Generalized probabilistic theories (GPTs) provide a framework in which a range of possible theories can be examined, including classical theory, quantum theory and those beyond. In general, enlarging the state space of a GPT leads to fewer possible measurements because the additional states give stronger constraints on the set of effects, the constituents of measurements. This can have implications for information processing. In boxworld, for example, a GPT in which any no-signalling distribution can be realised, there is no analogue of a measurement in the Bell basis and hence the analogue of entanglement swapping is impossible. A comprehensive study of measurements on multiple systems in boxworld has been lacking. Here we consider such measurements in detail, distinguishing those that can be performed by interacting with individual systems sequentially (termed wirings), and the more interesting set of those that cannot. We compute all the possible boxworld effects for cases with small numbers of inputs, outputs and parties, identifying those that are wirings. The large state space of boxworld leads to a small effect space and hence the effects of boxworld are widely applicable in GPTs. We also show some possible uses of non-wirings for information processing by studying state discrimination, nonlocality distillation and the boxworld analogue of nonlocality without entanglement. Finally, we connect our results to the study of logically consistent classical processes and to the composition of contextuality scenarios. By enhancing understanding of measurements in boxworld, our results could be useful in studies of possible underlying principles on which quantum theory can be based.
Quantum discord quantifies non-classical correlations in quantum states. We introduce discord for states in causal probabilistic theories, inspired by the original definition proposed in Ref. [17]. We show that the only probabilistic theory in which all states have null discord is classical probability theory. Non-null discord is then not just a quantum feature, but a generic signature of non-classicality.
As a part of the construction of an information theory based on general probabilistic theories, we propose and investigate the several distinguishability measures and "entropies" in general probabilistic theories. As their applications, no-cloning theorems, information-disturbance theorems are reformulated, and a bound of the accessible informations is discussed in any general probabilistic theories, not resorting to quantum theory.
In a previous paper, we showed that many important quantum information-theoretic phenomena, including the no-cloning and no-broadcasting theorems, are in fact generic in all non-classical probabilistic theories. An exception is teleportation, which most such theories do not support. In this paper, we investigate which probabilistic theories, and more particularly, which composite systems, {\em do} support a teleportation protocol. We isolate a natural class of composite systems that we term {\em regular}, and establish necessary and sufficient conditions for a regular tripartite system to supp…
The aim of this thesis project is to investigate the bit commitment protocol in the framework of operational probabilistic theories. In particular a careful study is carried on the feasibility of bit commitment in the non-local boxes theory. New aspects of the theory are also presented.
We introduce a new way of quantifying the degrees of incompatibility of two ob- servables in a probabilistic physical theory and, based on this, a global measure of the degree of incompatibility inherent in such theories, across all observable pairs. This opens up a novel and flexible way of comparing probabilistic theories with respect to the nonclassical feature of incompatibility, raising many interesting questions, some of which will be answered here. We show that quantum theory contains observables that are as incompatible as any probabilistic physical theory can have if arbitrary pairs o…
We consider optimal state discrimination in a general convex operational framework, so-called generalized probabilistic theories (GPTs), and present a general method of optimal discrimination by applying the complementarity problem from convex optimization. The method exploits the convex geometry of states but not other detailed conditions or relations of states and effects. We also show that properties in optimal quantum state discrimination are shared in GPTs in general: i) no measurement sometimes gives optimal discrimination, and ii) optimal measurement is not unique.
In this article we provide, from a novel perspective, arguments that support the idea that, in the wake of Cox' approach to probability theory, von Neumann's entropy should be the natural one in Quantum Mechanics. We also generalize the pertinent reasoning to more general orthomodular lattices, which reveals the structure of a general non-Boolean information theory.
Gleason-type theorems for quantum theory allow one to recover the quantum state space by assuming that (i) states consistently assign probabilities to measurement outcomes and that (ii) there is a unique state for every such assignment. We identify the class of general probabilistic theories which also admit Gleason-type theorems. It contains theories satisfying the no-restriction hypothesis as well as others which can simulate such an unrestricted theory arbitrarily well when allowing for post-selection on measurement outcomes. Our result also implies that the standard no-restriction hypothes…
Entropic uncertainty relations play an important role in both fundamentals and applications of quantum theory. Although they have been well-investigated in quantum theory, little is known about entropic uncertainty in generalized probabilistic theories (GPTs). The current study explores two types of entropic uncertainty relations, preparation and measurement uncertainty relations, in a class of GPTs which can be considered generalizations of quantum theory. Not only a method for obtaining entropic preparation uncertainty relations but also an entropic measurement uncertainty relation similar t…
In quantum theory, the no-information-without-disturbance and no-free-information theorems express that those observables that do not disturb the measurement of another observable and those that can be measured jointly with any other observable must be trivial, i.e., coin tossing observables. We show that in the framework of general probabilistic theories these statements do not hold in general and continue to completely specify these two classes of observables. In this way, we obtain characterizations of the probabilistic theories where these statements hold. As a particular class of state sp…
We call a probabilistic theory "complete" if it cannot be further refined by no-signaling hidden-variable models, and name a theory "spooky" if every equivalent hidden-variable model violates Shimony's Outcome Independence. We prove that a complete theory is spooky if and only if it admits a pure steering state in the sense of Schrödinger. Finally we show that steering of complementary states leads to a Schrödinger's cat-like paradox.
Coin-flipping is a fundamental cryptographic task where a spatially separated Alice and Bob wish to generate a fair coin-flip over a communication channel. It is known that ideal coin-flipping is impossible in both classical and quantum theory. In this work, we give a short proof that it is also impossible in generalized probabilistic theories under the Generalized No-Restriction Hypothesis. Our proof relies crucially on a formulation of cheating strategies as semi-infinite programs, i.e., cone programs with infinitely many constraints. This introduces a new formalism which may be of independe…
We investigate a state discrimination problem in operationally the most general framework to use a probability, including both classical, quantum theories, and more. In this wide framework, introducing closely related family of ensembles (which we call a {\it Helstrom family of ensembles}) with the problem, we provide a geometrical method to find an optimal measurement for state discrimination by means of Bayesian strategy. We illustrate our method in 2-level quantum systems and in a probabilistic model with square-state space to reproduce e.g., the optimal success probabilities for binary sta…
We quantify the intrinsic noise content of an observable in a general probabilistic theory and derive a noise content inequality for incompatible observables. We apply the derived inequality to standard quantum theory, the quantum theory of processes, and polytope state spaces. The noise content for positive operator-valued measures takes a particularly simple form and equals the sum of minimal eigenvalues of all the effects. We illustrate our findings with a number of examples including the introduced notion of reverse observables.
It is well-known that pure quantum states are typically almost maximally entangled, and thus have close to maximally mixed subsystems. We consider whether this is true for probabilistic theories more generally, and not just for quantum theory. We derive a formula for the expected purity of a subsystem in any probabilistic theory for which this quantity is well-defined. It applies to typical entanglement in pure quantum states, coin tossing in classical probability theory, and randomization in post-quantum theories; a simple generalization yields the typical entanglement in (anti)symmetric quan…
What singles out quantum mechanics as the fundamental theory of Nature? Here we study local measurements in generalised 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 generalisation of Dvoretzky's theorem for GPTs. The tripartite correlations can go beyond those exhibited by quantum mechanics, however.
We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence…
We discuss the possibility of creating money that is physically impossible to counterfeit. Of course, "physically impossible" is dependent on the theory that is a faithful description of nature. Currently there are several proposals for quantum money which have their security based on the validity of quantum mechanics. In this work, we examine Wiesner's money scheme in the framework of generalised probabilistic theories. This framework is broad enough to allow for essentially any potential theory of nature, provided that it admits an operational description. We prove that under a quantifiable …
Bit-commitment is a fundamental cryptographic task, in which Alice commits a bit to Bob such that she cannot later change the value of the bit, while, simultaneously, the bit is hidden from Bob. It is known that ideal bit-commitment is impossible within quantum theory. In this work, we show that it is also impossible in generalised probabilistic theories (under a small set of assumptions) by presenting a quantitative trade-off between Alice's and Bob's cheating probabilities. Our proof relies crucially on a formulation of cheating strategies as cone programs, a natural generalisation of semide…
We prove generic versions of the no-cloning and no-broadcasting theorems, applicable to essentially {\em any} non-classical finite-dimensional probabilistic model that satisfies a no-signaling criterion. This includes quantum theory as well as models supporting ``super-quantum'' correlations that violate the Bell inequalities to a larger extent than quantum theory. The proof of our no-broadcasting theorem is significantly more natural and more self-contained than others we have seen: we show that a set of states is broadcastable if, and only if, it is contained in a simplex whose vertices are …
The framework of generalized probabilistic theories (GPTs) is a popular approach for studying the physical foundations of quantum theory. The standard 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. We generalize the framework to account for systems that do not obey the no-restriction hypothesis. We then show how our framework can be used to describe new classes of probabilistic theories, for example those which include intrinsic noise. Relaxing the restricti…
We study incompatibility of measurements and its relation to steering and nonlocality in a class of finite dimensional general probabilistic theories (GPT).The basic idea is to represent finite collections of measurements as affine maps of a state space into a polysimplex and show that incompatibility is characterized by properties of these maps. We introduce the notion of an incompatibility witness and show its relation to incompatibility degree. We find the maximum incompatibility degree attainable by pairs of two-outcome quantum measurements and characterize state spaces for which incompati…
Information transfer in generalized probabilistic theories (GPT) is an important problem. We have dealt with the problem based on repeatability postulate, which generalizes Zurek's result to the GPT framework [Phys. Lett. A \textbf{379} (2015) 2694]. A natural question arises: can we deduce the information transfer result under weaker assumptions? In this paper, we generalize Zurek's result to the framework of GPT using weak repeatability postulate. We show that if distinguishable information can be transferred from a physical system to a series of apparatuses under the weak repeatability post…
This paper provides a concise summary of the framework of operational-probabilistic theories, aimed at emphasizing the interaction between category-theoretic and probabilistic structures. Within this framework, we review an operational version of the GNS construction, expressed by the so-called purification principle, which under mild hypotheses leads to an operational version of Stinespring's theorem.
In quantum theory, it is known for a pair of noncommutative observables that there is no state on which they take simultaneously definite values, and that there is no joint measurement of them. They are called preparation uncertainty and measurement uncertainty respectively, and research has unveiled that they are not independent from but related with each other in a quantitative way. This study aims to reveal whether similar relations to quantum ones hold also in generalized probabilistic theories (GPTs). In particular, a certain class of GPTs is considered which can be characterized by trans…
Analysing causal structures in generalised probabilistic theories
Institute for Quantum Optics and Quantum Information (IQOQI) Vienna, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所维也纳）；Department of Mathematics, University of York（约克大学数学系）
We investigate the connection between interference and computational power within the operationally defined framework of generalised probabilistic theories. To compare the computational abilities of different theories within this framework we show that any theory satisfying three natural physical principles possess a well-defined oracle model. Indeed, we prove a subroutine theorem for oracles in such theories which is a necessary condition for the oracle to be well-defined. The three principles are: causality (roughly, no signalling from the future), purification (each mixed state arises as th…
The formalism of general probabilistic theories provides a universal paradigm that is suitable for describing various physical systems including classical and quantum ones as particular cases. Contrary to the usual no-restriction hypothesis, the set of accessible meters within a given theory can be limited for different reasons, and this raises a question of what restrictions on meters are operationally relevant. We argue that all operational restrictions must be closed under simulation, where the simulation scheme involves mixing and classical post-processing of meters. We distinguish three c…
We address asymptotic decoupling in the context of Markovian quantum dynamics. Asymptotic decoupling is an asymptotic property on a bipartite quantum system, and asserts that the correlation between two quantum systems is broken after a sufficiently long time passes. The first goal of this paper is to show that asymptotic decoupling is equivalent to local mixing which asserts the convergence to a unique stationary state on at least one quantum system. In the study of Markovian dynamics, mixing and ergodicity are fundamental properties which assert the convergence and the convergence of the lon…
General probabilistic theories are designed to provide operationally the most general probabilistic models including both classical and quantum theories. In this letter, we introduce a systematic method to construct a series of entropies, all of which generalize Shannon entropy in classical system and von Neumann entropy in quantum system. Using these entropies, the Holevo bound, an upper bound of the accessible information from a quantum system, is generalized to hold in any general probabilistic theory.
These lecture notes provide a basic introduction to the framework of generalized probabilistic theories (GPTs) and a sketch of a reconstruction of quantum theory (QT) from simple operational principles. To build some intuition for how physics could be even more general than quantum, I present two conceivable phenomena beyond QT: superstrong nonlocality and higher-order interference. Then I introduce the framework of GPTs, generalizing both quantum and classical probability theory. Finally, I summarize a reconstruction of QT from the principles of Tomographic Locality, Continuous Reversibility,…
We study the entanglement structure, i.e., the structure of quantum composite system from operational aspects. The structure is not uniquely determined in General Probabilistic Theories (GPTs) even if we impose reasonable postulate about local systems. In this paper, we investigate the possibility that the standard entanglement structure can be determined uniquely by repeatability of measurement processing and its saturated situation called self-duality. Surprisingly, self-duality cannot determine the standard entanglement structure even if we additionally impose local unitary symmetry assumpt…
We present a simple categorical framework for the treatment of probabilistic theories, with the aim of reconciling the fields of Categorical Quantum Mechanics (CQM) and Operational Probabilistic Theories (OPTs). In recent years, both CQM and OPTs have found successful application to a number of areas in quantum foundations and information theory: they present many similarities, both in spirit and in formalism, but they remain separated by a number of subtle yet important differences. We attempt to bridge this gap, by adopting a minimal number of operationally motivated axioms which provide cle…
Generalized contextuality refers to our inability of explaining measurement statistics using a context-independent probabilistic and ontological model. On the other hand, measurement statistics can also be modeled using the framework of general probabilistic theories (GPTs). Here, starting from a construction of GPTs based on a Gleason-type theorem, we fully characterize these structures with respect to their permission and rejection of generalized (non)contextual ontological models. It follows that in any GPT construction the three insistence of (i) the no-restriction hypothesis, (ii) the ont…
Non-signalling quantum channels -- relevant in, e.g., the study of Bell and Einstein-Podolsky-Rosen scenarios -- may be simulated via affine combinations of local operations in bipartite scenarios. Moreover, when these channels correspond to stochastic maps between classical variables, such simulation is possible even in multipartite scenarios. These two results have proven useful when studying the properties of these channels, such as their communication and information processing power, and even when defining measures of the non-classicality of physical phenomena (such as Bell non-classicali…
Quantum theory is indeterministic, but not completely so. When a system is in a pure state there are properties it possesses with certainty, known as actual properties. The actual properties of a quantum system (in a pure state) fully determine the probability of finding the system to have any other property. We call this feature intermediate determinism. In dimensions of at least three, the intermediate determinism of quantum theory is guaranteed by the structure of its lattice of properties. This observation follows from Gleason's theorem, which is why it fails to hold in dimension two. In t…
In Ref. [1] one of the authors proposed postulates for axiomatizing Quantum Mechanics as a "fair operational framework", namely regarding the theory as a set of rules that allow the experimenter to predict future events on the basis of suitable tests, having local control and low experimental complexity. In addition to causality, the following postulates have been considered: PFAITH (existence of a pure preparationally faithful state), and FAITHE (existence of a faithful effect). These postulates have exhibited an unexpected theoretical power, excluding all known nonquantum probabilistic theor…
The existence of incompatibility is one of the most fundamental features of quantum theory, and can be found at the core of many of the theory's distinguishing features, such as Bell inequality violations and the no-broadcasting theorem. A scheme for obtaining new observables from existing ones via classical operations, the so-called simulation of observables, has led to an extension of the notion of compatibility for measurements. We consider the simulation of observables within the operational framework of general probabilistic theories and introduce the concept of simulation irreducibility.…
General Probabilistic Theories provide the most general mathematical framework for the theory of probability in an operationally natural manner, and generalize classical and quantum theories. In this article, we study state-discrimination problems in general probabilistic theories using a Bayesian strategy. After re-formulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that, for two-s…
I introduce a framework in which a variety of probabilistic theories can be defined, including classical and quantum theories, and many others. From two simple assumptions, a tensor product rule for combining separate systems can be derived. Certain features, usually thought of as specifically quantum, turn out to be generic in this framework, meaning that they are present in all except classical theories. These include the non-unique decomposition of a mixed state into pure states, a theorem involving disturbance of a system on measurement (suggesting that the possibility of secure key distri…
