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]

## History

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in Norway in February and March 1927, during which he received a letter from Werner Heisenberg regarding an as-yet-unpublished result, a thought experiment about a microscope using gamma rays.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] It was also covered in a 1953 essay by Bohr's collaborator Léon Rosenfeld.[13]

## Mathematical formalism

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

${\displaystyle \left[{\hat {A)),{\hat {B))\right]:={\hat {A)){\hat {B))-{\hat {B)){\hat {A))\neq {\hat {0)).}$

Observables corresponding to non-commuting operators are called incompatible observables. Incompatible observables cannot have a complete set of common eigenstates. Note that there can be some simultaneous eigenstates of ${\displaystyle {\hat {A))}$ and ${\displaystyle {\hat {B))}$, but not enough in number to constitute a complete basis.[14][15] The canonical commutation relation

${\displaystyle \left[{\hat {x)),{\hat {p))\right]=i\hbar }$

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.[16][17]

## References

1. ^ 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.
2. ^ 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.
3. ^ 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.
4. ^ 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.
5. ^ 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).
6. ^ 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.
7. ^ a b Fuchs, Christopher A. (2017). "Notwithstanding Bohr: The Reasons for QBism". Mind and Matter. 15: 245–300. arXiv:1705.03483. Bibcode:2017arXiv170503483F.
8. ^ Jammer, Max (1974). The Philosophy of Quantum Mechanics. John Wiley and Sons. ISBN 0-471-43958-4.
9. ^ Bohr, Niels (1939). "The causality problem in atomic physics". New theories in physics. Paris: International Institute of Intellectual Co-operation. pp. 11–38.
10. ^ 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.
11. ^ Bohr, Niels (1949). "Discussions with Einstein on Epistemological Problems in Atomic Physics". In Schilpp, Paul Arthur (ed.). Albert Einstein: Philosopher-Scientist. Open Court.
12. ^ 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.
13. ^ Rosenfeld, L. (1953). "Strife about Complementarity". Science Progress (1933- ). 41 (163): 393–410. ISSN 0036-8504.
14. ^ Griffiths, David J. (2017). Introduction to Quantum Mechanics. Cambridge University Press. p. 111. ISBN 978-1-107-17986-8.
15. ^ 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.
16. ^ 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.
17. ^ 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.