements. Examples include classical theories, which do not exhibit superposition and whose state space has the shape of a simplex, quantum mechanics, as well as more exotic models such as Popescu-Rohrlich boxes. We call two GPTs entangleable if their composite admits either entangled states or entangled measurements, and conjecture that any two non-classical theories are in fact entangleable. We present substantial evidence towards this conjecture by proving it (1) for the simplest case of 3-dimensional theories; (2) when the local state spaces are discrete, which covers foundationally relevant cases; (3) when one of the local theories is quantum mechanics. Furthermore, (4) we envision the existence of a quantitative relation between local non-classicality and global entangleability, explicitly describing it in the geometrically natural case where the local state spaces are centrally symmetric.
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.
This thesis explores various perspectives on quantum phenomena, and how our understanding of these phenomena is informed by the study of general probabilistic theories. Particular attention is given to quantum nonlocality, and its interaction with areas of physical and mathematical interest such as entropy, reversible dynamics, information-based games and the idea of negative probability. We begin with a review of non-signaling distributions and convex operational theories, including “black box” descriptions of experiments and the mathematics of convex vector spaces. In Chapter 3 we derive various classical and quantum-like quasiprobabilistic representations of arbitrary non-signaling distributions. Previously, results in which the density operator is allowed to become non-positive [1] have proved useful in derivations of quantum theory from physical requirements [2]; we derive a dual result in which the measurement operators instead are allowed to become non-positive, and show that the generation of any non-signaling distribution is possible using a fixed separable state with negligible correlation. We also derive two distinct “quasi-local” models of non-signaling correlations. Chapter 4 investigates non-local games, in particular the game known as Information Causality. By analysing the probability of success in this game, we prove the conjectured tightness of a bound given in [3] concerning how well entanglement allows us to perform the task of random access coding, and introduce a quadratic bias bound which seems to capture a great deal of information about the set of quantum-achievable correlations. By reformulating Information Causality in terms of entropies, we find that a sensible measure of entropy precludes many general probabilistic theories whose non-locality is stronger than that of quantum theory. Chapter 5 explores the role that reversible transitivity (the principle that any two pure states are joined by a reversible transformation) plays as a characteristic feature of quantum theory. It has previously been shown that in Boxworld, the theory allowing for the full set of non-signaling correlations, any reversible transformation on a restricted class of composite systems is merely a composition of relabellings of measurement choices and outcomes, and permutations of subsystems [4]. We develop a tabular description of Boxworld states and effects first introduced in [5], and use this to extend this reversibility result to any composite Boxworld system in which none of the subsystems are classical.
General probabilistic theories (GPTs) are theoretical models having states, measurements, and probabilities of obtaining measurement outcomes.For example, classical probability theory and quantum theory are GPTs with probability vectors and density matrices as states, respectively.In the study of GPTs, one often describes a finite number of GPTs A 1 , . . ., A n together, as one GPT A = A 1 • • • A n .In this case, we say that A is a whole system of subsystems A 1 , . . ., A n .If A 1 , . . ., A n are quantum systems, then A is called a locally quantum system.A quantum system is locally quantum, but there are many other locally quantum systems.In this thesis, we study three topics associated with quantum systems and locally quantum systems.The first one is capacity.For a GPT, the maximum number of simultaneously and perfectly distinguishable states is called the capacity.It is known that the capacity does not really depend on the GPT.In this thesis, we propose a statement S to determine the capacities of special locally quantum systems, and prove a weaker statement WS.Statement WS is a statement associated with tensor product spaces over C. Our proof works even if replacing the scalar field C with an arbitrary infinite field F. We also show a partial result for finite fields.The second one is differential privacy (DP).DP was born in the study to utilize private data while protecting the data (privacy-preserving data mining, PPDM).For this reason, DP has been studied by using classical probability theory, and has almost never been studied in quantum information theory.In this thesis, we define a quantum version of DP (called classical-quantum DP) and investigate its mathematical aspects.We define the set CQ n of all classical-quantum differentially private n-tuples of quantum states and the subset EC n of CQ n that is essentially classical.We prove that EC n = CQ n if n = 2 and EC n = CQ n if n ≥ 3.The third one is perfect discrimination of two states.In quantum theory, orthogonality is a necessary and sufficient condition to distinguish two states perfectly (denote this equivalence by E).However, equivalence E does not hold for a general GPT.In this thesis, we construct continuous one-parameter families of locally quantum systems in a certain natural manner, and investigate perfect discrimination of two states in the framework of those locally quantum systems.These one-parameter families contain a locally quantum system sufficiently close to a quantum system.Nevertheless, we show that they violate equivalence E.
This thesis deals with General Probabilistic Theories (GPTs) and Entanglement Structures (ESs). An ES is a possible structure of a quantum composite system in GPTs, which is not uniquely determined as the Standard Entanglement Structure (SES). It is an important problem to find reasonable postulates that determine various ESs as the SES. In order to solve this problem, this thesis explores the diversity of ESs. The topics of this thesis are roughly divided into two parts. First, this thesis considers state discrimination in ESs. As a main result, this thesis gives equivalent conditions for a given measurement in ESs to have a performance superior to standard quantum theory. Second, this thesis focuses on symmetry and self-duality. As a main result, this thesis gives derivations of the SES by symmetric conditions. On the other hand, this thesis clarifies that there are infinitely many self-dual ESs, even if they cannot be distinguished from the SES by a certain physical experiment with small errors.
A long-standing problem in the foundations of quantum mechanics is to identify a physical principle that explains why algebraically maximal violations of Bell inequalities can generally not be achieved in Nature. One recently proposed approach considers iterated Bell tests, where a Bell test is performed on a state that has undergone several rounds of entanglement swapping. Obtaining large violations in this scenario is more demanding, because it requires a theory to have both highly entangled states and highly entangled measurements. It has been conjectured that the maximal quantum mechanical Clauser-Horne-Shimony-Holt (CHSH)-value of 22 might be optimal for any probabilistic theory which, like quantum mechanics, maintains its CHSH-value after an arbitrary number of rounds of entanglement swapping. However, in a previous paper, we have exhibited a first example of a probabilistic theory that can sustain a CHSH value of 4 in this setting. In this work, further investigating this property, we give a classification of all general probabilistic theories (GPTs) whose CHSH value is stable in the above sense. The problem reduces to a representation-theoretic condition that allows for exactly seven solutions. The GPT from our previous work showed some counter-intuitive features, e.g. that the local state space had a higher dimension than seemed necessary to realize CHSH tests. The classification shows that this is necessarily so. Along the way, we generalize the concept of self-testing to GPTs.