# Pauli matrices

    The following is modified from w:Pauli matrices.
In physics, the Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices. Usually indicated by the Greek letter "sigma" (σ), they are occasionally denoted with a "tau" (τ) when used in connection with isospin symmetries. They are:
    $\sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$
    $\sigma _{2}=\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}$
    $\sigma _{3}=\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$
The name refers to Wolfgang Pauli.
The real (hence also, complex) subalgebra generated by the σ$_i$ (that is, the set of real or complex linear combinations of all the elements which can be built up as products of Pauli matrices) is the full set M$_2$(C) of complex 2×2 matrices. The σ$_i$ can also be seen as generating the real Clifford algebra of the real quadratic form with signature (3,0): this shows that this Clifford algebra Cℓ$_{3,0}$(R) is isomorphic to M$_2$(C), with the Pauli matrices providing an explicit isomorphism. (In particular, the Pauli matrices define a faithful representation of the real Clifford algebra Cℓ$_{3,0}$(R) on the complex vector space C$^2$ of dimension 2.)

## Algebraic properties

    $\sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=-i\sigma _{1}\sigma _{2}\sigma _{3}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=I$
where I is the identity matrix, i.e. the matrices are involutory.
* The determinants and traces of the Pauli matrices are:
    ${\begin{matrix}\det(\sigma _{i})&=&-1&\\[1ex]\operatorname {Tr} (\sigma _{i})&=&0&\quad {\hbox{for}}\ i=1,2,3.\end{matrix}}$
From above we can deduce that the eigenvalues of each σ$_i$ are ±1.
* Together with the identity matrix I (which is sometimes written as σ$_0$), the Pauli matrices form an orthogonal basis, in the sense of Hilbert-Schmidt, for the real Hilbert space of 2 × 2 complex Hermitian matrices, or the complex Hilbert space of all 2 × 2 matrices.

### Pauli vector

The Pauli vector is defined by
    ${\vec {\sigma }}=\sigma _{1}{\hat {x}}+\sigma _{2}{\hat {y}}+\sigma _{3}{\hat {z}}\,$
and provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows
    ${\begin{aligned}{\vec {a}}\cdot {\vec {\sigma }}&=(a_{i}{\hat {x}}_{i})\cdot (\sigma _{j}{\hat {x}}_{j})\\&=a_{i}\sigma _{j}{\hat {x}}_{i}\cdot {\hat {x}}_{j}\\&=a_{i}\sigma _{i}\end{aligned}}$
(summation over indices implied). Note that in this vector dotted with Pauli vector operation the Pauli matrices are treated in a scalar like fashion, commuting with the vector basis elements.

### Commutation relations

The Pauli matrices obey the following commutation and anticommutation relations:
    ${\begin{matrix}[\sigma _{a},\sigma _{b}]&=&2i\varepsilon _{abc}\,\sigma _{c}\\[1ex]\{\sigma _{a},\sigma _{b}\}&=&2\delta _{ab}\cdot I\\\end{matrix}}$
where $\varepsilon _{abc}$ is the Levi-Civita symbol, $\delta _{ab}$ is the Kronecker delta, and I is the identity matrix.
The above two relations are equivalent to:
    $\sigma _{a}\sigma _{b}=\delta _{ab}\cdot I+i\sum _{c}\varepsilon _{abc}\sigma _{c}\,$.
For example,
    ${\begin{matrix}\sigma _{1}\sigma _{2}&=&i\sigma _{3},\\\sigma _{2}\sigma _{3}&=&i\sigma _{1},\\\sigma _{2}\sigma _{1}&=&-i\sigma _{3},\\\sigma _{1}\sigma _{1}&=&I.\\\end{matrix}}$
and the summary equation for the commutation relations can be used to prove
    $({\vec {a}}\cdot {\vec {\sigma }})({\vec {b}}\cdot {\vec {\sigma }})=({\vec {a}}\cdot {\vec {b}})\,I+i{\vec {\sigma }}\cdot ({\vec {a}}\times {\vec {b}})\quad \quad \quad \quad (1)\,$
    (as long as the vectors a and b commute with the pauli matrices)
as well as
    $e^{i({\vec {a}}\cdot {\vec {\sigma }})}=\cos {a}+i({\hat {n}}\cdot {\vec {\sigma }})\sin {a}\quad \quad \quad \quad \quad \quad (2)\,$
for ${\vec {a}}=a{\hat {n}}$.
    $({\vec {a}}\cdot {\vec {\sigma }})({\vec {b}}\cdot {\vec {\sigma }})\,$ $=a_{i}\sigma _{i}b_{j}\sigma _{j}\,$ $=a_{i}b_{j}\sigma _{i}\sigma _{j}\,$ $=a_{i}b_{j}\left(\delta _{ij}\cdot I+i\varepsilon _{ijk}\sigma _{k}\right)\,$ $=a_{i}b_{j}\delta _{ij}\cdot I+i\sigma _{k}\varepsilon _{ijk}a_{i}b_{j}\,$ $=({\vec {a}}\cdot {\vec {b}})\cdot I+i{\vec {\sigma }}\cdot ({\vec {a}}\times {\vec {b}})\,$
First notice that for even powers
    $({\hat {n}}\cdot {\vec {\sigma }})^{2n}=I\,$
but for odd powers
    $({\hat {n}}\cdot {\vec {\sigma }})^{2n+1}={\hat {n}}\cdot {\vec {\sigma }}\,$
Combine these two facts with the knowledge of the relation of the exponential to sine and cosine:
    $e^{ix}\,$ $=\sum _{n=0}^{\infty }{\frac {i^{n}x^{n}}{n!}}\,$ $=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}+i\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\,$
Which, when we use $x=a({\hat {n}}\cdot {\vec {\sigma }})\,$ gives us
       $=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(a{\hat {n}}\cdot {\vec {\sigma }})^{2n}}{(2n)!}}+i\sum _{n=0}^{\infty }{\frac {(-1)^{n}(a{\hat {n}}\cdot {\vec {\sigma }})^{2n+1}}{(2n+1)!}}\,$
       $=\sum _{n=0}^{\infty }{\frac {(-1)^{n}a^{2n}}{(2n)!}}+i({\hat {n}}\cdot {\vec {\sigma }})\sum _{n=0}^{\infty }{\frac {(-1)^{n}a^{2n+1}}{(2n+1)!}}\,$
The sum on the left is cosine, and the sum on the right is sine so finally,
    $e^{ia({\hat {n}}\cdot {\vec {\sigma }})}=\cos {a}+i({\hat {n}}\cdot {\vec {\sigma }})\sin {a}\,$

### Completeness relation

An alternative notation that is commonly used for the Pauli matrices is to write the vector index $i$ in the superscript, and the matrix indices as subscripts, so that the element in row $\alpha$ and column $\beta$ of the $i$th Pauli matrix is $\sigma _{\alpha \beta }^{i}$.
In this notation, the completeness relation for the Pauli matrices can be written
    ${\vec {\sigma }}\cdot {\vec {\sigma }}=\sum _{i}\sigma _{\alpha \beta }^{i}\sigma _{\gamma \delta }^{i}=2\delta _{\alpha \delta }\delta _{\beta \gamma }-\delta _{\alpha \beta }\delta _{\gamma \delta }.\,$
The fact that the Pauli matrices, along with the identity matrix $I$, form an orthogonal basis for the complex Hilbert space of all 2 × 2 matrices mean that we can express any matrix $M$ as
    $M=c\mathbf {I} +\sum _{i}a_{i}\sigma ^{i}$
where $c$ is a complex number, and $a$ is a 3-component complex vector. It is straightforward to show, using the properties listed above, that
    $\mathrm {tr} \,\sigma ^{i}\sigma ^{j}=2\delta _{ij}$
where $\mathrm {tr}$ denotes the trace, and hence that $c={\frac {1}{2}}\mathrm {tr} \,M$ and $a_{i}={\frac {1}{2}}\mathrm {tr} \,\sigma ^{i}M$. This therefore gives
    $2M=I\mathrm {tr} \,M+\sum _{i}\sigma ^{i}\mathrm {tr} \,\sigma ^{i}M$
which can be rewritten in terms of matrix indices as
    $2M_{\alpha \beta }=\delta _{\alpha \beta }M_{\gamma \gamma }+\sum _{i}\sigma _{\alpha \beta }^{i}\sigma _{\gamma \delta }^{i}M_{\delta \gamma }$
where summation is implied over the repeated indices $\gamma$ and $\delta$. Since this is true for any choice of the matrix $M$, the completeness relation follows as stated above.

### Relation with the permutation operator

Let $P_{ij}$ be the permutation (transposition, actually) between two spins $\sigma _{i}$ and $\sigma _{j}$ living in the tensor product space $\mathbb {C} ^{2}\otimes \mathbb {C} ^{2}$, $P_{ij}|\sigma _{i}\sigma _{j}\rangle =|\sigma _{j}\sigma _{i}\rangle$. This operator can be written as $P_{ij}={\frac {1}{2}}({\vec {\sigma }}_{i}\cdot {\vec {\sigma }}_{j}+1)$, as the reader can easily verify.

## SU(2)

The matrix group SU(2) is a Lie group, and its Lie algebra is the set of the anti-Hermitian 2×2 matrices with trace 0. Direct calculation shows that the Lie algebra su(2) is the 3 dimensional real algebra spanned by the set {$i\sigma _{j}$}. In symbols,
    $\;\operatorname {su} (2)=\operatorname {span} \{i\sigma _{1},i\sigma _{2},i\sigma _{3}\}.$
As a result, $i\sigma _{j}$s can be seen as infinitesimal generators of SU(2).

### A Cartan decomposition of SU(2)

This relationship between the Pauli matrices and SU(2) can be explored further, as can be seen from the following simple example. We can write
    $\;\operatorname {su} (2)=\operatorname {span} \{i\sigma _{2}\}\oplus \operatorname {span} \{i\sigma _{1},i\sigma _{3}\}.$
We put
    $\;{\mathfrak {k}}=\operatorname {span} \{i\sigma _{3}\},$
and
    $\;{\mathfrak {p}}=\operatorname {span} \{i\sigma _{1},i\sigma _{2}\}.$
Using the algebraic identities listed in the previous section, it can be verified that ${\mathfrak {k}}$ and ${\mathfrak {p}}$ form a Cartan pair of the Lie algebra SU(2). Furthermore,
    $\;{\mathfrak {a}}=\operatorname {span} \{i\sigma _{2}\}$
is a maximal abelian subalgebra of ${\mathfrak {p}}$. Now, a version of Cartan decomposition states that any element U in the Lie group SU(2) can be expressed in the form
    $U=e^{k_{1}}e^{a}e^{k_{2}}\,\!$ where $k_{1},k_{2}\in {\mathfrak {k}}$ and $a\in {\mathfrak {a}}.$
In other words, any unitary U of determinant 1 is of the form
    $U=e^{i\alpha \sigma _{3}}e^{i\beta \sigma _{2}}e^{i\gamma \sigma _{3}}\,\!$
for some real numbers α, β, and γ.
Extending to unitary matrices gives that any unitary 2 × 2 U is of the form
    $U=e^{i\delta }e^{i\alpha \sigma _{3}}e^{i\beta \sigma _{2}}e^{i\gamma \sigma _{3}}\,\!$
where the additional parameter δ is also real (also compare with Leonhardt 2010, eq 5.22, pg. 99)

### SO(3)

The Lie algebra su(2) is isomorphic to the Lie algebra so(3), which corresponds to the Lie group SO(3), the group of rotations in three-dimensional space. In other words, one can say that $i\sigma _{j}$'s are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though su(2) and so(3) are isomorphic as Lie algebras, SU(2) and SO(3) are not isomorphic as Lie groups. SU(2) is actually a double cover of SO(3), meaning that there is a two-to-one homomorphism from SU(2) to SO(3).

### Quaternions

The real linear span of $\{I,i\sigma _{1},i\sigma _{2},i\sigma _{3}\}\,$ is isomorphic to the real algebra of quaternions H. The isomorphism from H to this set is given by the following map (notice that Pauli matrices are in reversed order):
    $1\mapsto I,\quad i\mapsto i\sigma _{3},\quad j\mapsto i\sigma _{2},\quad k\mapsto i\sigma _{1}.$
As the quaternions of unit norm is group-isomorphic to SU(2), this gives yet another way of describing SU(2) via the Pauli matrices. The two-to-one homomorphism from SU(2) to SO(3) can also be explicitly given in terms of the Pauli matrices in this formulation.
Quaternions form a division algebra—every non-zero element has an inverse—whereas Pauli matrices do not. For a quaternionic version of the algebra generated by Pauli matrices see biquaternions, which is a venerable algebra of eight real dimensions.

## Physics
### Quantum mechanics

* In quantum mechanics, each Pauli matrix represents an observable describing the spin of a spin ½ particle in the three spatial directions. Also, as an immediate consequence of the Cartan decomposition mentioned above, $i\sigma _{j}$ are the generators of rotation acting on non-relativistic particles with spin ½. The state of the particles are represented as two-component spinors. An interesting property of spin ½ particles is that they must be rotated by an angle of 4$\pi$ in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3) mentioned above, and the fact that, although one visualizes spin up/down as the north/south pole on the 2-sphere S$^2$, they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space.
* For a spin ½ particle, the spin operator is given by $\mathbf {J} ={\frac {\hbar }{2}}{\boldsymbol {\sigma }}$. The Pauli matrices can be generalized to describe higher spin systems in three spatial dimensions. The spin matrices for spin 1 and spin $\textstyle {\frac {3}{2}}$ are given below:
$j=1$:
    $J_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}$
    $J_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}}$
    $J_{z}=\hbar {\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}}$
$j=\textstyle {\frac {3}{2}}$:
    $J_{x}={\frac {\hbar }{2}}{\begin{pmatrix}0&{\sqrt {3}}&0&0\\{\sqrt {3}}&0&2&0\\0&2&0&{\sqrt {3}}\\0&0&{\sqrt {3}}&0\end{pmatrix}}$
    $J_{y}={\frac {\hbar }{2}}{\begin{pmatrix}0&-i{\sqrt {3}}&0&0\\i{\sqrt {3}}&0&-2i&0\\0&2i&0&-i{\sqrt {3}}\\0&0&i{\sqrt {3}}&0\end{pmatrix}}$
    $J_{z}={\frac {\hbar }{2}}{\begin{pmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3\end{pmatrix}}.$
* Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G$_n$ is defined to consist of all n-fold tensor products of Pauli matrices.
* The fact that any 2 × 2 complex Hermitian matrices can be expressed in terms of the identity matrix and the Pauli matrices also leads to the Bloch sphere representation of 2 × 2 mixed states (2 × 2 positive semidefinite matrices with trace 1). This can be seen by simply first writing a Hermitian matrix as a real linear combination of {σ$_0$, σ$_1$, σ$_2$, σ$_3$} then impose the positive semidefinite and trace 1 assumptions.

### Quantum information

* In quantum information, single-qubit quantum gates are 2 × 2 unitary matrices. The Pauli matrices are some of the most important single-qubit operations. In that context, the Cartan decomposition given above is called the Z-Y decomposition of a single-qubit gate. Choosing a different Cartan pair gives a similar X-Y decomposition of a single-qubit gate.