Editing Complementarity (physics) (section)

==Mathematical formalism==
For Bohr, complementarity was the "ultimate reason" behind the uncertainty principle. All attempts to grapple with atomic phenomena using classical physics were eventually frustrated, he wrote, leading to the recognition that those phenomena have "complementary aspects". But classical physics can be generalized to address this, and with "astounding simplicity", by describing physical quantities using non-commutative algebra.<ref name=":1" /> This mathematical expression of complementarity builds on the work of [[Hermann Weyl]] and [[Julian Schwinger]], starting with [[Hilbert spaces]] and [[unitary transformation]], leading to the theorems of [[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 |url=https://www.worldscientific.com/doi/abs/10.1142/S0219749910006502 |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 }}</ref>

In the [[mathematical formulation of quantum mechanics]], physical quantities that classical mechanics had treated as real-valued variables become [[self-adjoint operator]]s on a Hilbert space. These operators, called "[[Observable (physics)|observables]]", can fail to [[commutator (physics)|commute]], in which case they are called "incompatible":<math display="block">\left[\hat{A}, \hat{B}\right] := \hat{A}\hat{B} - \hat{B}\hat{A} \neq \hat{0}.</math>
Incompatible observables cannot have a complete set of common eigenstates; there can be some simultaneous eigenstates of <math>\hat{A}</math> and <math>\hat{B}</math>, but not enough in number to constitute a complete basis.<ref>{{Cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |title-link=Introduction to Quantum Mechanics (book) |date=2017 |publisher=Cambridge University Press |isbn=978-1-107-17986-8 |pages=111 |language=en |author-link=David J. Griffiths}}</ref><ref>{{Cite book |last1=Cohen-Tannoudji |first1=Claude |url=https://books.google.com/books?id=o6yftQEACAAJ |title=Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |date=2019-12-04 |publisher=Wiley |isbn=978-3-527-34553-3 |pages=232 |language=en |author-link1=Claude Cohen-Tannoudji}}</ref> The [[canonical commutation relation]]
<math display="block">\left[\hat{x}, \hat{p}\right] = i\hbar</math>
implies that this applies to position and momentum. In a Bohrian view, this is a mathematical statement that position and momentum are complementary aspects. Likewise, an analogous relationship holds for any two of the [[Spin (physics)|spin]] observables defined by the [[Pauli matrices]]; measurements of spin along perpendicular axes are complementary.<ref name=":0" /> The Pauli spin observables are defined for a quantum system described by a two-dimensional Hilbert space; mutually unbiased bases generalize these observables to Hilbert spaces of arbitrary finite dimension.<ref name="Klappenecker">{{Cite book |chapter=Mutually unbiased bases are complex projective 2-designs |url=https://ieeexplore.ieee.org/document/1523643 |title= Proceedings. International Symposium on Information Theory, 2005 |first1=A. |last1=Klappenecker |first2=M. |last2=Rotteler |date=2005 |pages=1740–1744 |language=en-US |doi=10.1109/isit.2005.1523643 |publisher=IEEE |isbn=0-7803-9151-9|s2cid=5981977 }}</ref> Two bases <math>\{|a_j\rangle\}</math> and <math>\{|b_k\rangle\}</math> for an <math>N</math>-dimensional Hilbert space are mutually unbiased when<math display="block">|\langle a_j|b_k \rangle|^2 = \frac{1}{N}\ \text{for all}\ j, k = 1, ... N-1.</math>

Here the basis vector <math>a_1</math>, for example, has the same overlap with every <math>b_k</math>; there is equal transition probability between a state in one basis and any state in the other basis. Each basis corresponds to an observable, and the observables for two mutually unbiased bases are complementary to each other.<ref name="Klappenecker"/> This leads to a definition of 'Principle of Complementarity' as:
{{blockquote|
For each degree of freedom the dynamical variables are a pair of complementary observables.<ref>{{Cite journal |last1=Scully |first1=Marian O. |last2=Englert |first2=Berthold-Georg |last3=Walther |first3=Herbert |date=May 1991 |title=Quantum optical tests of complementarity |url=https://www.nature.com/articles/351111a0 |journal=Nature |language=en |volume=351 |issue=6322 |pages=111–116 |doi=10.1038/351111a0 |bibcode=1991Natur.351..111S |s2cid=4311842 |issn=0028-0836|url-access=subscription }}</ref>}}The concept of complementarity has also been applied to quantum measurements described by [[POVM|positive-operator-valued measures]] (POVMs).<ref>{{Cite journal |last1=Busch |first1=P. |author-link=Paul Busch (physicist) |last2=Shilladay |first2=C. R. |date=2003-09-19 |title=Uncertainty reconciles complementarity with joint measurability |url=https://link.aps.org/doi/10.1103/PhysRevA.68.034102 |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 }}</ref><ref>{{Cite journal |last=Luis |first=Alfredo |date=2002-05-22 |title=Complementarity for Generalized Observables |url=https://link.aps.org/doi/10.1103/PhysRevLett.88.230401 |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-access=subscription }}</ref>