Bohr’s complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental set-up, a quantum system (or quanton) can exhibit either its wave-like character, denoted as W , or its particle-like character, denoted as P , but not both simultaneously. Modern interpretations of Bohr’s complementarity principle acknowledge the coexistence of these aspects in the same experiment while introducing the constraint W + P ≤ α . Notably, estimations of W or P frequently rely on indirect retrodiction methods, a practice that has led to the claim of the violation of Bohr’s complementarity principle. By taking a different route, recent advancements demonstrate that quantum complementarity relations can be rigorously derived from the axioms of quantum mechanics. To reconcile these observations and eliminate potential paradoxes or violations, we propose an updated formulation for the quantum complementarity principle, which is stated as follows: For a given quantum state preparation ρ t at a specific instant of time t , the wave and particle behaviours of a quanton are constrained by a complementarity relation 𝔚 ( ρ t ) + 𝔓 ( ρ t ) ≤ α ( d ) , which is derived directly from the axioms of quantum mechanics.
Bohr’s complementarity principle has long been a fundamental concept in quantum mechanics, positing that, within a given experimental set-up, a quantum system (or quanton) can exhibit either its wave-like character, denoted as W , or its particle-like character, denoted as P , but not both simultaneously. Modern interpretations of Bohr’s complementarity principle acknowledge the coexistence of these aspects in the same experiment while introducing the constraint W + P ≤ α . Notably, estimations of W or P frequently rely on indirect retrodiction methods, a practice that has led to the claim of the violation of Bohr’s complementarity principle. By taking a different route, recent advancements demonstrate that quantum complementarity relations can be rigorously derived from the axioms of quantum mechanics. To reconcile these observations and eliminate potential paradoxes or violations, we propose an updated formulation for the quantum complementarity principle, which is stated as follows: For a given quantum state preparation ρ t at a specific instant of time t , the wave and particle behaviours of a quanton are constrained by a complementarity relation 𝔚 ( ρ t ) + 𝔓 ( ρ t ) ≤ α ( d ) , which is derived directly from the axioms of quantum mechanics.
Bohr's complementarity principle is of fundamental historic and conceptual importance for quantum mechanics (QM), and states that, with a given experimental apparatus configuration, one can observe either the wave-like or the particle-like character of a quantum system, but not both. However, it was eventually realized that these dual behaviors can both manifest partially in the same experimental setup, and, using ad hoc proposed measures for the wave and particle aspects of the quanton, complementarity relations were proposed limiting how strong these manifestations can be. Recently, a formalism was developed and quantifiers for the particleness and waveness of a quantum system were derived from the mathematical structure of QM entailed in the density matrix's basic properties ($\ensuremath{\rho}\ensuremath{\ge}0, \mathrm{Tr}\ensuremath{\rho}=1$). In this article, using IBM Quantum Experience quantum computers, we perform experimental tests of these complementarity relations applied to a particular class of one-qubit quantum states and also for random quantum states of one, two, and three qubits.
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.
Bohr's Complementarity Principle is a core concept of quantum mechanics. In this article, an updated complementarity relation for the wave and ondulatory aspects of a quantum system is presented and discussed. Two interferometric experiments are implemented in one and two qubit circuits and executed on real hardware. The final state density matrices are reconstructed using quantum state tomography and the complementarity relation is tested via direct computation. Results of the executions are presented both graphically and with a mean squared error analysis for a better comprehension.
Updating the wave-particle duality Maria C. Bosca e-mail: bosca@ugr.es Departamento de Fisica Atomica, Molecular y Nuclear Universidad de Granada E-18071. Granada Spain The wave-particle-duality, the fundamental component of the new quantum formalism in Bohr’s opinion, must be reformulated by incorporating the results of some experiments accomplished in the last decades of twentieth century. The Bohr´s complementarity principle stated the mutual exclusiveness and joint full completeness of the two (classical) descriptions of quantum systems; after Einstein-Podolsky-Rosen’paper, the wave-particle duality, or wave-particle complementarity, could be expressed by stating that it is impossible to build up an experimental arrangement in which we observe at the same time both corpuscular and wave aspects. In a two-slit experiment, they would correspond, respectively, to the which-way knowledge and the observation of interference pattern. Bohr showed this mutual exclusivity in numerous examples , and linked it to the unavoidable disturbance inherent in any measurement event. In quantum mechanical formalism, the complementarity principle has a clear mathematical expression: two observables are complementary if precise knowledge of one of them implies that all possible outcomes of the other are equally probable; their extension to classical concepts (as wave and particle) is not concerned. In 1991 Scully et al published a variant of the two-slit experiment that incorporates two micromasers cavities and a laser beam to provide which-path information without net momentum transferred during the interaction ; the impossibility of know which slit an atom went through and still observe the interference fringes is preserved by the establishing of quantum correlations between the measuring apparatus and the system being observed. They claimed that complementarity, of which wave-particle duality would be to them a manifestation, is more fundamental than the uncertainty principle . In 1996 B-G. Englert, following an approach originally due to Wooters and Zurek , derived , without making use of Heisenberg’s uncertainty relation, an inequality that quantifies the mutual compatibility relation between fringe visibility and which-way information. The inequality, that they denominated as “interferometric duality”, has the expression D^2 + V^2 ≤ 1, where D stands for the distinguishability of the ways and V for the fringe visibility; both of them are mathematical expressions that can be measured to check experimentally the inequality . Today, it is clear that intermediate particle-wave behaviours exist and, in addition to that, there are single experiments in which both classical wave-like and particle-like behaviours are showed total and simultaneously on an individual system. For instance, in the Bose’s double-prism experiment , tunnelling and perfect anticoincidence were observed in single photon states. Consequently, the meaning of the wave-particle duality must incorporate the simultaneous use of the two classical descriptions in the interpretation of experiments, loosing their original mutual exclusivity , which is incorporated as an extreme case in the new interferometric duality, a continuous quantum concept . The wave-particle duality, therefore, must be formulated as an interpretative addition to quantum mechanics, to which is possible to renounce if any pretension of visualize quantum phenomena in terms of classical concepts and intuitions is abandoned.
Ninety years ago in 1927, at an international congress in Como, Italy, Niels Bohr gave an address which is recognized as the first instance in which the term "complementarity", as a physical concept, was spoken publicly [1], revealing Bohr's own thinking about Louis de Broglie's "duality". Bohr had very slowly accepted duality as a principle of physics: close observation of any quantum object will reveal either wave-like or particle-like behavior, one or the other of two fundamental and complementary features. Little disagreement exists today about complementarity's importance and broad applicability in quantum science. Book-length scholarly examinations even provide speculations about the relevance of complementarity in fields as different from physics as biology, psychology and social anthropology, connections which were apparently of interest to Bohr himself (see Jammer [2], Murdoch [3] and Whitaker [4]). Confusion evident in Como following his talk was not eliminated by Bohr's article [1], and complementarity has been subjected to nine decades of repeated examination ever since with no agreed resolution. Semi-popular treatments [5] as well as expert examinations [6-9] show that the topic cannot be avoided, and complementarity retains its central place in the interpretation of quantum mechanics. However, recent approaches by our group [10-13] and others [14-20] to the underlying notion of coherence now allow us to present a universal formulation of complementarity that may signal the end to the confusion. We demonstrate a new relationship that constrains the behavior of an electromagnetic field (quantum or classical) in the fundamental context of two-slit experiments. We show that entanglement is the ingredient needed to complete Bohr's formulation of complementarity, debated for decades because of its incompleteness.
Niels Bohr, as is well known, introduced the notion of complementarity into physics, as a fundamental principle of quantum mechanics. It holds that objects have complementary properties that cannot be measured accurately at the same time. For example, the particle and wave aspects of physical objects are such complementary phenomena. Both concepts are borrowed from classical mechanics, where it is impossible to be both, a particle and a wave at the same time. Particle and Wave represent the complementarity of the Discrete and the Continuous. Humans reason by means of concepts (meanings) and language, as well as, by means of logical or arithmetic symbolism. Meanings are continua, whereas logic and arithmetic are based on relations of identity and difference.