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{{Short description|Quantum physics concept}}

{{Quantum book backlink|Conceptual and interpretations}}
'''Complementarity principle''' is a conceptual aspect of [[Physics:Quantum mechanics|quantum mechanics]] that [[Biography:Niels Bohr|Niels Bohr]] regarded as an essential feature of the theory.<ref>{{Cite magazine|last=Wheeler|first=John A.|date=January 1963|title="No Fugitive and Cloistered Virtue"—A tribute to Niels Bohr|magazine=Physics Today |volume=16 |issue=1 |page=30 |bibcode=1963PhT....16a..30W |doi=10.1063/1.3050711}}</ref><ref name="Howard2004">{{cite journal |title=Who invented the Copenhagen Interpretation? A study in mythology |year=2004 |last1=Howard |first1=Don |journal=Philosophy of Science |pages=669–682 |volume=71 |issue=5 |jstor=10.1086/425941 |doi=10.1086/425941 |url=http://www.nd.edu/~dhoward1/Copenhagen%20Myth%20A.pdf |citeseerx=10.1.1.164.9141 |s2cid=9454552 }}</ref> The principle holds that quantum objects have pairs of complementary properties that cannot all be observed or measured simultaneously, such as position and momentum or wave-like and particle-like behavior. In modern terms, complementarity is closely related to both the [[Physics:Uncertainty principle|uncertainty principle]] and [[Physics:Wave–particle duality|wave–particle duality]].

Bohr held that setting up an experiment to measure one quantity of a complementary pair excludes the possibility of measuring the other in the same arrangement, yet both experimental contexts are needed for a full account of the system under study. In this view, the behavior of atomic and subatomic objects cannot be separated from the measuring instruments that define the experimental context. There is therefore no single classical picture that unifies all results; only the ''totality of the phenomena'' provides a complete description.<ref name="Bohr1996a">{{cite book |first1=Niels |last1=Bohr |first2=Léon |last2=Rosenfeld |title=Foundations of Quantum Physics II (1933–1958) |series=Niels Bohr Collected Works |volume=7 |year=1996 |publisher=Elsevier |isbn=978-0-444-89892-0 |pages=284–285 |chapter=Complementarity: Bedrock of the Quantal Description |chapter-url=https://books.google.com/books?id=yet5P7f_63oC&pg=PA284 }}</ref>

[[File:Quantum_complementarity_principle_yellow.png|thumb|400px|Complementarity in quantum mechanics: different experimental arrangements reveal mutually exclusive but jointly necessary aspects of quantum systems, such as wave-like interference or particle-like path information.]]

== Historical background ==
Complementarity as a physical principle derives from Bohr’s 1927 presentation in Como, Italy, given during a celebration of the work of [[Biography:Alessandro Volta|Alessandro Volta]].<ref>{{Cite book |last=Baggott |first=J. E. |title=The quantum story: a history in 40 moments |date=2013 |publisher=Oxford Univ. Press |isbn=978-0-19-965597-7 |edition=Impression: 3 |location=Oxford}}</ref> Bohr’s subject was the idea that quantum measurements provide complementary information through apparently contradictory results.<ref name="BohrComo">{{Cite journal |last=Bohr |first=N. |title=The Quantum Postulate and the Recent Development of Atomic Theory |journal=Nature |volume=121 |issue=3050 |pages=580–590 |year=1928 |doi=10.1038/121580a0 |bibcode=1928Natur.121..580B |doi-access=free}}</ref> Although his presentation was not initially well received, it crystallized the issues that would become central to the modern interpretation of wave–particle duality.<ref name="Kumar2011">{{cite book |last=Kumar |first=Manjit |title=Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality |publisher=W. W. Norton & Company |edition=Reprint |year=2011 |pages=242, 375–376 |isbn=978-0-393-33988-8 |url=https://archive.org/details/quantumeinsteinb00manj/page/242 }}</ref>

The contradictions that motivated complementarity had accumulated from studies of both light and electrons. The wave theory of light, highly successful for more than a century, was challenged by [[Biography:Max Planck|Planck]]’s explanation of black-body radiation and [[Biography:Albert Einstein|Einstein]]’s interpretation of the [[Physics:Photoelectric effect|photoelectric effect]], both of which required discrete quanta of energy. The photon concept remained controversial until [[Biography:Arthur Compton|Arthur Compton]] demonstrated that light also carries momentum.<ref name="Whittaker">{{Cite book |last=Whittaker |first=Edmund T. |title=A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York}}</ref>

For electrons, the sequence was reversed. Experiments by [[Biography:J. J. Thomson|J. J. Thomson]], Robert Millikan, and [[Biography:Charles Thomson Rees Wilson|Charles Wilson]] showed particle-like properties, while [[Biography:Louis de Broglie|Louis de Broglie]] proposed in 1924 that electrons possess an associated wave, and [[Biography:Erwin Schrödinger|Schrödinger]] then showed that wave equations could account for electron behavior in atoms. Thus both light and matter displayed apparently incompatible but experimentally necessary descriptions.

== Bohr’s formulation ==
Bohr’s mature formulation of complementarity emerged in 1927, partly in response to [[Biography:Werner Heisenberg|Werner Heisenberg]]’s microscope thought experiment. Bohr believed that Heisenberg’s discussion of measurement disturbance did not yet fully capture the deeper point: in a context designed to measure position, momentum is not merely disturbed but is not sharply definable in the same sense, and vice versa.<ref name="Baggott2011">{{cite book|title=The Quantum Story: A History in 40 moments|last=Baggott|first=Jim|publisher=Oxford University Press|year=2011|isbn=978-0-19-956684-6|series=Oxford Landmark Science|location=Oxford|page=97}}</ref>

He publicly introduced complementarity in September 1927 at the International Physics Congress in Como, and again one month later at the Fifth Solvay Congress in Brussels.<ref name="Bohr1928English">{{cite journal |last=Bohr |first=N. |title=The Quantum Postulate and the Recent Development of Atomic Theory |journal=Nature |volume=121 |issue=3050 |pages=580–590 |year=1928 |doi=10.1038/121580a0 |bibcode=1928Natur.121..580B |doi-access=free}}</ref> In these lectures, Bohr emphasized that just as the finite speed of light prevents a sharp classical separation of space and time in relativity, the finite quantum of action prevents a sharp separation between a system and the measuring apparatus in quantum theory. This led him to the idea that different experimental setups reveal different but mutually necessary aspects of atomic reality.

Physicists F. A. M. Frescura and [[Biography:Basil Hiley|Basil Hiley]] later summarized Bohr’s point by noting that quantum mechanics undermines the classical assumption that all aspects of a system can be viewed simultaneously. Instead, one apparatus reveals one aspect at the expense of another, and a different apparatus reveals a different complementary aspect.<ref>{{cite journal|first1=F. A. M. |last1=Frescura |first2=B. J. |last2=Hiley |title=Algebras, quantum theory and pre-space |journal=Revista Brasileira de Física |volume=Special volume "Os 70 anos de Mario Schonberg" |pages=49–86 |date=July 1984 |url=http://www.bbk.ac.uk/tpru/BasilHiley/P12FrescandHiley3.pdf }}</ref>

== Complementary observables ==
Complementarity is most often illustrated by pairs of observables such as position and momentum. A measurement arrangement that sharply determines position excludes a simultaneous sharp determination of momentum, and the reverse is also true. Likewise, experimental arrangements that display wave-like interference do not yield which-path information, while path-detecting arrangements destroy the interference pattern.

In Bohr’s view, these are not merely practical limitations but reflect a basic feature of quantum description. Experimental context matters essentially, and different contexts reveal different aspects of the same system. Complementarity therefore does not assert that one description is true and the other false; rather, both are necessary, though they cannot be realized simultaneously in a single classical picture.<ref name="Bohr1996a" />

This idea became central in Bohr’s response to the [[Physics:EPR paradox|EPR paradox]], where [[Biography:Albert Einstein|Einstein]], [[Biography:Boris Podolsky|Boris Podolsky]], and [[Biography:Nathan Rosen|Nathan Rosen]] argued that quantum mechanics must be incomplete if it does not assign simultaneous precise values to quantities such as position and momentum. Bohr replied that the meaning of such quantities depends on the full experimental arrangement, so a value inferred in one context cannot simply be transferred to another incompatible context.<ref name="Fuchs2017">{{cite journal|first=Christopher A. |last=Fuchs |title=Notwithstanding Bohr: The Reasons for QBism |journal=Mind and Matter |volume=15 |pages=245–300 |year=2017 |arxiv=1705.03483 |bibcode=2017arXiv170503483F}}</ref><ref>{{cite book|last=Jammer|first=Max|title=The Philosophy of Quantum Mechanics|publisher=John Wiley and Sons|year=1974|isbn=0-471-43958-4}}</ref>

Later statements by Bohr, including his 1938 Warsaw lecture and his 1949 essay written for a volume honoring Einstein, continued to emphasize complementarity as a central principle of quantum theory.<ref name="BohrWarsaw1939">{{cite book|first=Niels |last=Bohr |chapter=The causality problem in atomic physics |title=New theories in physics |publisher=International Institute of Intellectual Co-operation |location=Paris |year=1939 |pages=11–38}}</ref><ref name="chevalley1999">{{cite book|first=Catherine |last=Chevalley |chapter=Why Do We Find Bohr Obscure? |title=Epistemological and Experimental Perspectives on Quantum Physics |editor-first1=Daniel |editor-last1=Greenberger |editor-first2=Wolfgang L. |editor-last2=Reiter |editor-first3=Anton |editor-last3=Zeilinger |publisher=Springer Science+Business Media |doi=10.1007/978-94-017-1454-9 |isbn=978-9-04815-354-1 |year=1999 |pages=59–74}}</ref><ref name="Bohr1949">{{cite book|title=Albert Einstein: Philosopher-Scientist|last=Bohr|first=Niels|publisher=Open Court|year=1949|editor=Schilpp|editor-first=Paul Arthur|chapter=Discussions with Einstein on Epistemological Problems in Atomic Physics}}</ref><ref>{{Cite journal|last=Rosenfeld|first=L.|date=1953|title=Strife about Complementarity|url=https://www.jstor.org/stable/43414997|journal=Science Progress (1933- )|volume=41|issue=163|pages=393–410|jstor=43414997 |issn=0036-8504}}</ref>

== Mathematical formalism ==
For Bohr, complementarity was the deeper reason behind the uncertainty principle. In the modern mathematical formulation of quantum mechanics, physical quantities are represented by [[Self-adjoint operator|self-adjoint operators]] acting on a Hilbert space. Two observables are incompatible when their operators fail to commute:
<math display="block">[\hat{A},\hat{B}] := \hat{A}\hat{B} - \hat{B}\hat{A} \neq \hat{0}.</math>

In such cases, the observables do not possess a complete common eigenbasis, so they cannot in general be assigned simultaneous sharp values.<ref>{{Cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |date=2017 |publisher=Cambridge University Press |isbn=978-1-107-17986-8 |pages=111}}</ref><ref>{{Cite book |last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications |date=2019-12-04 |publisher=Wiley |isbn=978-3-527-34553-3 |pages=232 |url=https://books.google.com/books?id=o6yftQEACAAJ }}</ref>

The most familiar example is the canonical commutation relation for position and momentum:
<math display="block">[\hat{x},\hat{p}] = i\hbar.</math>
This relation expresses mathematically that position and momentum are complementary. Similar statements hold for spin components defined by the [[Pauli matrices]]; spin measured along perpendicular axes is complementary as well.<ref name="Fuchs2017" />

Modern treatments generalize complementarity using [[Mutually unbiased bases|mutually unbiased bases]].<ref>{{Cite journal |last1=Durt |first1=Thomas |last2=Englert |first2=Berthold-Georg |last3=Bengtsson |first3=Ingemar |last4=Życzkowski |first4=Karol |date=2010-06-01 |title=On Mutually Unbiased Bases |journal=International Journal of Quantum Information |language=en |volume=08 |issue=4 |pages=535–640 |doi=10.1142/S0219749910006502 |issn=0219-7499 |arxiv=1004.3348 |s2cid=118551747 |url=https://www.worldscientific.com/doi/abs/10.1142/S0219749910006502 }}</ref> Two orthonormal bases <math>\{|a_j\rangle\}</math> and <math>\{|b_k\rangle\}</math> in an <math>N</math>-dimensional Hilbert space are mutually unbiased when
<math display="block">|\langle a_j|b_k\rangle|^2 = \frac{1}{N}</math>
for all basis states. If a system is sharp in one basis, it is maximally indeterminate in the other. This gives a precise mathematical sense in which the corresponding observables are complementary.<ref name="Klappenecker">{{Cite book |chapter=Mutually unbiased bases are complex projective 2-designs |title=Proceedings. International Symposium on Information Theory, 2005 |first1=A. |last1=Klappenecker |first2=M. |last2=Rötteler |date=2005 |pages=1740–1744 |doi=10.1109/isit.2005.1523643 |publisher=IEEE |isbn=0-7803-9151-9 |s2cid=5981977 |url=https://ieeexplore.ieee.org/document/1523643 }}</ref>

Complementarity has also been extended to generalized quantum measurements described by [[POVM|positive-operator-valued measures]].<ref>{{Cite journal |last1=Busch |first1=P. |last2=Shilladay |first2=C. R. |date=2003-09-19 |title=Uncertainty reconciles complementarity with joint measurability |journal=Physical Review A |language=en |volume=68 |issue=3 |page=034102 |doi=10.1103/PhysRevA.68.034102 |issn=1050-2947 |arxiv=quant-ph/0207081 |bibcode=2003PhRvA..68c4102B |s2cid=119482431 |url=https://link.aps.org/doi/10.1103/PhysRevA.68.034102 }}</ref><ref>{{Cite journal |last=Luis |first=Alfredo |date=2002-05-22 |title=Complementarity for Generalized Observables |journal=Physical Review Letters |language=en |volume=88 |issue=23 |page=230401 |doi=10.1103/PhysRevLett.88.230401 |pmid=12059339 |bibcode=2002PhRvL..88w0401L |issn=0031-9007 |url=https://link.aps.org/doi/10.1103/PhysRevLett.88.230401 }}</ref>

== Continuous complementarity ==
Complementarity need not be discussed only in terms of two idealized extremes. In many experiments one can continuously trade off wave-like interference against particle-like path information. This is captured by the wave–particle duality relation
<math display="block">D^2 + V^2 \leq 1,</math>
where <math>D</math> is path distinguishability and <math>V</math> is interference visibility.<ref name="Zeilinger">{{Cite journal |last=Zeilinger |first=Anton |date=1999-03-01 |title=Experiment and the foundations of quantum physics |journal=Reviews of Modern Physics |language=en |volume=71 |issue=2 |pages=S288–S297 |doi=10.1103/RevModPhys.71.S288 |bibcode=1999RvMPS..71..288Z |issn=0034-6861 |url=https://link.aps.org/doi/10.1103/RevModPhys.71.S288}}</ref><ref>{{Cite journal |last=Englert |first=Berthold-Georg |date=1999-01-01 |title=Remarks on Some Basic Issues in Quantum Mechanics |journal=Zeitschrift für Naturforschung A |language=en |volume=54 |issue=1 |pages=11–32 |doi=10.1515/zna-1999-0104 |issn=1865-7109 |doi-access=free |bibcode=1999ZNatA..54...11E }}</ref><ref name="Sen2014">{{Cite journal |last1=Sen |first1=D. |title=The Uncertainty relations in quantum mechanics |journal=Current Science |volume=107 |issue=2 |year=2014 |pages=203–218 |jstor=24103129 |url=https://www.jstor.org/stable/24103129 }}</ref>

Both <math>D</math> and <math>V</math> range between 0 and 1, but experiments that increase path knowledge necessarily reduce fringe visibility. This quantitative relation shows complementarity not only as a philosophical principle but also as an experimentally testable constraint on quantum phenomena.

== Modern role ==
Modern experiments have made complementarity far more concrete than in the 1920s. Quantum eraser and delayed-choice experiments test in detail the relation between interference, path information, and measurement context.<ref name="Zeilinger" /> These developments also connect complementarity to [[Quantum entanglement|entanglement]], quantum information, and the foundations of measurement theory.

[[Biography:Julian Schwinger|Julian Schwinger]] linked complementarity to the structure of [[Physics:Quantum field theory|quantum field theory]], remarking that relativistic quantum mechanics may be viewed as the union of Bohr’s complementarity and Einstein’s relativity principle.<ref>{{cite journal|last=Schwinger |first=Julian |title=Relativistic Quantum Field Theory |journal=Science |volume=153 |number=3739 |year=1966 |pages=949–953 |doi=10.1126/science.153.3739.949 |jstor=1719338 |pmid=17837239 |bibcode=1966Sci...153..949S }}</ref> The [[Physics:Consistent histories|consistent histories]] interpretation likewise uses a generalized form of complementarity as one of its defining ideas.<ref name="Hohenberg2010">{{Cite journal |last=Hohenberg |first=P. C. |date=2010-10-05 |title=Colloquium : An introduction to consistent quantum theory |journal=Reviews of Modern Physics |language=en |volume=82 |issue=4 |pages=2835–2844 |doi=10.1103/RevModPhys.82.2835 |issn=0034-6861 |arxiv=0909.2359 |bibcode=2010RvMP...82.2835H |s2cid=20551033 |url=https://link.aps.org/doi/10.1103/RevModPhys.82.2835 }}</ref>

Complementarity therefore remains one of the central conceptual tools for understanding how quantum mechanics departs from classical intuition, not by replacing one picture with another, but by requiring several experimentally grounded descriptions that are mutually exclusive yet jointly indispensable.

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}

==Further reading==
* [[Biography:Berthold-Georg Englert|Berthold-Georg Englert]], Marlan O. Scully & [[Biography:Herbert Walther|Herbert Walther]], ''Quantum Optical Tests of Complementarity'', Nature, Vol 351, pp 111–116 (9 May 1991); and the same authors, ''The Duality in Matter and Light'', ''Scientific American'', pp. 56–61 (December 1994).
* [[Biography:Niels Bohr|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.
* {{cite book|first=Richard |last=Rhodes |title=The Making of the Atomic Bomb |publisher=Simon & Schuster |year=1986 |isbn=0-671-44133-7 |oclc=231117096 |title-link=The Making of the Atomic Bomb}}

[[Category:Quantum mechanics]]
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Physics:Quantum Complementarity principle - Revision history

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