Any pure state |𝚿⟩ of a compound system S = (S₁, S₂ ) with the state space H = H₁ ⊗ H₂ determines a kind of covariance operator D_𝚿 acting in the Cartesian product H = H × H₂. If this operator is positively defined, then it determines a random field valued in H. In this case compound quantum system S can be treated as a classical random field system whose configuration space is not tensor, but Cartesian product space. It happensthat̂ D_𝚿 ≥ 0 and a subquantum process exists if and only if quantum state |𝚿⟩ is not entangled. The technical framework used in this note is already presented by von Neumann

We develop a contextual measurement model (CMM) that is used for the clarification of the quantum foundations. This model matches Bohr’s views on the role of experimental contexts. CMM is based on a contextual probability theory that is connected with generalized probability theory. CMM covers measurements in classical, quantum and semi-classical physics. The CMM formalism is illustrated by a few examples. We consider the CMM framing of classical probability, the von Neumann measurement theory and the quantum instrument theory. CMM can also be applied outside of physics, e.g. in cognition, decision-making and psychology, the so-called quantum-like modelling.

This is a brief review on modeling genetic codes with the aid of 2-adic dynamical systems. In this model amino acids are encoded by the attractors of such dynamical systems. Each genetic code is coupled to the special class of 2-adic dynamics. We consider the discrete dynamical systems, These are the iterations of a function F : Z₂ -> Z₂, where Z₂ is the ring of 2-adic numbers (2-adic tree). A genetic code is characterized by the set of attractors of a function belonging to the code generating functional class. The main mathematical problem is to reduce degeneration of dynamic representation and select the optimal generating function. Here optimality can be treated in many ways. One possibility is to consider the Lipschitz functions playing the crucial role in general theory of iterations. Then we minimize the Lip-constant. The main issue is to find the proper biological interpretation of code-functions. One can speculate that the evolution of the genetic codes can be described in information space of the nucleotide-strings endowed with ultrametric (treelike) geometry. A code-function is a fitness function; the solutions of the genetic code optimization problem are attractors of the code-function. We illustrate this approach by generation of the standard nuclear and (vertebrate) mitochondrial genetics codes.

QBism’s foundational statement that “the outcome of a measurement of an observable is personal” is in the straight contraversion with Ozawa’s Intersubjectivity Theorem (OIT). The latter (proven within the quantum formalism) states that two observers, agents within the QBism terminology, performing joint measurements of the same observable A on a system S in the state  should get the same outcome  In Ozawa’s terminology, this outcome is intersubjective and it can’t be treated as personal. This is the strong objection to QBism which can’t survive without updating its principles. The essential aspect in understanding of the OIT-impact on QBism’s foundations takes the notion of quantum observable. This paper comprises the complementary discussion highlighting the difference between the accurate, von Neumann, and inaccurate, noisy, quantum observables which are represented by PVMs and POVMs respectively. Moreover, we discuss the OIT-impact on the Copenhagen interpretation of quantum mechanics.

Quantum-inspired algorithms represent an important direction in modern software information technologies that use heuristic methods and approaches of quantum science. This work presents a quantum approach for document search, retrieval, and ranking based on the Bell-like test, which is well-known in quantum physics. We propose quantum probability theory in the hyperspace analog to language (HAL) framework exploiting a Hilbert space for word and document vector specification. The quantum approach allows for accounting for specific user preferences in different contexts. To verify the algorithm proposed, we use a dataset of synthetic advertising text documents from travel agencies generated by the OpenAI GPT-4 model. We show that the "entanglement" in two-word document search and retrieval can be recognized as the frequent occurrence of two words in incompatible query contexts. We have found that the user preferences and word ordering in the query play a significant role in relatively small sizes of the HAL window. The comparison with the cosine similarity metrics demonstrates the key advantages of our approach based on the user-enforced contextual and semantic relationships between words and not just their superficial occurrence in texts. Our approach to retrieving and ranking documents allows for the creation of new information search engines that require no resource-intensive deep machine learning algorithms.

Distributed intelligence systems (DIS) containing natural and artificial intelligence agents (NIA and AIA) for decision making (DM) belong to promising interdisciplinary studies aimed at digitalization of routine processes in industry, economy, management, and everyday life. In this work, we suggest a novel quantum-inspired approach to investigate the crucial features of DIS consisting of NIAs (users) and AIAs (digital assistants, or avatars). We suppose that N users and their avatars are located in N nodes of a complex avatar - avatar network. The avatars can receive information from and transmit it to each other within this network, while the users obtain information from the outside. The users are associated with their digital assistants and cannot communicate with each other directly. Depending on the meaningfulness/uselessness of the information presented by avatars, users show their attitude making emotional binary "like"/"dislike" responses. To characterize NIA cognitive abilities in a simple DM process, we propose a mapping procedure for the Russell's valence-arousal circumplex model onto an effective quantum-like two-level system. The DIS aims to maximize the average satisfaction of users via AIAs long-term adaptation to their users. In this regard, we examine the opinion formation and social impact as a result of the collective emotional state evolution occurring in the DIS. We show that generalized cooperativity parameters G_i, i = 1,...,N introduced in this work play a significant role in DIS features reflecting the users activity in possible cooperation and responses to their avatar suggestions. These parameters reveal how frequently AIAs and NIAs communicate with each other accounting the cognitive abilities of NIAs and information losses within the network. We demonstrate that conditions for opinion formation and social impact in the DIS are relevant to the second-order non-equilibrium phase transition. The transition establishes a non-vanishing average information field inherent to information diffusion and long-term avatar adaptation to their users. It occurs above the phase transition threshold, i.e. at G_i > 1, which implies small (residual) social polarization of the NIAs community. Below the threshold, at weak AIA-NIA coupling (G_i ≤ 1), many uncertainties in the DIS inhibit opinion formation and social impact for the DM agents due to the information diffusion suppression; the AIAs self-organization within the avatar-avatar network is elucidated in this limit. To increase the users' impact, we suggest an adaptive approach by establishing a network-dependent coupling rate with their digital assistants. In this case, the mechanism of AIA control helps resolve the DM process in the presence of some uncertainties resulting from the variety of users' preferences. Our findings open new perspectives in different areas where AIAs become effective teammates for humans to solve common routine problems in network organizations.

The textbook conceptualization of phenotype creation, "genotype (G) + environment (E) + genotype & environment interactions (GE) phenotype (Ph)", is modeled with open quantum systems theory (OQST) or more generally with adaptive dynamics theory (ADT). The model is quantum-like , i.e., it is not about quantum physical processes in biosystems. Generally such modeling is about applications of the quantum formalism and methodology outside of physics. Macroscopic biosystems, in our case genotypes and phenotypes, are treated as information processors which functioning matches the laws of quantum information theory. Phenotypes are the outputs of the E-adaptation processes described by the quantum master equation, Gorini- Kossakowski-Sudarshan-Lindblad equation (GKSL). Its stationary states correspond to phenotypes. We highlight the class of GKSL dynamics characterized by the camel-like graphs of (von Neumann) entropy: in the process of E-adaptation phenotype's state entropy (disorder) first increases and then falls down - a stable and wellordered phenotype is created. Traits, an organism's phenotypic characteristics, are modeled within the quantum measurement theory, as generally unsharp observables given by positive operator valued measures (POVMs. This paper is also a review on the methods and mathematical apparatus of quantum information biology.

The Ozawa's intersubjectivity theorem (OIT) proved within quantum measurement theory supports the new postulate of relational quantum mechanics (RQM), the postulate on internally consistent descriptions. But from OIT viewpoint postulate's formulation should be completed by the assumption of probability reproducibility. We remark that this postulate was proposed only recently to resolve the problem of intersubjectivity of information in RQM. In contrast to RQM for which OIT is a supporting theoretical statement, QBism is challenged by OIT.

The genetic code is a map which gives the correspondence between codons in DNA and amino acids. In the attractor dynamical model (ADM), genetic codes can be described as the sets of the cyclic attractors of discrete dynamical systems - the iterations of functions acting in the ring of 2-adic integers Z₂. This ring arises from representation of nucleotides by binary vectors and hence codons by triples of binary vectors. We construct a Universal Function B such that the dynamical functions for all known genetic codes can be obtained from B by simple transformations on the set of codon cycles- the "Addition"and "Division"operations. ADM can be employed for study of phylogenetic dynamics of genetic codes. One can speculate that the "common ancestor genetic code"was caused by B. We remark that this function has 24 cyclic attractors which distribution coincides with the distribution for the hypothetical pre-LUCA code. This coupling of the Universal Function with the pre-LUCA code assigns the genetic codes evolution perspective to ADM. All genetic codes are generated from B through the special chains of the "Addition"and "Division"operations. The challenging problem is to assign the biological meaning to these mathematical operations.

This paper is devoted to an experimental investigation of cognitive contextuality inspired by quantum contextuality research. This contextuality is related to, but not identical to context-sensitivity which is well-studied in cognitive psychology and decision making. This paper is a part of quantum-like modeling, i.e., exploring the methodology of quantum theory outside of physics. We examined the bistable perception of cup-like objects, which strongly depends on experimental contexts. Our experimental data confirmed the existence of cognitive hysteresis, the important role of memory, and the non-commutative structure of cognitive observables. In physics, quantum contextuality is assessed using Bell-CHSH inequalities, and their violation is incorrectly believed to imply the nonlocality of Nature. The violation of Bell-type inequalities in cognitive and social science strongly indicates that the metaphysical implications of these inequalities are quite limited. In our experiments, modified Leggett-Garg inequalities were also significantly violated, but this only means that experimental data from experiments performed in different contexts cannot be modeled by a unique set of noncontextual, jointly distributed random variables. In our experiments, we know the empirical probability distributions measured in different contexts; thus, we can obtain much more detailed and reliable information about contextuality in human cognition by performing nonparametric compatibility tests.