Bell-type inequalities for bivariate maps on orthomodular lattices

Bell-type inequalities on orthomodular lattices, in which conjunctions of propositions are not modeled by meets but by maps for simultaneous measurements (s-maps), are studied. It is shown that the most simple of these inequalities, that involves only two propositions, is always satisfied, contrary to what happens in the case of traditional version of this inequality in which conjunctions of propositions are modeled by meets. Equivalence of various Bell-type inequalities formulated with the aid of bivariate maps on orthomodular lattices is studied. Our invesigations shed new light on the interpretation of various multivariate maps defined on orthomodular lattices already studied in the literature. The paper is concluded by showing the possibility of using s-maps and j-maps to represent counterfactual conjunctions and disjunctions of non-compatible propositions about quantum systems.Comment: 14 pages, no figure

Copulas in Classical Probability Sense

            دوال الكوبلة بشكل مبسط بناءات مكافئة لدوال التوزيع المشتركة. في هذه الدراسة اقترحنا بناءات معدلة تعتمد على الاحتمالية التقليدية ومفاهيم بدلالة دوال الكوبلة. قُدمت دوال الكوبلة بصيغ قياسية احتمالية مكافئة للصيغ التقليدية من اجل اختبار أي علاقات احتمالية جديدة. قمنا بتقديم الاحتمالية للاحداث كعناصر لدالة الكوبلة بدلا من المتغيرات العشوائية مع المعرفة بان كل احتمالية لحدث تنتمي للفترة [0,1]. كذلك، تم اثبات بعض البناءات الاحتمالية من خلال مفاهيم الاحتمالية الاستقلالية والشرطية. ناقشت الدراسة علاقة بيز الاحتمالية وخواصها بالنسبة لدوال الكوبلة. بالإضافة الى ذلك، قُدم التوسع للبناءات المتعددة لكل دالة كوبلة. أخيرا، وضحنا عدد من الأمثلة التي بينت كل علاقة للكوبلة بدلالة الفضاء الاحتمالي بدلا من دوال التوزيع.               Copulas are simply equivalent structures to joint distribution functions. Then, we propose modified structures that depend on classical probability space and concepts with respect to copulas. Copulas have been presented in equivalent probability measure forms to the classical forms in order to examine any possible modern probabilistic relations. A probability of events was demonstrated as elements of copulas instead of random variables with a knowledge that each probability of an event belongs to [0,1]. Also, some probabilistic constructions have been shown within independent, and conditional probability concepts. A Bay's probability relation and its properties were discussed with respect to copulas. Moreover, an extension of multivariate constructions of each probabilistic copula has been presented. Finally, we have shown some examples that explain each relation of copula in terms of probability space instead of distribution functions

Pattern Recognition In Non-Kolmogorovian Structures

We present a generalization of the problem of pattern recognition to arbitrary probabilistic models. This version deals with the problem of recognizing an individual pattern among a family of different species or classes of objects which obey probabilistic laws which do not comply with Kolmogorov's axioms. We show that such a scenario accommodates many important examples, and in particular, we provide a rigorous definition of the classical and the quantum pattern recognition problems, respectively. Our framework allows for the introduction of non-trivial correlations (as entanglement or discord) between the different species involved, opening the door to a new way of harnessing these physical resources for solving pattern recognition problems. Finally, we present some examples and discuss the computational complexity of the quantum pattern recognition problem, showing that the most important quantum computation algorithms can be described as non-Kolmogorovian pattern recognition problems

Eventology versus contemporary theories of uncertainty

The development of probability theory together with the Bayesian approach in the three last centuries is caused by two factors: the variability of the physical phenomena and partial ignorance about them. As now it is standard to believe [Dubois, 2007], the nature of these key factors is so various, that their descriptions are required special uncertainty theories, which differ from the probability theory and the Bayesian credo, and provide a better account of the various facets of uncertainty by putting together probabilistic and set-valued representations of information to catch a distinction between variability and ignorance. Eventology [Vorobyev, 2007], a new direction of probability theory and philosophy, offers the original event approach to the description of variability and ignorance, entering an agent, together with his/her beliefs, directly in the frameworks of scientific research in the form of eventological distribution of his/her own events. This allows eventology, by putting together probabilistic and set-event representation of information and philosophical concept of event as co-being [Bakhtin, 1920], to provide a unified strong account of various aspects of uncertainty catching distinction between variability and ignorance and opening an opportunity to define imprecise probability as a probability of imprecise event in the mathematical frameworks of Kolmogorov's probability theory [Kolmogorov, 1933].uncertainty, probability, event, co-being, eventology, imprecise event

Application-Oriented Benchmarking of Quantum Generative Learning Using QUARK

Benchmarking of quantum machine learning (QML) algorithms is challenging due to the complexity and variability of QML systems, e.g., regarding model ansatzes, data sets, training techniques, and hyper-parameters selection. The QUantum computing Application benchmaRK (QUARK) framework simplifies and standardizes benchmarking studies for quantum computing applications. Here, we propose several extensions of QUARK to include the ability to evaluate the training and deployment of quantum generative models. We describe the updated software architecture and illustrate its flexibility through several example applications: (1) We trained different quantum generative models using several circuit ansatzes, data sets, and data transformations. (2) We evaluated our models on GPU and real quantum hardware. (3) We assessed the generalization capabilities of our generative models using a broad set of metrics that capture, e.g., the novelty and validity of the generated data.Comment: 10 pages, 10 figure

Self-move and Other-move: Quantum Categorical Foundations of Japanese

The purpose of this work is to contribute toward the larger goal of creating a Quantum Natural Language Processing (QNLP) translator program. This work contributes original diagrammatic representations of the Japanese language based on prior work that accomplished on the English language based on category theory. The germane differences between the English and Japanese languages are emphasized to help address English language bias in the current body of research. Additionally, topological principles of these diagrams and many potential avenues for further research are proposed. Why is this endeavor important? Hundreds of languages have developed over the course of millennia coinciding with the evolution of human interaction across time and geographic location. These languages are foundational to human survival, experience, flourishing, and living the good life. They are also, however, the strongest barrier between people groups. Over the last several decades, advancements in Natural Language Processing (NLP) have made it easier to bridge the gap between individuals who do not share a common language or culture. Tools like Google Translate and DeepL make it easier than ever before to share our experiences with people globally. Nevertheless, these tools are still inadequate as they fail to convey our ideas across the language barrier fluently, leaving people feeling anxious and embarrassed. This is particularly true of languages born out of substantially different cultures, such as English and Japanese. Quantum computers offer the best chance to achieve translation fluency in that they are better suited to simulating the natural world and natural phenomenon such as natural speech. Keywords: category theory, DisCoCat, DisCoCirc, Japanese grammar, English grammar, translation, topology, Quantum Natural Language Processing, Natural Language ProcessingComment: 104 pages; 31 figures; 9 table