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.
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.
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.
For a collection of states indexed by [n]=1,…,n, let F(H) denote the optimal unnormalized minimum discrimination error for each nonempty subfamily H⊆[n]. We call F the discrimination profile and characterize exactly which profiles can arise in a general probabilistic theory (GPT). For fixed n, the GPT region Gn is a bounded rational polytope, giving a hierarchy Cn⊆Qn⊆Gn of classical, quantum, and GPT profiles, analogous to the local--quantum--no-signalling hierarchy in Bell theory.
For n=3 we determine the classical and GPT polytopes explicitly. Within G3, the classical region is characterized by β:=F(123)−F(12)−F(13)−F(23)≤0, while the GPT maximum is 1. For qubit triples, the maximal value is attained by the trine, βtr=33/2−2, and we prove the dimension-independent quantum bound βQ≤2/3, with a slight further improvement. Hence 0=βC<33/2−2≤βQ≤2/3<1=βGPT. We conjecture βQ=βtr and prove this for arbitrary pure-state triples, together with further supporting evidence.
For arbitrary n, every classical facet admitting a GPT violation already admits a quantum violation in dimension at most 3. A four-state qubit example shows that quantum success profiles need not be submodular, unlike for n=3. Finally, we exhibit a four-dimensional quantum triple whose discrete profile is classical for all subfamilies but becomes nonclassical when unequal priors are allowed.
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.
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.
An operational probabilistic theory where all systems are classical, and all pure states of composite systems are entangled, is constructed. The theory is endowed with a rule for composing an arbitrary number of systems, and with a nontrivial set of transformations. Hence, we demonstrate that the presence of entanglement is independent of the existence of incompatible measurements. We then study a variety of phenomena occurring in the theory, some of them contradicting both classical and quantum theories, including cloning, entanglement swapping, dense coding, additivity of classical capacities, nonmonogamous entanglement, hypersignaling. We also prove the existence, in the theory, of a universal processor. The theory is causal and satisfies the no-restriction hypothesis. At the same time, it violates a number of information-theoretic principles enjoyed by quantum theory, most notably, local discriminability, purity of parallel composition of states, and purification. Moreover, we introduce an exhaustive procedure to construct generic operational probabilistic theories, and a sufficient set of conditions to verify their consistency. In addition, we prove a characterization theorem for the parallel composition rules of arbitrary theories, and specialize it to the case of bilocal-tomographic theories. We conclude pointing out some open problems. In particular, on the basis of the fact that every separable state of the theory is a statistical mixture of entangled states, we formulate a no-go conjecture for the existence of a local-realistic ontological model.
Abstract It is a key issue to characterize the model of standard quantum theory out of general models by an operational condition. The framework of general probabilistic theories (GPTs) is a new information theoretical approach to single out standard quantum theory. It is known that traditional properties, for example, Bell-CHSH inequality, are not sufficient to single out standard quantum theory among possible models in GPTs. As a more precise property, we focus on the bound of the performance for an information task called state discrimination in general models. We give an equivalent condition for outperforming the minimum discrimination error probability under the standard quantum theory given by the trace norm. Besides, by applying the equivalent condition, we characterize standard quantum theory out of general models in GPTs by the bound of the performance for state discrimination.