What singles out quantum mechanics as the fundamental theory of nature? Here we study local measurements in generalized probabilistic theories (GPTs) and investigate how observational limitations affect the production of correlations. We find that if only a subset of typical local measurements can be made then all the bipartite correlations produced in a GPT can be simulated to a high degree of accuracy by quantum mechanics. Our result makes use of a generalization of Dvoretzky's theorem for GPTs. The tripartite correlations can go beyond those exhibited by quantum mechanics, however.
What singles out quantum mechanics as the fundamental theory of nature? Here we study local measurements in generalized probabilistic theories (GPTs) and investigate how observational limitations affect the production of correlations. We find that if only a subset of typical local measurements can be made then all the bipartite correlations produced in a GPT can be simulated to a high degree of accuracy by quantum mechanics. Our result makes use of a generalization of Dvoretzky's theorem for GPTs. The tripartite correlations can go beyond those exhibited by quantum mechanics, however.
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.
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.
We introduce the class of genuinely local operations and shared randomness multipartite nonlocal correlations, that is, correlations between N parties that cannot be obtained from unlimited shared randomness supplemented by any composition of $(N\ensuremath{-}1)$-shared causal generalized probabilistic theory (GPT) resources. We then show that noisy N-partite Greenberger-Horne-Zeilinger (GHZ) quantum states as well as the three-partite W quantum state can produce such correlations. This proves, if the operational predictions of quantum theory are correct, that nature's nonlocality must be boundlessly multipartite in any causal GPT. We develop a computational method which certifies that a noisy N=3 GHZ quantum state with fidelity 85 satisfies this property, making an experimental demonstration of our results within reach. We provide our definition and contrast it with preexisting notions of genuine multipartite nonlocality. This work extends a more compact parallel Letter [X. Coiteux-Roy et al., No Bipartite-Nonlocal Causal Theory Can Explain Nature's Correlations, Phys. Rev. Lett. 127, 200401 (2021)] on the same subject and provides all the required technical proofs.
The framework of generalized probabilistic theories (GPT) is a widely-used approach for studying the physical foundations of quantum theory. The standard GPT framework assumes the no-restriction hypothesis, in which the state space of a physical theory determines the set of measurements. However, this assumption is not physically motivated. In Janotta and Lal [Phys. Rev. A 87, 052131 (2013)], it was shown how this assumption can be relaxed, and how such an approach can be used to describe new classes of probabilistic theories. This involves introducing a new, more general, definition of maximal joint state spaces, which we call the generalised maximal tensor product. Here we show that the generalised maximal tensor product recovers the standard maximal tensor product when at least one of the systems in a bipartite scenario obeys the no-restriction hypothesis. We also show that, under certain conditions, relaxing the no-restriction hypothesis for a given state space does not allow for stronger non-locality, although the generalized maximal tensor product may allow new joint states.
Generalized probabilistic theories (GPTs) provide a unified framework for describing probabilistic physical theories, encompassing classical and quantum theories as well as hypothetical theories beyond quantum mechanics. Since most GPTs are highly unrealistic and far removed from known physical theories, it is important to constrain them by physically reasonable principles. One of the most important such principles is consistency with thermodynamics, which has been extensively studied through toy models involving semipermeable membranes (SPMs) implementing measurements. On the other hand, information thermodynamics, which plays a central role in understanding the relationship between measurement and thermodynamics in classical and quantum theory, has remained largely undeveloped in GPTs. In this work, we construct information thermodynamics in GPTs and provide a unified framework for analyzing the relationship between measurement, feedback, information erasure, and the second law of thermodynamics. We also formulate a general framework for SPM models and analyze the thermodynamic cost of measurements implemented by SPMs. As a result, we show that no work can be extracted in contradiction with the second law as long as the measurement processes are consistent with entropy nondecrease, and derive sufficient conditions for this property for several entropy definitions proposed in GPTs. Moreover, by considering measurement processes violating these conditions, we construct explicit GPT systems realizing isothermal SPM cycles from which positive work can be extracted. These results demonstrate that violations of the second law can arise from the lack of fundamental entropy properties or discrepancies between entropy definitions, and provide a unified and model-independent foundation for understanding the relationship between thermodynamics and measurement in GPTs.
Self-testing usually refers to the task of taking a given set of observed correlations that are assumed to arise via a process that is accurately described by quantum theory, and trying to infer the quantum state and measurements. In other words it is concerned with the question of whether we can tell what quantum black-box devices are doing by looking only at their input-output behavior and is known to be possible in several cases. Here we introduce a more general question: is it possible to self-test a theory, and, in particular, quantum theory? More precisely, we ask whether within a particular causal structure there are tasks that can only be performed in theories that have the same correlations as quantum mechanics in any scenario. We present a candidate task for such a correlation self-test and analyze it in a range of generalized probabilistic theories (GPTs), showing that none of these perform better than quantum theory. A generalization of our results showing that all nonquantum GPTs are strictly inferior to quantum mechanics for this task would point to a new way to axiomatize quantum theory, and enable an experimental test that simultaneously rules out such GPTs.