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In physics, **complementarity** is a conceptual aspect of quantum mechanics that Niels Bohr regarded as an essential feature of the theory.^{ [1] }^{ [2] } The complementarity principle holds that objects have certain pairs of complementary properties which cannot all be observed or measured simultaneously. An example of such a pair is position and momentum. Bohr considered one of the foundational truths of quantum mechanics to be the fact that setting up an experiment to measure one quantity of a pair, for instance the position of an electron, excludes the possibility of measuring the other, yet understanding both experiments is necessary to characterize the object under study. In Bohr's view, the behavior of atomic and subatomic objects cannot be separated from the measuring instruments that create the context in which the measured objects behave. Consequently, there is no "single picture" that unifies the results obtained in these different experimental contexts, and only the "totality of the phenomena" together can provide a completely informative description.^{ [3] }

Niels Bohr apparently conceived of the principle of complementarity during a skiing vacation in Norway in February and March 1927, during which he received a letter from Werner Heisenberg regarding an as-yet-unpublished result, a thought experiment about a microscope using gamma rays. This thought experiment implied a tradeoff between uncertainties that would later be formalized as the uncertainty principle. To Bohr, Heisenberg's paper did not make clear the distinction between a position measurement merely disturbing the momentum value that a particle carried and the more radical idea that momentum was meaningless or undefinable in a context where position was measured instead. Upon returning from his vacation, by which time Heisenberg had already submitted his paper for publication, Bohr convinced Heisenberg that the uncertainty tradeoff was a manifestation of the deeper concept of complementarity.^{ [4] } Heisenberg duly appended a note to this effect to his paper, before its publication, stating:

Bohr has brought to my attention [that] the uncertainty in our observation does not arise exclusively from the occurrence of discontinuities, but is tied directly to the demand that we ascribe equal validity to the quite different experiments which show up in the [particulate] theory on one hand, and in the wave theory on the other hand.

Bohr publicly introduced the principle of complementarity in a lecture he delivered on 16 September 1927 at the International Physics Congress held in Como, Italy, attended by most of the leading physicists of the era, with the notable exceptions of Einstein, Schrödinger, and Dirac. However, these three were in attendance one month later when Bohr again presented the principle at the Fifth Solvay Congress in Brussels, Belgium. The lecture was published in the proceedings of both of these conferences, and was republished the following year in *Naturwissenschaften* (in German) and in *Nature* (in English).^{ [5] }

In his original lecture on the topic, Bohr pointed out that just as the finitude of the speed of light implies the impossibility of a sharp separation between space and time (relativity), the finitude of the quantum of action implies the impossibility of a sharp separation between the behavior of a system and its interaction with the measuring instruments and leads to the well-known difficulties with the concept of 'state' in quantum theory; the notion of complementarity is intended to capture this new situation in epistemology created by quantum theory. Physicists F.A.M. Frescura and Basil Hiley have summarized the reasons for the introduction of the principle of complementarity in physics as follows:^{ [6] }

In the traditional view, it is assumed that there exists a reality in space-time and that this reality is a given thing, all of whose aspects can be viewed or articulated at any given moment. Bohr was the first to point out that quantum mechanics called this traditional outlook into question. To him the "indivisibility of the quantum of action" [...] implied that not all aspects of a system can be viewed simultaneously. By using one particular piece of apparatus only certain features could be made manifest at the expense of others, while with a different piece of apparatus another complementary aspect could be made manifest in such a way that the original set became non-manifest, that is, the original attributes were no longer well defined. For Bohr, this was an indication that the principle of complementarity, a principle that he had previously known to appear extensively in other intellectual disciplines but which did not appear in classical physics, should be adopted as a universal principle.

Complementarity was a central feature of Bohr's reply to the EPR paradox, an attempt by Albert Einstein, Boris Podolsky and Nathan Rosen to argue that quantum particles must have position and momentum even without being measured and so quantum mechanics must be an incomplete theory.^{ [7] } The thought experiment proposed by Einstein, Podolsky and Rosen involved producing two particles and sending them far apart. The experimenter could choose to measure either the position or the momentum of one particle. Given that result, they could in principle make a precise prediction of what the corresponding measurement on the other, faraway particle would find. To Einstein, Podolsky and Rosen, this implied that the faraway particle must have precise values of both quantities whether or not that particle is measured in any way. Bohr argued in response that the deduction of a position value could not be transferred over to the situation where a momentum value is measured, and vice versa.^{ [8] }

Later expositions of complementarity by Bohr include a 1938 lecture in Warsaw ^{ [9] }^{ [10] } and a 1949 article written for a festschrift honoring Albert Einstein.^{ [11] }^{ [12] }

Complementarity is mathematically expressed by the operators that represent the observable quantities being measured failing to commute:

Observables corresponding to non-commutative operators are called *incompatible observables*. Incompatible observables cannot have a complete set of common eigenstates. Note that there can be some simultaneous eigenstates of and , but not enough in number to constitute a complete basis.^{ [13] }^{ [14] } The canonical commutation relation

implies that this applies to position and momentum. Likewise, an analogous relationship holds for any two of the spin observables defined by the Pauli matrices; measurements of spin along perpendicular axes are complementary.^{ [7] } This has been generalized to discrete observables with more than two possible outcomes using mutually unbiased bases, which provide complementary observables defined on finite-dimensional Hilbert spaces.^{ [15] }^{ [16] }

The **Copenhagen interpretation** is a collection of views about the meaning of quantum mechanics principally attributed to Niels Bohr and Werner Heisenberg. It is one of the oldest of numerous proposed interpretations of quantum mechanics, as features of it date to the development of quantum mechanics during 1925–1927, and it remains one of the most commonly taught.

The **Einstein–Podolsky–Rosen paradox** is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen (EPR), with which they argued that the description of physical reality provided by quantum mechanics was incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing them. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

**Quantum mechanics** is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

In quantum mechanics, **Schrödinger's cat** is a thought experiment that illustrates a paradox of quantum superposition. In the thought experiment, a hypothetical cat may be considered simultaneously both alive and dead as a result of its fate being linked to a random subatomic event that may or may not occur.

In quantum mechanics, the **uncertainty principle** is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, *x*, and momentum, *p*, can be predicted from initial conditions.

**Wave–particle duality** is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the behaviour of quantum-scale objects. As Albert Einstein wrote:

It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.

An **interpretation of quantum mechanics** is an attempt to explain how the mathematical theory of quantum mechanics "corresponds" to reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, which elements of quantum mechanics can be considered real, and what is the nature of measurement, among other matters.

In quantum mechanics, **wave function collapse** occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an "observation". It is the essence of a measurement in quantum mechanics which connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is the continuous evolution via the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment. Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions *apparently* reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation. More importantly, this is not enough to explain wave function collapse, as decoherence does not reduce it to a single eigenstate.

In physics, **hidden-variable theories** are proposals to provide explanations of quantum mechanical phenomena through the introduction of unobservable hypothetical entities. The existence of fundamental indeterminacy for some measurements is assumed as part of the mathematical formulation of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle. Most hidden-variable theories are attempts at a deterministic description of quantum mechanics, to avoid quantum indeterminacy, but at the expense of requiring the existence of nonlocal interactions.

In quantum physics, a **measurement** is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.

In quantum mechanics, the **measurement problem** considers how, or whether, wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer.

A **Bell test**, also known as **Bell inequality test** or **Bell experiment**, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. The experiments test whether or not the real world satisfies local realism, which requires the presence of some additional local variables to explain the behavior of particles like photons and electrons. To date, all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave.

The **Bohr–Einstein debates** were a series of public disputes about quantum mechanics between Albert Einstein and Niels Bohr. Their debates are remembered because of their importance to the philosophy of science, since the disagreements and the outcome of Bohr's version of quantum mechanics that became the prevalent view form the root of the modern understanding of physics. Most of Bohr's version of the events held in Solvay in 1927 and other places was first written by Bohr decades later in an article titled, "Discussions with Einstein on Epistemological Problems in Atomic Physics". Based on the article, the philosophical issue of the debate was whether Bohr's Copenhagen Interpretation of quantum mechanics, which centered on his belief of complementarity, was valid in explaining nature. Despite their differences of opinion and the succeeding discoveries that helped solidify quantum mechanics, Bohr and Einstein maintained a mutual admiration that was to last the rest of their lives.

**Quantum mechanics** is the study of very small things. It explains the behavior of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the Moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original scientific paradigm: the *theory of relativity* and the development of *quantum mechanics*. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century. It describes these concepts in roughly the order in which they were first discovered. For a more complete history of the subject, see *History of quantum mechanics*.

**Heisenberg's microscope** is a thought experiment proposed by Werner Heisenberg that has served as the nucleus of some commonly held ideas about quantum mechanics. In particular, it provides an argument for the uncertainty principle on the basis of the principles of classical optics.

The **ensemble interpretation** of quantum mechanics considers the quantum state description to apply only to an ensemble of similarly prepared systems, rather than supposing that it exhaustively represents an individual physical system.

**Popper's experiment** is an experiment proposed by the philosopher Karl Popper to put to the test different interpretations of quantum mechanics (QM). In fact, as early as 1934, Popper started criticising the increasingly more accepted Copenhagen interpretation, a popular subjectivist interpretation of quantum mechanics. Therefore, in his most famous book *Logik der Forschung* he proposed a first experiment alleged to empirically discriminate between the Copenhagen Interpretation and a realist interpretation, which he advocated. Einstein, however, wrote a letter to Popper about the experiment in which he raised some crucial objections and Popper himself declared that this first attempt was "a gross mistake for which I have been deeply sorry and ashamed of ever since".

In physics, the **observer effect** is the disturbance of an observed system by the act of observation. This is often the result of instruments that, by necessity, alter the state of what they measure in some manner. A common example is checking the pressure in an automobile tire; this is difficult to do without letting out some of the air, thus changing the pressure. Similarly, it is not possible to see any object without light hitting the object, and causing it to reflect that light. While the effects of observation are often negligible, the object still experiences a change. This effect can be found in many domains of physics, but can usually be reduced to insignificance by using different instruments or observation techniques.

Some interpretations of quantum mechanics posit a central role for an **observer** of a quantum phenomenon. The quantum mechanical observer is tied to the issue of observer effect, where a measurement necessarily requires interacting with the physical object being measured, affecting its properties through the interaction. The term "observable" has gained a technical meaning, denoting a Hermitian operator that represents a measurement.

A hallmark of Albert Einstein's career was his use of visualized thought experiments as a fundamental tool for understanding physical issues and for elucidating his concepts to others. **Einstein's thought experiments** took diverse forms. In his youth, he mentally chased beams of light. For special relativity, he employed moving trains and flashes of lightning to explain his most penetrating insights. For general relativity, he considered a person falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his debates with Niels Bohr on the nature of reality, he proposed imaginary devices intended to show, at least in concept, how the Heisenberg uncertainty principle might be evaded. In a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as quantum entanglement.

- ↑ Wheeler, John A. (January 1963). ""No Fugitive and Cloistered Virtue"—A tribute to Niels Bohr".
*Physics Today*. Vol. 16 no. 1. p. 30. Bibcode:1963PhT....16a..30W. doi:10.1063/1.3050711. - ↑ Howard, Don (2004). "Who invented the Copenhagen Interpretation? A study in mythology" (PDF).
*Philosophy of Science*.**71**(5): 669–682. CiteSeerX 10.1.1.164.9141 . doi:10.1086/425941. JSTOR 10.1086/425941. S2CID 9454552. - ↑ Bohr, Niels; Rosenfeld, Léon (1996). "Complementarity: Bedrock of the Quantal Description".
*Foundations of Quantum Physics II (1933–1958)*. Niels Bohr Collected Works.**7**. Elsevier. pp. 284–285. ISBN 978-0-444-89892-0. - ↑ Baggott, Jim (2011).
*The Quantum Story: A History in 40 moments*. Oxford Landmark Science. Oxford: Oxford University Press. p. 97. ISBN 978-0-19-956684-6. - ↑ Bohr, N. (1928). "The Quantum Postulate and the Recent Development of Atomic Theory".
*Nature*.**121**(3050): 580–590. Bibcode:1928Natur.121..580B. doi: 10.1038/121580a0 . Available in the collection of Bohr's early writings,*Atomic Theory and the Description of Nature*(1934). - ↑ Frescura, F. A. M.; Hiley, B. J. (July 1984). "Algebras, quantum theory and pre-space" (PDF).
*Revista Brasileira de Física*. Special volume "Os 70 anos de Mario Schonberg": 49–86, 2. - 1 2 Fuchs, Christopher A. (2017). "Notwithstanding Bohr: The Reasons for QBism".
*Mind and Matter*.**15**: 245–300. arXiv: 1705.03483 . Bibcode:2017arXiv170503483F. - ↑ Jammer, Max (1974).
*The Philosophy of Quantum Mechanics*. John Wiley and Sons. ISBN 0-471-43958-4. - ↑ Bohr, Niels (1939). "The causality problem in atomic physics".
*New theories in physics*. Paris: International Institute of Intellectual Co-operation. pp. 11–38. - ↑ Chevalley, Catherine (1999). "Why Do We Find Bohr Obscure?". In Greenberger, Daniel; Reiter, Wolfgang L.; Zeilinger, Anton (eds.).
*Epistemological and Experimental Perspectives on Quantum Physics*. Springer Science+Business Media. pp. 59–74. doi:10.1007/978-94-017-1454-9. ISBN 978-9-04815-354-1. - ↑ Bohr, Niels (1949). "Discussions with Einstein on Epistemological Problems in Atomic Physics". In Schilpp, Paul Arthur (ed.).
*Albert Einstein: Philosopher-Scientist*. Open Court. - ↑ Saunders, Simon (2005). "Complementarity and Scientific Rationality".
*Foundations of Physics*.**35**(3): 417–447. arXiv: quant-ph/0412195 . Bibcode:2005FoPh...35..417S. doi:10.1007/s10701-004-1982-x. S2CID 17301341. - ↑ Griffiths, David J. (2017).
*Introduction to Quantum Mechanics*. Cambridge University Press. p. 111. ISBN 978-1-107-17986-8. - ↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019-12-04).
*Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications*. Wiley. p. 232. ISBN 978-3-527-34553-3. - ↑ Bengtsson, Ingemar; Ericsson, Åsa (June 2005). "Mutually Unbiased Bases and the Complementarity Polytope".
*Open Systems & Information Dynamics*.**12**(2): 107–120. arXiv: quant-ph/0410120 . Bibcode:2004quant.ph.10120B. doi:10.1007/s11080-005-5721-3. ISSN 1230-1612. S2CID 37108528. - ↑ Blanchfield, Kate (2014-04-04). "Orbits of mutually unbiased bases".
*Journal of Physics A: Mathematical and Theoretical*.**47**(13): 135303. arXiv: 1310.4684 . Bibcode:2014JPhA...47m5303B. doi:10.1088/1751-8113/47/13/135303. ISSN 1751-8113. S2CID 118340150.

- Berthold-Georg Englert, Marlan O. Scully & Herbert Walther,
*Quantum Optical Tests of Complementarity*, Nature, Vol 351, pp 111–116 (9 May 1991) and (same authors)*The Duality in Matter and Light*Scientific American, pg 56–61, (December 1994). - Niels Bohr,
*Causality and Complementarity: supplementary papers edited by Jan Faye and Henry J. Folse. The Philosophical Writings of Niels Bohr, Volume IV*. Ox Bow Press. 1998. - Rhodes, Richard (1986).
*The Making of the Atomic Bomb*. Simon & Schuster. ISBN 0-671-44133-7. OCLC 231117096.

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