The notion of a partition on a set is mathematically dual to the notion of a subset of a set, so there is a logic of partitions dual to Boole's logic of subsets (Boolean logic is usually mis-specified as "propositional" logic). The notion of an element of a subset has as its dual the notion of a distinction of a partition (a pair of elements in different blocks). Boole developed finite logical probability as the normalized counting measure on elements of (...) subsets so there is a dual concept of logical entropy which is the normalized counting measure on distinctions of partitions. Thus the logical notion of information is a measure of distinctions. Classical logical entropy naturally extends to the notion of quantum logical entropy which provides a more natural and informative alternative to the usual Von Neumann entropy in quantum information theory. The quantum logical entropy of a post-measurement density matrix has the simple interpretation as the probability that two independent measurements of the same state using the same observable will have different results. The main result of the paper is that the increase in quantum logical entropy due to a projective measurement of a pure state is the sum of the absolute squares of the off-diagonal entries ("coherences") of the pure state density matrix that are zeroed ("decohered") by the measurement, i.e., the measure of the distinctions ("decoherences") created by the measurement. (shrink)

We present a quantum mechanical analysis of Bell’s approach to quantum foundations based on his hidden-variable model. We claim and try to justify that the Bell model contradicts to the Heinsenberg’s uncertainty and Bohr’s complementarity principles. The aim of this note is to point to the physical seed of the aforementioned principles. This is the Bohr’s quantum postulate: the existence of indivisible quantum of action given by the Planck constant h. By contradicting these basic principles of QM, Bell’s model implies (...) rejection of this postulate as well. Thus, this hidden-variable model contradicts not only the QM-formalism, but also the fundamental feature of the quantum world discovered by Planck. (shrink)

In this paper we study the problem of a possibility to use quantum observables to describe a possible combination of the order effect with sequential reproducibility for quantum measurements. By the order effect we mean a dependence of probability distributions on the order of measurements. We consider two types of the sequential reproducibility: adjacent reproducibility ) and separated reproducibility). The first one is reproducibility with probability 1 of a result of measurement of some observable A measured twice, one A measurement (...) after the other. The second one, \, is reproducibility with probability 1 of a result of A measurement when another quantum observable B is measured between two A’s. Heuristically, it is clear that the second type of reproducibility is complementary to the order effect. We show that, surprisingly, this may not be the case. The order effect can coexist with a separated reproducibility as well as adjacent reproducibility for both observables A and B. However, the additional constraint in the form of separated reproducibility of the \ type makes this coexistence impossible. The problem under consideration was motivated by attempts to apply the quantum formalism outside of physics, especially, in cognitive psychology and psychophysics. However, it is also important for foundations of quantum physics as a part of the problem about the structure of sequential quantum measurements. (shrink)

This is a textbook on quantum mechanics. It is addressed to graduates and advanced undergraduates. The book presents quantum theory as a logically coherent system, placing stronger emphasis on the theory' s probabilistic structure and on the role of symmetries. It makes students aware of foundational problems from the very beginning, but at the same time, it urges them to adopt a pragmatic attitude towards the quantum formalism. The book consists of five parts. Part I is a review of classical (...) physics, including probability theory, and a historical introduction to quantum theory. Part II presents quantum theory in terms of six fundamental principles. The principles are presented together with the necessary mathematical background and with applications to simple systems. Part III analyses the properties of quantum particles. It starts with the description of symmetries, continues to non-relativistic particles, the introduction of spin, and particle statistics. It concludes with a symmetries-based introduction to relativistic quantum systems and with the first steps towards relativistic quantum field theory. Part IV presents techniques adapted to specific problems: the structure and spectra of composite systems, decays and transitions, particle scattering, and open quantum systems. Finally, Part V revisits quantum foundations: quantum measurement theory, the major interpretations of quantum theory, and the frontiers of quantum theory in contemporary research. (shrink)

The paper describes an algorithm for semantic representation of behavioral contexts relative to a dichotomic decision alternative. The contexts are represented as quantum qubit states in two-dimensional Hilbert space visualized as points on the Bloch sphere. The azimuthal coordinate of this sphere functions as a one-dimensional semantic space in which the contexts are accommodated according to their subjective relevance to the considered uncertainty. The contexts are processed in triples defined by knowledge of a subject about a binary situational factor. The (...) obtained triads of context representations function as stable cognitive structure at the same time allowing a subject to model probabilistically-variative behavior. The developed algorithm illustrates an approach for quantitative subjectively-semantic modeling of behavior based on conceptual and mathematical apparatus of quantum theory. (shrink)

In this paper we wish to raise the following question: which conceptual obstacles need to be overcome to arrive at a scientific and theoretical understanding of the mind? In the course of this examination, we shall encounter methodological and explanatory challenges and discuss them from the point of view of the philosophy of chemistry and quantum mechanics. This will eventually lead us to a discussion of emergence and metaphysics, thereby focusing on the status of objects. The question remains whether this (...) could be interpreted in terms of a re-description or dissolution of seemingly troubling problems in the philosophy of mind, or whether it further emphasizes the problematic: the ubiquitous and irreducible role of mind and consciousness in scientific activities. (shrink)

The paper introduces mathematical encoding for subjective experience and meaning in natural cognition. The code is based on a quantum-theoretic qubit structure supplementing classical bit with circular dimension, functioning as a process-causal template for representation of contexts relative to the basis decision. The qubit state space is demarcated in categories of emotional experience of animals and humans. Features of the resulting spherical map align with major theoreties in cognitive and emotion science, modeling of natural language, and semiotics, suggesting several generalizations (...) and improvements. The developed model bridges psychological, quantum-theoretic, and semiotic perspectives, allowing for an integrative account of subjectivity, agency, and meaning in living Nature.Graphical abstract. (shrink)

We analyze the interrelation of quantum and classical entanglement. The latter notion is widely used in classical optic simulation of some quantum-like features of light. We criticize the common interpretation that “quantum nonlocality” is the basic factor differing quantum and classical realizations of entanglement. Instead, we point to the breakthrough Grangier et al. experiment on coincidence detection which was done in 1986 and played the crucial role in rejection of classical field models in favor of quantum mechanics. Classical entanglement sources (...) produce light beams with the coefficient of second order coherence \} \ge 1.\) This feature of classical entanglement is obscured by using intensities of signals in different channels, instead of counting clicks of photo-detectors. The interplay between intensity and clicks counting is not just a technicality. We elevate this issue to the high foundational level. (shrink)