In this study, we consider generalized probabilistic theories (GPTs) and focus on a class of theories called regular polygon theories, which can be regarded as natural generalizations of a two-level quantum system (a qubit system). In the usual CHSH setting for quantum theory, the CHSH value is known to be optimized by maximally entangled states. This research will reveal that the same observations are obtained also in regular polygon theories. Our result gives a physical meaning to the concept of ``maximal entanglement" in regular polygon theories.
Various inequalities (Boole inequality, Chung-Erdös inequality, Frechet inequality) for Kolmogorov (classical) probabilities are considered. Quantum counterparts of these inequalities are introduced, which have an extra `quantum correction' term, and which hold for all quantum states. When certain sufficient conditions are satisfied, the quantum correction term is zero, and the classical version of these inequalities holds for all states. But in general, the classical version of these inequalities is violated by some of the quantum states. For example in bipartite systems, classical Boole ineq…
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…
The thermodynamical entropy of a system which consists of different kinds of ideal gases is known to be defined successfully in the case when the differences are described by classical or quantum theory. Since these theories are special examples in the framework of generalized probabilistic theories (GPTs), it is natural to generalize the notion of thermodynamical entropy to systems where the internal degrees of particles are described by other possible theories. In this paper, we consider thermodynamical entropy of mixing in a specific series of theories of GPTs called the regular polygon the…
In a recent preprint Gill and Lambare, criticize our paper published in Frontiers in Physics. Their criticism is unfounded and misleading. They define a probabilistic coupling, in which BI-CHSH hold for all finite samples. It does not mean, that BI-CHSH hold in our model, in which four incompatible experiments are described by setting dependent random variables implemented on 4 disjoint dedicated probability spaces. A joint probability distribution of these random variables does not exist and may not be used to derive inequalities. Moreover, their probabilistic coupling is useless, for a subse…
Bell's theorem shows that no hidden-variable model can explain the measurement statistics of a quantum system shared between two parties, thus ruling out a classical (local) understanding of nature. In this work we demonstrate that by relaxing the positivity restriction in the hidden-variable probability distribution it is possible to derive quasiprobabilistic Bell inequalities whose sharp upper bound is written in terms of a negativity witness of said distribution. This provides an analytic solution for the amount of negativity necessary to violate the CHSH inequality by an arbitrary amount, …
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…
We propose the gentle measurement principle (GMP) as one of the principles at the foundation of quantum mechanics. It asserts that if a set of states can be distinguished with high probability, they can be distinguished by a measurement that leaves the states almost invariant, including correlation with a reference system. While GMP is satisfied in both classical and quantum theories, we show, within the framework of general probabilistic theories, that it imposes strong restrictions on the law of physics. First, the measurement uncertainty of a pair of observables cannot be significantly larg…
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.…
Many experiments in the field of quantum foundations seek to adjudicate between quantum theory and speculative alternatives to it. This requires one to analyze the experimental data in a manner that does not presume the correctness of the quantum formalism. The mathematical framework of generalized probabilistic theories (GPTs) provides a means of doing so. We present a scheme for determining which GPTs are consistent with a given set of experimental data. It proceeds by performing tomography on the preparations and measurements in a self-consistent manner, i.e., without presuming a prior char…
To find the essential nature of quantum theory has been an important problem for not only theoretical interest but also applications to quantum technologies. In those studies on quantum foundations, the notion of uncertainty plays a primary role among several stunning features of quantum theory. The purpose of this thesis is to investigate fundamental aspects of uncertainty. In particular, we address this problem focusing on convexity, which has an operational origin. We first try to reveal why in quantum theory similar bounds are often obtained for two types of uncertainty relations, namely, …
