Andrei Khrennikov

Andrei Khrennikov, Masanao Ozawa, Felix Benninger et al.

The past few years have seen a surge in the application of quantum theory methodologies and quantum-like modeling in fields such as cognition, psychology, and decision-making. Despite the success of this approach in explaining various psychological phenomena such as order, conjunction, disjunction, and response replicability effects there remains a potential dissatisfaction due to its lack of clear connection to neurophysiological processes in the brain. Currently, it remains a phenomenological approach. In this paper, we develop a quantum-like representation of networks of communicating neurons. This representation is not based on standard quantum theory but on generalized probability theory (GPT), with a focus on the operational measurement framework. Specifically, we use a version of GPT that relies on ordered linear state spaces rather than the traditional complex Hilbert spaces. A network of communicating neurons is modeled as a weighted directed graph, which is encoded by its weight matrix. The state space of these weight matrices is embedded within the GPT framework, incorporating effect observables and state updates within the theory of measurement instruments a critical aspect of this model. This GPT based approach successfully reproduces key quantum-like effects, such as order, non-repeatability, and disjunction effects (commonly associated with decision interference). Moreover, this framework supports quantum-like modeling in medical diagnostics for neurological conditions such as depression and epilepsy. While this paper focuses primarily on cognition and neuronal networks, the proposed formalism and methodology can be directly applied to a wide range of biological and social networks.

Miho Fuyama, Andrei Khrennikov, Masanao Ozawa

We characterize the class of quantum measurements that matches the applications of quantum theory to cognition (and decision making) - quantum-like modeling. Projective measurements describe the canonical measurements of the basic observables of quantum physics. However, the combinations of the basic cognitive effects, such as the question order and response replicability effects, cannot be described by projective measurements. We motivate the use of the special class of quantum measurements, namely {\it sharp repeatable non-projective measurements} - ${\cal SR\bar{P}}. $ This class is practically unused in quantum physics. Thus, physics and cognition explore different parts of quantum measurement theory. Quantum-like modeling isn't automatic borrowing of the quantum formalism. Exploring the class ${\cal SR\bar{P}}$ highlights the role of {\it noncommutativity of the state update maps generated by measurement back action.} Thus, ``non-classicality'' in quantum physics as well as quantum-like modeling for cognition is based on two different types of noncommutativity, of operators (observables) and instruments (state update maps): {\it observable-noncommutativity} vs. {\it state update-noncommutativity}. We speculate that distinguishing quantum-like properties of the cognitive effects are the expressions of the latter, or possibly both.

Alexander Lebedev, Andrei Khrennikov

Recently people started to understand that applications of the mathematical formalism of quantum theory are not reduced to physics. Nowadays, this formalism is widely used outside of quantum physics, in particular, in cognition, psychology, decision making, information processing, especially information retrieval. The latter is very promising. The aim of this brief introductory review is to stimulate research in this exciting area of information science. This paper is not aimed to present a complete review on the state of art in quantum information retrieval.

Ekaterina Yurova Axelsson, Andrei Khrennikov

In this paper, we study groups of automorphisms of algebraic systems over a set of $p$-adic integers with different sets of arithmetic and coordinate-wise logical operations and congruence relations modulo $p^k,$ $k\ge 1.$ The main result of this paper is the description of groups of automorphisms of $p$-adic integers with one or two arithmetic or coordinate-wise logical operations on $p$-adic integers. To describe groups of automorphisms, we use the apparatus of the $p$-adic analysis and $p$-adic dynamical systems. The motive for the study of groups of automorphism of algebraic systems over $p$-adic integers is the question of the existence of a fully homomorphic encryption in a given family of ciphers. The relationship between these problems is based on the possibility of constructing a "continuous" $p$-adic model for some families of ciphers (in this context, these ciphers can be considered as "discrete" systems). As a consequence, we can apply the "continuous" methods of $p$-adic analysis to solve the "discrete" problem of the existence of fully homomorphic ciphers.

Andrei Khrennikov, Ekaterina Yurova

In this paper we consider the description of homomorphic and fully homomorphic ciphers in the $p$-adic model of encryption. This model describes a wide class of ciphers, but certainly not all. Homomorphic and fully homomorphic ciphers are used to ensure the credibility of remote computing, including cloud technology. The model describes all homomorphic ciphers with respect to arithmetic and coordinate-wise logical operations in the ring of $p$-adic integers $Z_p$. We show that there are no fully homomorphic ciphers for each pair of the considered set of arithmetic and coordinate-wise logical operations on $Z_p$. We formulate the problem of constructing a fully homomorphic cipher as follows. We consider a homomorphic cipher with respect to operation "$*$" on $Z_p$. Then, we describe the complete set of operations "$G$", for which the cipher is homomorphic. As a result, we construct a fully homomorphic cipher with respect to the operations "$*$" and "$G$". We give a description of all operations "$G$", for which we obtain fully homomorphic ciphers with respect to the operations "$+$" and "$G$" from the homomorphic cipher constructed with respect to the operation "$+$". We also present examples of such "new" operations.