Editing Unitary matrix (section)

==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]] 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. |author1-link=Michael Nielsen |last2=Chuang |first2=Isaac |author2-link=Isaac Chuang |year=2010 |title=Quantum Computation and Quantum Information |publisher=[[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 | bibcode=1995PhRvA..52.3457B | 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 |bibcode=2022NatPh..18..283M |s2cid=245840243 |issn=1745-2481 |lang=en |url=https://www.nature.com/articles/s41567-021-01464-0}}</ref><ref>{{cite journal |last=Jarlskog |first = Cecilia |date=2006 |title=Recursive parameterisation and invariant phases of unitary matrices |journal = Journal of Mathematical Physics |volume = 47 |issue = 1 |page = 013507 |doi = 10.1063/1.2159069 |arxiv=math-ph/0510034|bibcode = 2006JMP....47a3507J }}</ref><ref>{{cite journal |author=Alhambra, Álvaro M. |date=10 January 2022 |title=Forbidden by symmetry |journal=[[Nature (journal)|Nature Physics]] |volume=18 |issue=3 |pages=235–236 |issn=1745-2481 |doi=10.1038/s41567-021-01483-x |bibcode=2022NatPh..18..235A |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>