Editing Pauli matrices (section)

===Exponential of a Pauli vector===
For 
:<math>\vec{a} = a\hat{n},  \quad |\hat{n}| = 1,</math>

one has, for even powers, {{math|1=2''p'', ''p'' = 0, 1, 2, 3, ...}}
:<math>(\hat{n} \cdot \vec{\sigma})^{2p} = I ,</math>

which can be shown first for the {{math|1=''p'' = 1}} case using the anticommutation relations. For convenience, the case {{math|1=''p'' = 0}} is taken to be {{mvar|I}} by convention.

For odd powers, {{math|1=2''q'' + 1, ''q'' = 0, 1, 2, 3, ...}}
:<math>\left(\hat{n} \cdot \vec{\sigma}\right)^{2q+1} = \hat{n} \cdot \vec{\sigma} \, .</math>

[[Matrix exponential|Matrix exponentiating]], and using the [[Taylor series#List of Maclaurin series of some common functions|Taylor series for sine and cosine]],
:<math>\begin{align}
  e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)}
    &= \sum_{k=0}^\infty{\frac{i^k \left[a \left(\hat{n} \cdot \vec{\sigma}\right)\right]^k}{k!}} \\
    &= \sum_{p=0}^\infty{\frac{(-1)^p (a\hat{n}\cdot \vec{\sigma})^{2p}}{(2p)!}} + i\sum_{q=0}^\infty{\frac{(-1)^q (a\hat{n}\cdot \vec{\sigma})^{2q + 1}}{(2q + 1)!}} \\
    &= I\sum_{p=0}^\infty{\frac{(-1)^p a^{2p}}{(2p)!}} + i (\hat{n}\cdot \vec{\sigma}) \sum_{q=0}^\infty{\frac{(-1)^q a^{2q+1}}{(2q + 1)!}}\\
\end{align}</math>.

In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,
{{NumBlk||{{Equation box 1
  |indent =:
  |equation = <math>~~e^{i a\left(\hat{n} \cdot \vec{\sigma}\right)} = I\cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} ~~</math> 
  |cellpadding= 6
  |border
  |border colour = #0073CF
  |bgcolor=#F9FFF7
 }}
 |{{EquationRef|2}}
}}

which is [[quaternions and spatial rotation#Using quaternion as rotations|analogous]] to [[Euler's formula]], extended to [[quaternions]]. In particular,

<math>e^{i a \sigma_1} = \begin{pmatrix} \cos a & i \sin a \\ i \sin a & \cos a \end{pmatrix}, \quad
e^{i a \sigma_2} = \begin{pmatrix} \cos a & \sin a \\ - \sin a & \cos a \end{pmatrix}, \quad
e^{i a \sigma_3} = \begin{pmatrix} e^{ia} & 0  \\ 0 & e^{-ia} \end{pmatrix}.</math>

Note that
:<math>\det[i a(\hat{n} \cdot \vec{\sigma})] = a^2</math>,

while the determinant of the exponential itself is just {{math|1}}, which makes it the '''generic group element of [[SU(2)]]'''.

A more abstract version of formula {{EquationNote|(2)}} for a general {{math|2 × 2}} matrix can be found in the article on [[Matrix exponential#Evaluation by Laurent series|matrix exponentials]]. A general version of {{EquationNote|(2)}} for an analytic (at ''a'' and −''a'') function  is provided by application of  [[Sylvester's formula]],<ref>
{{cite book |title=Quantum Computation and Quantum Information |last1=Nielsen |first1=Michael A. |author-link1=Michael Nielsen |last2=Chuang |first2=Isaac L. |author-link2=Isaac Chuang |year=2000 |publisher=Cambridge University Press |location=Cambridge, UK |isbn=978-0-521-63235-5 |oclc=43641333}}
</ref>
:<math>f(a(\hat{n} \cdot \vec{\sigma})) = I\frac{f(a) + f(-a)}{2} +  \hat{n} \cdot \vec{\sigma} \frac{f(a) - f(-a)}{2}.</math>

====The group composition law of {{math|SU(2)}}====
A straightforward application of  formula {{EquationNote|(2)}}  provides a parameterization of the composition law of  the group {{math|SU(2)}}.{{efn|The relation among {{math|''a, b, c,'' ''' n, m, k '''}} derived here in the {{math|2 × 2}} representation holds for ''all representations'' of {{math|SU(2)}}, being a ''group identity''. Note that, by virtue of the standard normalization of that group's generators as ''half'' the Pauli matrices, the parameters ''a'',''b'',''c'' correspond to ''half'' the rotation angles of the rotation group. That is, the Gibbs formula linked amounts to <math>\hat k \tan c/2=  (\hat n \tan a/2+ \hat m \tan b/2 -\hat m \times \hat n \tan a/2 ~ \tan b/2 )/(1-\hat m\cdot \hat n \tan a/2 ~\tan b/2 )</math>.}} One may directly solve for {{mvar|c}} in 
<math display=block>\begin{align}
  e^{ia\left(\hat{n} \cdot \vec{\sigma}\right)} e^{ib\left(\hat{m} \cdot \vec{\sigma}\right)}
    &= I\left(\cos a \cos b - \hat{n} \cdot \hat{m} \sin a \sin b\right) + i\left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n} \times \hat{m} ~ \sin a \sin b \right) \cdot \vec{\sigma} \\
    &= I\cos{c} + i \left(\hat{k} \cdot \vec{\sigma}\right) \sin c \\
    &= e^{ic \left(\hat{k} \cdot \vec{\sigma}\right)},
\end{align}</math>

which specifies the generic group multiplication, where, manifestly,  
<math display=block>\cos c = \cos a \cos b - \hat{n} \cdot \hat{m} \sin a \sin b~,</math>
the [[spherical law of cosines]].  Given {{mvar|c}}, then, 
<math display=block>\hat{k} = \frac{1}{\sin c}\left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} \sin a \sin b\right).</math>

Consequently, the composite rotation parameters in this group element (a closed form of the respective [[Baker–Campbell–Hausdorff formula|BCH expansion]] in this case) simply amount to<ref>{{cite book |first=J.W. |last=Gibbs |year=1884 |title=Elements of Vector Analysis |place=New Haven, CT |page=67 |author-link=J. W. Gibbs |chapter=4. Concerning the differential and integral calculus of vectors |chapter-url={{GBurl|VurzAAAAMAAJ|p=67}} |publisher=Tuttle, Moorehouse & Taylor }} In fact, however, the formula goes back to [[Olinde Rodrigues]] (1840), replete with half-angle: {{cite journal |first=Olinde |last=Rodrigues |author-link=Olinde Rodrigues |year=1840 |title=Des lois géometriques qui regissent les déplacements d' un systéme solide dans l' espace, et de la variation des coordonnées provenant de ces déplacement considérées indépendant des causes qui peuvent les produire |journal=[[J. Math. Pures Appl.]] |volume=5 |pages=380–440 |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1840_1_5_A39_0.pdf}}</ref>

<math display=block> e^{ic \hat{k} \cdot \vec{\sigma}} =
  \exp \left( i\frac{c}{\sin c} \left(\hat{n} \sin a \cos b + \hat{m} \sin b \cos a - \hat{n}\times\hat{m} ~ \sin a \sin b\right) \cdot \vec{\sigma}\right). </math>

(Of course, when <math>\hat{n}</math> is parallel to <math>\hat{m}</math>, so is <math>\hat{k}</math>, and {{math|1=''c'' = ''a + b''}}.)
{{see also|Rotation formalisms in three dimensions#Rodrigues vector|Spinor#Three dimensions}}

====Adjoint action====
It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle <math>a</math> along any axis <math>\hat n</math>:
<math display=block>  
  R_n(-a) ~ \vec{\sigma} ~ R_n(a) =
  e^{i \frac{a}{2}\left(\hat{n} \cdot \vec{\sigma}\right)} ~ \vec{\sigma} ~  e^{-i \frac{a}{2}\left(\hat{n} \cdot \vec{\sigma}\right)} =
  \vec{\sigma}\cos (a) + \hat{n} \times \vec{\sigma} ~ \sin(a) + \hat{n} ~ \hat{n} \cdot \vec{\sigma} ~ (1 - \cos(a)) ~ .
</math>

Taking the dot product of any unit vector with the above formula generates the expression of any single qubit operator under any rotation. For example, it can be shown that <math display="inline">R_y\mathord\left(-\frac{\pi}{2}\right)\, \sigma_x\, R_y\mathord\left(\frac{\pi}{2}\right) = \hat{x} \cdot \left(\hat{y} \times \vec{\sigma}\right) = \sigma_z</math>.

{{see also|Rodrigues' rotation formula}}