View source for Unitary matrix

{{Short description|Complex matrix whose conjugate transpose equals its inverse}}
{{for multi|matrices with  orthogonality  over the real number field|orthogonal matrix|the restriction on the allowed evolution of quantum systems that ensures the sum of probabilities of all possible outcomes of any event always equals 1|unitarity}}

In [[Linear algebra|linear algebra]], an [[Invertible matrix|invertible]] [[Complex number|complex]] [[Matrix (mathematics)|square matrix]] {{mvar|U}} is '''unitary''' if its [[Invertible matrix|matrix inverse]] {{math|''U''<sup>−1</sup>}} equals its [[Conjugate transpose|conjugate transpose]] {{math|''U''<sup>*</sup>}}, that is, if

<math display=block>U^* U = UU^* = I,</math>

where {{mvar|I}} is the [[Identity matrix|identity matrix]].

In physics, especially in quantum mechanics, the conjugate transpose is referred to as the [[Hermitian adjoint]] of a matrix and is denoted by a dagger (†), so the equation above is written

<math display=block>U^\dagger U = UU^\dagger = I.</math>

A complex matrix {{mvar|U}} is '''special unitary''' if it is unitary and its matrix determinant equals {{math|1}}.

For [[Real number|real number]]s, the analogue of a unitary matrix is an [[Orthogonal matrix|orthogonal matrix]]. Unitary matrices have significant importance in quantum mechanics because they preserve [[Norm (mathematics)|norms]], and thus, [[Physics:Probability amplitude|probability amplitude]]s.

==Properties==
For any unitary matrix {{mvar|U}} of finite size, the following hold:
* Given two complex vectors {{math|'''x'''}} and {{math|'''y'''}}, multiplication by {{mvar|U}} preserves their inner product; that is, {{math|1=⟨''U'''''x''', ''U'''''y'''⟩ = ⟨'''x''', '''y'''⟩}}.
* {{mvar|U}} is [[Normal matrix|normal]] (<math>U^* U = UU^*</math>).
* {{mvar|U}} is [[Diagonalizable matrix|diagonalizable]]; that is, {{mvar|U}} is unitarily similar to a diagonal matrix, as a consequence of the [[Spectral theorem|spectral theorem]]. Thus, {{mvar|U}} has a decomposition of the form <math>U = VDV^*,</math> where {{mvar|V}} is unitary, and {{mvar|D}} is diagonal and unitary.
* <math>\left|\det(U)\right| = 1</math>.  That is, <math>\det(U)</math> will be on the unit circle of the complex plane.
* Its eigenspaces are orthogonal.
* {{mvar|U}} can be written as {{math|1=''U'' = ''e''<sup>''iH''</sup>}}, where {{mvar|e}} indicates the [[Matrix exponential|matrix exponential]], {{mvar|i}} is the imaginary unit, and {{mvar|H}} is a [[Hermitian matrix]].

For any nonnegative [[Integer|integer]] {{math|''n''}}, the set of all {{math|''n'' × ''n''}} unitary matrices with matrix multiplication forms a [[Group (mathematics)|group]], called the [[Unitary group|unitary group]] {{math|U(''n'')}}.

Any square matrix with unit Euclidean norm is the average of two unitary matrices.<ref>{{cite journal| first1=Chi-Kwong|last1= Li |first2= Edward|last2= Poon|doi=10.1080/03081080290025507|title=Additive decomposition of real matrices| year=2002| journal=Linear and Multilinear Algebra| volume=50| issue=4| pages=321–326|s2cid= 120125694 }}</ref>

==Equivalent conditions==
If ''U'' is a square, complex matrix, then the following conditions are equivalent:<ref>{{Cite book | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher={{wipe|Cambridge University Press}} | isbn=9781139020411 | year=2013 |doi=10.1017/CBO9781139020411 }}</ref>
# <math>U</math> is unitary.
# <math>U^*</math>  is unitary.
# <math>U</math> is invertible with <math>U^{-1} = U^*</math>.
# The columns of <math>U</math> form an [[Orthonormal basis|orthonormal basis]] of <math>\Complex^n</math> with respect to the usual inner product. In other words, <math>U^*U = I</math>. 
# The rows of <math>U</math> form an orthonormal basis of <math>\Complex^n</math> with respect to the usual inner product. In other words, <math>UU^* = I</math>. 
# <math>U</math> is an [[Isometry|isometry]] with respect to the usual norm. That is, <math>\|Ux\|_2 = \|x\|_2</math> for all <math>x \in \Complex^n</math>, where <math display="inline">\|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2}</math>.
# <math>U</math> is a [[Normal matrix|normal matrix]] (equivalently, there is an orthonormal basis formed by eigenvectors of <math>U</math>) with eigenvalues lying on the [[Unit circle|unit circle]].

==Elementary constructions==

=== 2 × 2 unitary matrix ===
One general expression of a {{nobr|2 × 2}} unitary matrix is

<math display=block>U = \begin{bmatrix}
 a & b \\
 -e^{i\varphi} b^* & e^{i\varphi} a^* \\
\end{bmatrix},
\qquad
\left| a \right|^2 + \left| b \right|^2 = 1\ ,</math>

which depends on 4&nbsp;real parameters (the phase of {{mvar|a}}, the phase of {{mvar|b}}, the relative magnitude between {{mvar|a}} and {{mvar|b}}, and the angle {{mvar|φ}}). The form is configured so the [[Determinant|determinant]] of such a matrix is
<math display=block> \det(U) = e^{i \varphi} ~. </math>

The sub-group of those elements <math>\ U\ </math> with <math>\ \det(U) = 1\ </math> is called the [[Special unitary group#The group SU(2)|special unitary group]] SU(2).

Among several alternative forms, the matrix {{mvar|U}} can be written in this form:
<math display=block>\ U = e^{i\varphi / 2} \begin{bmatrix}
 e^{i\alpha} \cos \theta & e^{i\beta} \sin \theta \\
 -e^{-i\beta} \sin \theta & e^{-i\alpha} \cos \theta \\
\end{bmatrix}\ ,</math>

where <math>\ e^{i\alpha} \cos \theta = a\ </math> and <math>\ e^{i\beta} \sin \theta = b\ ,</math> above, and the angles <math>\ \varphi, \alpha, \beta, \theta\ </math> can take any values.

By introducing <math>\ \alpha = \psi + \delta\ </math> and <math>\ \beta = \psi - \delta\ ,</math> has the following factorization:

<math display=block> U = e^{i\varphi /2} \begin{bmatrix}
 e^{i\psi} & 0 \\
 0 & e^{-i\psi}
\end{bmatrix}
\begin{bmatrix}
 \cos \theta  & \sin \theta \\
 -\sin \theta & \cos \theta \\
\end{bmatrix} 
\begin{bmatrix}
 e^{i\delta} & 0 \\
 0 & e^{-i\delta}
\end{bmatrix} ~.
</math>

This expression highlights the relation between {{nobr|2 × 2}} unitary matrices and {{nobr|2 × 2}} [[Orthogonal matrix|orthogonal matrices]] of angle {{mvar|θ}}.

Another factorization is<ref>{{cite journal |last1=Führ |first1=Hartmut |last2=Rzeszotnik |first2=Ziemowit |year=2018 |title=A note on factoring unitary matrices |journal=Linear Algebra and Its Applications |volume=547 |pages=32–44 |doi=10.1016/j.laa.2018.02.017 |s2cid=125455174 |issn=0024-3795|doi-access=free }}</ref>

<math display=block>U = \begin{bmatrix}
 \cos \rho  &   -\sin \rho \\
 \sin \rho  &   \;\cos \rho \\
\end{bmatrix} 
\begin{bmatrix}
 e^{i\xi} & 0 \\
 0 & e^{i\zeta}
\end{bmatrix}
\begin{bmatrix}
 \;\cos \sigma  &   \sin \sigma \\
 -\sin \sigma   &   \cos \sigma \\
\end{bmatrix} ~.
</math>

Many other factorizations of a unitary matrix in basic matrices are possible.<ref>{{cite book |last=Williams |first=Colin P. |year=2011 |section=Quantum gates |title=Explorations in Quantum Computing |pages=82 |editor-last=Williams |editor-first=Colin P. |series=Texts in Computer Science |place=London, UK |publisher=Springer |lang=en |doi=10.1007/978-1-84628-887-6_2 |isbn=978-1-84628-887-6}}</ref><ref>{{cite book |last1=Nielsen |first1=M.A. |last2=Chuang |first2=Isaac |year=2010 |title=Quantum Computation and Quantum Information |publisher={{wipe|Cambridge University Press}} |isbn=978-1-10700-217-3 |place=Cambridge, UK |oclc=43641333 |url=https://www.cambridge.org/9781107002173 |page=20}}</ref><ref name=Barenco>{{cite journal | last1=Barenco | first1=Adriano | last2=Bennett | first2=Charles H. | last3=Cleve | first3=Richard | last4=DiVincenzo | first4=David P. | last5=Margolus | first5=Norman | last6=Shor | first6=Peter | last7=Sleator | first7=Tycho | last8=Smolin | first8=John A. | last9=Weinfurter | first9=Harald | display-authors=6 | date=1995-11-01 | df=dmy-all | title=Elementary gates for quantum computation | journal=[[Physical Review A]] | publisher=American Physical Society (APS) | volume=52 | issue=5 | issn=1050-2947 | doi=10.1103/physreva.52.3457 | pages=3457–3467, esp.p. 3465 | pmid=9912645 | arxiv=quant-ph/9503016 | s2cid=8764584 }}</ref><ref>{{cite journal |last=Marvian |first=Iman |date=2022-01-10 |df=dmy-all  |title=Restrictions on realizable unitary operations imposed by symmetry and locality  |journal=Nature Physics |volume=18 |issue=3 |pages=283–289 |arxiv=2003.05524 |doi=10.1038/s41567-021-01464-0 |s2cid=245840243 |issn=1745-2481 |lang=en |url=https://www.nature.com/articles/s41567-021-01464-0}}</ref><ref>{{cite arXiv |last=Jarlskog |first = Cecilia |date=2006 |title=Recursive parameterisation and invariant phases of unitary matrices |eprint=math-ph/0510034}}</ref>
:See also:
{{cite journal |author=Alhambra, Álvaro M. |date=10 January 2022 |title=Forbidden by symmetry |journal=Nature Physics |volume=18 |issue=3 |pages=235–236 |issn=1745-2481 |doi=10.1038/s41567-021-01483-x |s2cid=256745894 |department=News & Views |url=https://www.nature.com/articles/s41567-021-01483-x.epdf?sharing_token=cb9JltmO0c_GuA_zyl_Hn9RgN0jAjWel9jnR3ZoTv0N2eMl-wQgGXVDdGkt0dHblV7Y2XiScmBn7eBbLkk2wN8fTlUuAcjP8wOfRS37lCMALVlmwQ72SNethITLikGw1OaeWVi_dwhQkvNW-wS5wsbz_fc5pIxAQO3XEghzc25Y%3D |quote=The physics of large systems is often understood as the outcome of the local operations among its components. Now, it is shown that this picture may be incomplete in quantum systems whose interactions are constrained by symmetries.}}</ref>

==See also==
{{div col begin |colwidth=15em}}
* [[Hermitian matrix]] and
: [[Skew-Hermitian matrix]]
* [[Matrix decomposition]]
* [[Orthogonal group|Orthogonal group O(''n'')]]
* [[Orthogonal group|Special orthogonal group SO(''n'')]]
* [[Orthogonal matrix]]
* [[Semi-orthogonal matrix]]
* [[Quantum logic gate]]
* [[Special unitary group|Special Unitary group SU(''n'')]]
* [[Symplectic matrix]]
* [[Unitary group|Unitary group U(''n'')]]
* [[Unitary operator]]
{{div col end}}

== References ==
{{reflist|25em}}

== External links ==
* {{MathWorld|urlname=UnitaryMatrix |title=Unitary Matrix |others=Todd Rowland}}
* {{SpringerEOM|id=U/u095540|title=Unitary matrix |first=O. A. |last=Ivanova}}
* {{cite web |title=Show that the eigenvalues of a unitary matrix have modulus 1 |work=[[Company:Stack Exchange|Stack Exchange]] |date=March 28, 2016 |url=https://math.stackexchange.com/q/1717713 }}

{{Matrix classes}}

{{DEFAULTSORT:Unitary Matrix}}
[[Category:Matrices]]
[[Category:Unitary operators]]

{{Sourceattribution|Unitary matrix}}