Pauli matrices: Information from Answers.com

In physics, the Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices (See also ref.^[1]). 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.

Taking the set of all the linear combinations of all the elements which can be built up as products of Pauli matrices gives a representation of an algebra called the Pauli algebra, also known as the Clifford algebra Cℓ_3,0(R).

1 Algebraic properties
2 SU(2)
3 Physics
- 3.1 Quantum mechanics
- 3.2 Quantum information
4 See also
5 Notes
6 References

Algebraic properties

$\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I$

where I is the identity matrix, i.e. the matrices are involutary.

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 σ₀), 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

$\begin{align} \vec{a} \cdot \vec{\sigma} &= (a_i \hat{x}_i) \cdot (\sigma_j \hat{x}_j ) \\ &= a_i \sigma_j \hat{x}_i \cdot \hat{x}_j \\ &= a_i \sigma_i \end{align}$

(summation over indexes implied). Note that in this vector dotted with Pauli vector operation the Pauli matrixes 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:

$\begin{matrix} [\sigma_a, \sigma_b] &=& 2 i \varepsilon_{a b c}\,\sigma_c \\[1ex] \{\sigma_a, \sigma_b\} &=& 2 \delta_{a b} \cdot I \end{matrix}$

where $\varepsilon_{abc}$ is the Levi-Civita symbol, $δ a b$ 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 $\vec{a} = a \hat{n}$ .

Proof of (1)

$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) \,$	$= a_i \sigma_i b_j \sigma_j \,$
	$= a_i b_j \sigma_i \sigma_j \,$
	$= a_i b_j \left( \delta_{ij} \cdot I+ i \varepsilon_{ijk} \sigma_k \right) \,$
	$= a_i b_j \delta_{ij} \cdot I+ i \sigma_k \varepsilon_{ijk} a_i b_j \,$
	$= ( \vec{a} \cdot \vec{b} ) \cdot I+ i \vec{\sigma} \cdot ( \vec{a} \times \vec{b} )\,$

Proof of (2)

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,

$e^{i a(\hat{n} \cdot \vec{\sigma})} = \cos{a} + i (\hat{n} \cdot \vec{\sigma}) \sin{a} \,$

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 $α$ and column $β$ of the $i$ ^th Pauli matrix is $\sigma^i_{\alpha\beta}$ .

In this notation, the completeness relation for the Pauli matrices can be written

$\sum_i \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma} - \delta_{\alpha\beta}\delta_{\gamma\delta}\,$ .

Proof

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 $tr$ 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

$2M_{\alpha\beta} = \delta_{\alpha\beta} M_{\gamma\gamma} + \sum_i \sigma^i_{\alpha\beta} \sigma^i_{\gamma\delta} M_{\delta\gamma}$

where summation is implied over the repeated indices $γ$ and $δ$ . Since this is true for any choice of the matrix $M$ , the completeness relation follows as stated above.

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 { $i σ j$ }. In symbols,

$\; \operatorname{su}(2) = \operatorname{span} \{ i \sigma_1, i \sigma_2 , i \sigma_3 \}.$

As a result, $i σ j$ 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

$U = e^{i \alpha \sigma_1} e^{i \beta \sigma_2} e^{i \gamma \sigma_3}\,\!$

for some real numbers α, β, and γ.

Extending to unitary matrices gives that any unitary 2 × 2 U is of the form

$U = e^{i \delta} e^{i \alpha \sigma_1} e^{i \beta \sigma_2} e^{i \gamma \sigma_3}\,\!$

where the additional parameter δ is also real.

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 $i σ j$ '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

Consider the real linear span S of {I, σ₁ σ₂, σ₂ σ₃, σ₃ σ₁}. S is isomorphic to the real algebra of quaternions H. The isomorphism from H to S is given by

$1 \simeq 1, i \simeq \sigma_1 \sigma_2, j \simeq \sigma_3 \sigma_1, k \simeq \sigma_2 \sigma_3.$

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 - there always is an inverse - whereas Pauli matrices do not.

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, $i σ j$ 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 $π$ 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 S², they are actually represented by orthogonal vectors in the two dimensional complex Hilbert space.

For a spin ¹⁄₂ particle, the spin operator is given by $\mathbf{J} =\frac\hbar2\boldsymbol{\sigma}$ . The Pauli matrices can be generalized to describe higher spin systems in three spatial dimensions. The spin matrices for spin 1 and spin 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=³⁄₂:

$J_x = \frac\hbar2 \begin{pmatrix} 0&\sqrt{3}&0&0\\ \sqrt{3}&0&2&0\\ 0&2&0&\sqrt{3}\\ 0&0&\sqrt{3}&0 \end{pmatrix}$

$J_y = \frac\hbar2 \begin{pmatrix} 0&-i\sqrt{3}&0&0\\ i\sqrt{3}&0&-2i&0\\ 0&2i&0&-i\sqrt{3}\\ 0&0&i\sqrt{3}&0 \end{pmatrix}$

$J_z = \frac\hbar2 \begin{pmatrix} 3&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-3 \end{pmatrix}$

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 {σ₀, σ₁, σ₂, σ₃} 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.

Notes

^ http://planetmath.org/encyclopedia/PauliMatrices.html

References

Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.
Schiff, Leonard I. (1968). Quantum Mechanics. McGraw-Hill. ISBN 007-Y85643-5.

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