This chapter is devoted to the Bohr complementarity principle.This is one of the basic quantum principles. We dissolve it into separate subprincipleson contextuality, incompatibility, complementary-completeness,and individuality. We emphasize the role of the contexuatlity component. It is not highlighted in the foundational discussions. ByBohr, the outputs of measurements are resulted from the complexinteraction between a system and measurement context, the values ofquantum observables cannot be treated as objective properties of systems.Such Bohr contextuality is more general than joint measurement contextuality(JMC) considered in the discussions on the Bellinequality. JMC is a very special form of the Bohr contextuality. The incompatibility component is always emphasized and often referred as the wave-particle duality. The principle ofinformation complementary-completeness represents Bohr’s claim on completenessof quantum theory. The individuality principle is basic for thenotion of phenomenon used by Bohr to emphasize the individuality and discreteness of outputs of measurements. Individuality plays the crucial role in distinguishing quantum and classical optics entanglements.
This chapter is devoted to the Bohr complementarity principle.This is one of the basic quantum principles. We dissolve it into separate subprincipleson contextuality, incompatibility, complementary-completeness,and individuality. We emphasize the role of the contexuatlity component. It is not highlighted in the foundational discussions. ByBohr, the outputs of measurements are resulted from the complexinteraction between a system and measurement context, the values ofquantum observables cannot be treated as objective properties of systems.Such Bohr contextuality is more general than joint measurement contextuality(JMC) considered in the discussions on the Bellinequality. JMC is a very special form of the Bohr contextuality. The incompatibility component is always emphasized and often referred as the wave-particle duality. The principle ofinformation complementary-completeness represents Bohr’s claim on completenessof quantum theory. The individuality principle is basic for thenotion of phenomenon used by Bohr to emphasize the individuality and discreteness of outputs of measurements. Individuality plays the crucial role in distinguishing quantum and classical optics entanglements.
This is a review devoted to the complementarity–contextuality interplay with connection to the Bell inequalities. Starting the discussion with complementarity, I point to contextuality as its seed. Bohr contextuality is the dependence of an observable’s outcome on the experimental context; on the system–apparatus interaction. Probabilistically, complementarity means that the joint probability distribution (JPD) does not exist. Instead of the JPD, one has to operate with contextual probabilities. The Bell inequalities are interpreted as the statistical tests of contextuality, and hence, incompatibility. For context-dependent probabilities, these inequalities may be violated. I stress that contextuality tested by the Bell inequalities is the so-called joint measurement contextuality (JMC), the special case of Bohr’s contextuality. Then, I examine the role of signaling (marginal inconsistency). In QM, signaling can be considered as an experimental artifact. However, often, experimental data have signaling patterns. I discuss possible sources of signaling—for example, dependence of the state preparation on measurement settings. In principle, one can extract the measure of “pure contextuality” from data shadowed by signaling. This theory is known as contextuality by default (CbD). It leads to inequalities with an additional term quantifying signaling: Bell–Dzhafarov–Kujala inequalities.
Our aim is to compare the fundamental notions of quantum physics - contextuality vs. incompatibility. One has to distinguish two different notions of contextuality, {\it Bohr-contextuality} and {\it Bell-contextuality}. The latter is defined operationally via violation of noncontextuality (Bell type) inequalities. This sort of contextuality will be compared with incompatibility. It is easy to show that, for quantum observables, there is {\it no contextuality without incompatibility.} The natural question arises: What is contextuality without incompatibility? (What is "dry-residue"?) Generally this is the very complex question. We concentrated on contextuality for four quantum observables. We shown that in the CHSH-scenarios (for "natural quantum observables") {\it contextuality is reduced to incompatibility.} However, generally contextuality without incompatibility may have some physical content. We found a mathematical constraint extracting the contextuality component from incompatibility. However, the physical meaning of this constraint is not clear. In appendix 1, we briefly discuss another sort of contextuality based on the Bohr's complementarity principle which is treated as the {\it contextuality-incompatibility principle}. Bohr-contextuality plays the crucial role in quantum foundations. Incompatibility is, in fact, a consequence of Bohr-contextuality. Finally, we remark that outside of physics, e.g., in cognitive psychology and decision making Bell-contextuality cleaned of incompatibility can play the important role.
Abstract Our aim is to compare the fundamental notions of quantum physics - contextuality vs. incompatibility. One has to distinguish two different notions of contextuality,Bohr-contextualityandBell-contextuality. The latter is defined operationally via violation of noncontextuality (Bell type) inequalities. This sort of contextuality will be compared with incompatibility. It is easy to show that, for quantum observables, there isno contextuality without incompatibility.The natural question arises: What is contextuality without incompatibility? (What is “dry-residue”?) Generally this is the very complex question. We concentrated on contextuality for four quantum observables. We shown that, for “natural quantum observables” ,contextuality is reduced to incompatibility.But, generally contextuality without incompatibility may have some physical content. We found a mathematical constraint extracting the contextuality component from incompatibility. However, the physical meaning of this constraint is not clear. In Appendix 1, we briefly discuss another sort of contextuality based on Bohr’s contextuality-incompatibility principle. Bohr-contextuality plays the crucial role in quantum foundations. Incompatibility is, in fact, a consequence of Bohr-contextuality. Finally, we remark that outside of physics, e.g., in cognitive psychology and decision making Bell-contextuality distilled of incompatibility can play the important role.
Quantum mechanics challenges classical intuitions of space, time, and causality via the superposition principle, which allows systems to exist in multiple states simultaneously. Niels Bohr addressed these paradoxes through his Complementarity Principle, asserting that mutually exclusive properties-such as wave and particle behaviour-are jointly necessary for a complete description of quantum systems. Bohr also emphasized contextuality, the idea that measurement outcomes depend on the broader experimental setup. Despite his profound insights, Bohr did not offer a formal logical framework for these concepts. We address this gap by generalizing Reichenbach's three-valued logic into a seven-valued, formally contextual system, where contextuality is explicitly encoded via existential quantifiers. This logic retains Reichenbach's indeterminate value while modelling conditional truths across incompatible measurements, resolving key quantum paradoxes without logical inconsistency or nonlocality. Our approach draws inspiration from Jain logic which embraces multiple, context-dependent truths. Beyond quantum theory, this framework has potential applications in perceptual psychology and cognitive science, where context and contradiction also play central roles. The aim is to extend Bohr's legacy by equipping his epistemological vision with rigorous, many-valued logical tools.
The origin and basis of the notion of quantum contextuality is identified in the Copenhagen approach to quantum mechanics, where context is automatically invoked by its requirement that the experimental arrangement involved in any measurements or set of measurements be taken into account while, in general, the outcome of a measurement may depend on other measurements immediately preceding or jointly performed on the same system. For Bohr, the specification of the experimental situation of any measurement is essential to its significance in light of complementarity and the omnipresence of the quantum of action in physics; for Heisenberg, the incompatibility of pairs of sharp measurements belonging to different situations coheres with both the completeness of the quantum state as an objective physical description and the principle of indeterminacy. Here, context in the Copenhagen approach is taken to be the equivalence class of experimental arrangements corresponding to a set of compatible measurements of quantum observables in standard quantum mechanics; the associated form of contextuality in quantum mechanics arises via the non-commutativity in general of sharp observables, proven by von Neumann, that can appear, providing different contexts. This notion is related to theoretical situations explored later by Bell, by Kochen and Specker, and by others in relation to the classification of hidden-variables theories and elsewhere in physics. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.
Quantum complementarity is a fundamental feature of quantum systems and has captivated the physics research community for nearly a century, with significant advancements emerging in recent decades. This review traces the historical evolution of the concept of complementarity, beginning with Bohr's original formulation. It then explores its modern quantification through complementarity relations and its profound connection to the foundational postulates of quantum theory. Furthermore, it delves into key related developments, such as the operational definition of complementarity in the context of incompatible observables, its potential links with quantum uncertainty relations and contextuality, its various applications, and other pertinent topics. This review aims to serve physicists interested in quantum resources, quantum correlations, and the foundational principles of quantum mechanics.