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Abstract Irregular spatial distribution of photon transmission through a photochromic crystal photoisomerized by a local optical near-field excitation was previously reported, which manifested complex branching processes via the interplay of material deformation and near-field photon transfer therein. Furthermore, by combining such naturally constructed complex photon transmission with a simple photon detection protocol, Schubert polynomials, the foundation of versatile permutation operations in mathematics, have been generated. In this study, we demonstrated an order recognition algorithm inspired by Schubert calculus using optical near-field statistics via nanometre-scale photochromism. More specifically, by utilizing Schubert polynomials generated via optical near-field patterns, we showed that the order of slot machines with initially unknown reward probability was successfully recognized. We emphasized that, unlike conventional algorithms, the proposed principle does not estimate the reward probabilities but exploits the inversion relations contained in the Schubert polynomials. To quantitatively evaluate the impact of Schubert polynomials generated from an optical near-field pattern, order recognition performances were compared with uniformly distributed and spatially strongly skewed probability distributions, where the optical near-field pattern outperformed the others. We found that the number of singularities contained in Schubert polynomials and that of the given problem or considered environment exhibited a clear correspondence, indicating that superior order recognition is attained when the singularity of the given situations is presupposed. This study paves way for physical computing through the interplay of complex natural processes and mathematical insights gained by Schubert calculus.

Human consciousness is characterized by constant transitions in time. On the other hand, what is consciously experienced always possesses the temporal feature of “now.” In consciousness, “now” constantly holds different contents, yet it remains “now” no matter how far it goes. This duality is thematized in Husserlian phenomenology as “the standing-streaming now.” Although this phrase appears contradictory in everyday language, it has a structure that can be clearly understood and formalized. In this paper, we show that this structure can be described as a monoid in category theory. Furthermore, monoids can be transformed into the coslice category, which corresponds to the way of perceiving present moments as juxtaposed in succession. The seemingly contradictory nature of the “now” as both flowing and standing can be precisely structured and comprehended through the monoid, while the perspective of the “now” as discrete points on a timeline can be effectively formalized using the coslice category. This framework helps us more precisely understand the differences between ordinary consciousness and meditative consciousness, specifically the experience of the “eternal now.” We illustrate how the meditative states of consciousness presented in the early Buddhist scriptures (Pali Canon) and Dōgen’s Shōbōgenzō remarkably reflect a monoid structure.

What are the nature of the relationship among qualia, contents of consciousness, and behavioral reports? Traditionally, this type of question has been only addressed via a qualitative and philosophical approach. Some theorists emphasize an incomplete and inaccurate nature of reports of one's own qualia to discourage formal research programs on qualia. Other empirical researchers, however, have made substantial progress in understanding the structure of qualia from such limited reports. What is the precise relationship between the two? To answer this question, we introduce the concept of “adjoint” or “adjunction” from the category theory in mathematics. We claim that the adjunction captures some aspects of the nuanced relationships between qualia and reports. The concept of adjunction allows us to clarify the conceptual issues with a precise mathematical formulation. In particular, adjunction establishes coherence between two categories that cannot be considered equivalent, yet has an important relationship. This rises in empirical experimental situations between qualia and reports. More importantly, an idea of adjunction naturally leads to various proposals of new empirical experiments to test the predictions about the nature of their relationship as well as other issues in consciousness research.

Abstract Collective decision making is important for maximizing total benefits while preserving equality among individuals in the competitive multi-armed bandit (CMAB) problem, wherein multiple players try to gain higher rewards from multiple slot machines. The CMAB problem represents an essential aspect of applications such as resource management in social infrastructure. In a previous study, we theoretically and experimentally demonstrated that entangled photons can physically resolve the difficulty of the CMAB problem. This decision-making strategy completely avoids decision conflicts while ensuring equality. However, decision conflicts can sometimes be beneficial if they yield greater rewards than non-conflicting decisions, indicating that greedy actions may provide positive effects depending on the given environment. In this study, we demonstrate a mixed strategy of entangled- and correlated-photon-based decision-making so that total rewards can be enhanced when compared to the entangled-photon-only decision strategy. We show that an optimal mixture of entangled- and correlated-photon-based strategies exists depending on the dynamics of the reward environment as well as the difficulty of the given problem. This study paves the way for utilizing both quantum and classical aspects of photons in a mixed manner for decision making and provides yet another example of the supremacy of mixed strategies known in game theory, especially in evolutionary game theory.

Abstract Situations involving competition for resources among entities can be modeled by the competitive multi-armed bandit (CMAB) problem, which relates to social issues such as maximizing the total outcome and achieving the fairest resource repartition among individuals. In these respects, the intrinsic randomness and global properties of quantum states provide ideal tools for obtaining optimal solutions to this problem. Based on the previous study of the CMAB problem in the two-arm, two-player case, this paper presents the theoretical principles necessary to find polarization-entangled N-photon states that can optimize the total resource output while ensuring equality among players. These principles were applied to two-, three-, four-, and five-player cases by using numerical simulations to reproduce realistic configurations and find the best strategies to overcome potential misalignment between the polarization measurement systems of the players. Although a general formula for the N-player case is not presented here, general derivation rules and a verification algorithm are proposed. This report demonstrates the potential usability of quantum states in collective decision making with limited, probabilistic resources, which could serve as a first step toward quantum-based resource allocation systems.

In the present paper, we propose a new approach to quantum fields in terms of category algebras and states on categories. We define quantum fields and their states as category algebras and states on causal categories with partial involution structures. By utilizing category algebras and states on categories instead of simply considering categories, we can directly integrate relativity as a category theoretic structure and quantumness as a noncommutative probabilistic structure. Conceptual relationships with conventional approaches to quantum fields, including Algebraic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT), are also be discussed.

Several interesting physical phenomena and industrial applications explained by the dressed photon have been reported in recent years. These require a novel concept in an off-shell science that deviates from the conventional optics, satisfying energy and momentum conservation laws. In this paper, starting from an original model that captures dressed-photon characteristics phenomenologically, the dynamics of the dressed photon in a nanomatter system and the mechanism for extracting internal degrees of freedom of the dressed photon to an external space have been examined by theoretical and numerical approaches. Our proposal is that basis states of the dressed photon can be transformed to the form that reflects the spatial distribution of the dressed-photon steady state in the system, and some of basis states with predetermined spatial distribution can relate to the dissipation components in the external space by means of the renormalization technique. From the results of numerical simulation, it is found that quasi-static states are regarded as the photon with light mass or massless, and the extraction of active states strongly affects the spatial distribution in a new steady state. The concept for extracting dressed-photon energy to an external space will contribute to a detailed understanding of dressed-photon physics and future industrial applications.

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor U on a topos of sets S together with a natural transformation υ, instead of the terms as “standard”, “internal”, or “external”. Moreover, we propose a general notion of a space called U-space, and the category USpace whose objects are U-spaces and morphisms are functions called U-spatial morphisms. The category USpace, which is shown to be Cartesian closed, gives a unified viewpoint toward topological and coarse geometric structure. It will also be useful to further study symmetries/asymmetries of the systems with infinite degrees of freedom, such as quantum fields.

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups.

Soft robotics is an emerging field of research where the robot body is composed of flexible and soft materials. It allows the body to bend, twist, and deform to move or to adapt its shape to the environment for grasping, all of which are difficult for traditional hard robots with rigid bodies. However, the theoretical basis and design principles for soft robotics are not well-founded despite their recognized importance. For example, the control of soft robots is outsourced to morphological attributes and natural processes; thus, the coupled relations between a robot and its environment are particularly crucial. In this paper, we propose a mathematical foundation for soft robotics based on category theory, a branch of abstract mathematics where any notions can be described by objects and arrows. It allows for a rigorous description of the inherent characteristics of soft robots and their relation to the environment as well as the differences compared to conventional hard robots. We present a notion called the category of mobility to well describe the subject matter. The theory has been applied to a model system and analysis to highlight the adaptation behavior observed in universal grippers, which are a typical example of soft robotics. The aim of the present study is not to offer concrete engineering solutions to existing robotics but to provide clear mathematical description of soft robots by category theory and to imply its potential abilities by a simple soft gripper demonstration. This paper paves the way to developing a theoretical background and design principles for soft robotics.

In the light of the Newton–Wigner–Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this example model is extracted in such a form as quantum field ontology associated with the eventualization principle, which enables us to explain the mutual relations, back and forth, between quantum fields and various forms of particles in the localized form of the former.

Decision making based on behavioral and neural observations of living systems has been extensively studied in brain science, psychology, neuroeconomics, and other disciplines. Decision-making mechanisms have also been experimentally implemented in physical processes, such as single photons and chaotic lasers. The findings of these experiments suggest that there is a certain common basis in describing decision making, regardless of its physical realizations. In this study, we propose a local reservoir model to account for choice-based learning (CBL). CBL describes decision consistency as a phenomenon where making a certain decision increases the possibility of making that same decision again later. This phenomenon has been intensively investigated in neuroscience, psychology, and other related fields. Our proposed model is inspired by the viewpoint that a decision is affected by its local environment, which is referred to as a local reservoir. If the size of the local reservoir is large enough, consecutive decision making will not be affected by previous decisions, thus showing lower degrees of decision consistency in CBL. In contrast, if the size of the local reservoir decreases, a biased distribution occurs within it, which leads to higher degrees of decision consistency in CBL. In this study, an analytical approach for characterizing local reservoirs is presented, as well as several numerical demonstrations. Furthermore, a physical architecture for CBL based on single photons is discussed, and the effects of local reservoirs are numerically demonstrated. Decision consistency in human decision-making tasks and in recruiting empirical data is evaluated based on the local reservoir. This foundation based on a local reservoir offers further insights into the understanding and design of decision making.

Solution searching that accompanies combinatorial explosion is one of the most important issues in the age of artificial intelligence. Natural intelligence, which exploits natural processes for intelligent functions, is expected to help resolve or alleviate the difficulties of conventional computing paradigms and technologies. In fact, we have shown that a single-celled organism such as an amoeba can solve constraint satisfaction problems and related optimization problems as well as demonstrate experimental systems based on non-organic systems such as optical energy transfer involving near-field interactions. However, the fundamental mechanisms and limitations behind solution searching based on natural processes have not yet been understood. Herein, we present a theoretical background of solution searching based on optical excitation transfer from a category-theoretic standpoint. One important indication inspired by the category theory is that the satisfaction of short exact sequences is critical for an adequate computational operation that determines the flow of time for the system and is termed as “short-exact-sequence-based time.” In addition, the octahedral and braid structures known in triangulated categories provide a clear understanding of the underlying mechanisms, including a quantitative indication of the difficulties of obtaining solutions based on homology dimension. This study contributes to providing a fundamental background of natural intelligence.