Pauli matrices explained

In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries.

These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left).

Each Pauli matrix is Hermitian, and together with the identity matrix (sometimes considered as the zeroth Pauli matrix), the Pauli matrices form a basis for the real vector space of Hermitian matrices. This means that any Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.

R ^{3} .

The Pauli matrices (after multiplication by to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices form a basis for the real Lie algebra

ak{su}(2) , which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices is isomorphic to the Clifford algebra of R ^{3,} ^[1] and the (unital) associative algebra generated by functions identically (is isomorphic) to that of quaternions (H ).

Algebraic properties

\sigma _{x} ! style=width:3em

\sigma _{y}	style=width:3em	\sigma _{z}
\sigma _{x}	I	i \sigma _{z}	- i \sigma _{y}
\sigma _{y}	- i \sigma _{z}	I	i \sigma _{x}
\sigma _{z}	i \sigma _{y}	- i \sigma _{x}	I

All three of the Pauli matrices can be compacted into a single expression:

\sigma _{j} = \begin{pmatrix} \delta _{j3} & \delta _{j1} - i\delta _{j2} \\ \delta _{j1} + i\delta _{j2} & - \delta _{j3} \end{pmatrix}, where the solution to is the "imaginary unit", and is the Kronecker delta, which equals if and 0 otherwise. This expression is useful for "selecting" any one of the matrices numerically by substituting values of in turn useful when any of the matrices (but no particular one) is to be used in algebraic manipulations.

The matrices are involutory:

\sigma ^{2} _{1} = \sigma ^{2} _{2} = \sigma ^{2} _{3} = - i\sigma _{1} \sigma _{2} \sigma _{3} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} = I, where is the identity matrix.

The determinants and traces of the Pauli matrices are

\begin{align} \det \sigma _{j} & = -1 , \\ \operatorname{tr} \sigma _{j} & = 0 , \end{align} from which we can deduce that each matrix has eigenvalues +1 and −1.l{H} _{2} of Hermitian matrices over R , and the Hilbert space l{M} _{2,2} ( C ) of all complex matrices over C .

Commutation and anti-commutation relations

Commutation relations

The Pauli matrices obey the following commutation relations:

[\sigma _{j,} \sigma _{k]} = 2 i \varepsilon _{j} \sigma _{l,} where the structure constant is the Levi-Civita symbol, and Einstein summation notation is used.

These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra

( R ^{3,} x ) \cong ak{su}(2) \cong ak{so}(3) .

Anticommutation relations

They also satisfy the anticommutation relations:

\{\sigma _{j,} \sigma _{k\}} = 2 \delta _{j} I,

where

\{\sigma _{j,} \sigma _{k\}} is defined as \sigma _{j} \sigma _{k} + \sigma _{k} \sigma _{j,} and is the Kronecker delta. denotes the identity matrix.

These anti-commutation relations make the Pauli matrices the generators of a representation of the Clifford algebra for

R ^{3,} denoted Cl _{3(R)} .

The usual construction of generators

\sigma _{jk} = \tfrac{ 1 }{ 4 } [\sigma _{j,} \sigma _{k]} of ak{so}(3) using the Clifford algebra recovers the commutation relations above, up to unimportant numerical factors.

A few explicit commutators and anti-commutators are given below as examples:

Eigenvectors and eigenvalues

Each of the (Hermitian) Pauli matrices has two eigenvalues: and . The corresponding normalized eigenvectors are

\begin{align} \psi _{x+} & = (1)/(\sqrt{ 2 }) \begin{bmatrix} 1 \ 1 \end{bmatrix}, & \psi _{x-} & = (1)/(\sqrt{ 2 }) \begin{bmatrix} 1 \ -1 \end{bmatrix}, \\ \psi _{y+} & = (1)/(\sqrt{ 2 }) \begin{bmatrix} 1 \ i \end{bmatrix}, & \psi _{y-} & = (1)/(\sqrt{ 2 }) \begin{bmatrix} 1 \ - i \end{bmatrix}, \\ \psi _{z+} & = \begin{bmatrix} 1 \ 0 \end{bmatrix}, & \psi _{z-} & = \begin{bmatrix} 0 \ 1 \end{bmatrix}. \end{align}

Pauli vectors

The Pauli vector is defined by$$\backslash vec\; =\; \backslash sigma\_1\; \backslash hat\_1\; +\; \backslash sigma\_2\; \backslash hat\_2\; +\; \backslash sigma\_3\; \backslash hat\_3,$$where

\hat{x} _{1} , \hat{x} _{2} , and \hat{x} _{3} are an equivalent notation for the more familiar \hat{x} , \hat{y} , and \hat{z} .

The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis^[2] as follows:using Einstein's summation convention.

More formally, this defines a map from

R ^{3} to the vector space of traceless Hermitian 2 x 2 matrices. This map encodes structures of R ^{3} as a normed vector space and as a Lie algebra (with the cross-product as its Lie bracket) via functions of matrices, making the map an isomorphism of Lie algebras. This makes the Pauli matrices intertwiners from the point of view of representation theory.

Another way to view the Pauli vector is as a

2 x 2 Hermitian traceless matrix-valued dual vector, that is, an element of Mat _{2 x} ( C ) ⊗ ( R ^{3)} ^{*} that maps \vec a \mapsto \vec a ⋅ \vec \sigma.

Completeness relation

Each component of

\vec a can be recovered from the matrix (see completeness relation below) $$\backslash frac\; \backslash operatorname\; \backslash Bigl(\backslash bigl(\backslash vec\; \backslash cdot\; \backslash vec\; \backslash bigr)\; \backslash vec\; \backslash Bigr)\; =\; \backslash vec.$$This constitutes an inverse to the map \vec a \mapsto \vec a ⋅ \vec \sigma , making it manifest that the map is a bijection.

Determinant

The norm is given by the determinant (up to a minus sign)$$\backslash det\; \backslash bigl(\backslash vec\; \backslash cdot\; \backslash vec\; \backslash bigr)\; =\; -\backslash vec\; \backslash cdot\; \backslash vec\; =\; -|\backslash vec|^2.$$Then, considering the conjugation action of an

SU ( 2 ) matrix U on this space of matrices,U * \vec a ⋅ \vec \sigma : = U \vec a ⋅ \vec \sigma U ^{-1} , we find \det(U * \vec a ⋅ \vec\sigma) = \det(\vec a ⋅ \vec \sigma), and that U * \vec a ⋅ \vec \sigma is Hermitian and traceless. It then makes sense to define U * \vec a ⋅ \vec\sigma = \vec a' ⋅ \vec\sigma, where \vec a' has the same norm as \vec a, and therefore interpret U as a rotation of three-dimensional space. In fact, it turns out that the special restriction on U implies that the rotation is orientation preserving. This allows the definition of a map R: SU ( 2 ) \to SO ( 3 ) given byU * \vec a ⋅ \vec \sigma = \vec a' ⋅ \vec \sigma = : (R(U) \vec a) ⋅ \vec \sigma, where R(U) \in SO ( 3 ). This map is the concrete realization of the double cover of SO ( 3 ) by SU ( 2 ), and therefore shows that SU ( 2 ) \cong Spin ( 3 ). The components of R(U) can be recovered using the tracing process above:R(U) _{ij} = (1)/(2) \operatorname{tr} \left( \sigma _{i} U \sigma _{j} U ^{-1} \right).

Cross-product

The cross-product is given by the matrix commutator (up to a factor of

2 i )$$[\backslash vec\; a\; \backslash cdot\; \backslash vec\; \backslash sigma,\; \backslash vec\; b\; \backslash cdot\; \backslash vec\; \backslash sigma]\; =\; 2i\backslash ,(\backslash vec\; a\; \backslash times\; \backslash vec\; b)\; \backslash cdot\; \backslash vec\; \backslash sigma.$$In fact, the existence of a norm follows from the fact that R ^{3} is a Lie algebra (see Killing form).

This cross-product can be used to prove the orientation-preserving property of the map above.

Eigenvalues and eigenvectors

The eigenvalues of

\vec a ⋅ \vec \sigma are \pm |\vec{a}|. This follows immediately from tracelessness and explicitly computing the determinant.

More abstractly, without computing the determinant, which requires explicit properties of the Pauli matrices, this follows from

(\vec a ⋅ \vec \sigma) ^{2} - |\vec a| ^{2} = 0 , since this can be factorised into (\vec a ⋅ \vec \sigma - |\vec a|)(\vec a ⋅ \vec \sigma + |\vec a|) = 0 . A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is diagonal) means this implies \vec a ⋅ \vec \sigma is diagonal with possible eigenvalues \pm |\vec a|. The tracelessness of \vec a ⋅ \vec \sigma means it has exactly one of each eigenvalue.

Its normalized eigenvectors are $$\backslash psi\_+\; =\; \backslash frac\; \backslash begin\; a\_3\; +\; \backslash left|\backslash vec\backslash right|\; \backslash \backslash \; a\_1\; +\; ia\_2\; \backslash end;\; \backslash qquad\; \backslash psi\_-\; =\; \backslash frac\; \backslash begin\; ia\_2\; -\; a\_1\; \backslash \backslash \; a\_3\; +\; |\backslash vec|\; \backslash end\; ~\; .$$These expressions become singular for

a _{3\to} - \left|\vec{a} \right| . They can be rescued by letting \vec{a} = \left|\vec{a} \right|(\epsilon, 0 , - ( 1- \epsilon ^{2/2))} and taking the limit \epsilon\to 0 , which yields the correct eigenvectors (0,1) and (1,0) of \sigma _{z} .

Alternatively, one may use spherical coordinates

\vec{a} = a(\sin\vartheta\cos\varphi, \sin\vartheta\sin\varphi, \cos\vartheta) to obtain the eigenvectors \psi _{+=(\cos(\vartheta/2),} \sin(\vartheta/ 2 )\exp(i\varphi)) and \psi _{-=(-\sin(\vartheta/2)\exp(-i\varphi),} \cos(\vartheta/ 2 )) .

Pauli 4-vector

The Pauli 4-vector, used in spinor theory, is written

\sigma ^{\mu} with components\sigma ^{\mu} = (I, \vec\sigma). This defines a map from R ^{1,3} to the vector space of Hermitian matrices,x _{\mu} \mapsto x ^{\mu ,} _{\mu\sigma} which also encodes the Minkowski metric (with mostly minus convention) in its determinant:\det (x ^{\mu)} _{\mu\sigma} = η (x,x).

This 4-vector also has a completeness relation. It is convenient to define a second Pauli 4-vector

\bar\sigma ^{\mu} = (I, - \vec\sigma). and allow raising and lowering using the Minkowski metric tensor. The relation can then be written$$x\_\backslash nu\; =\; \backslash tfrac\; \backslash operatorname\; \backslash Bigl(\backslash bar\backslash sigma\_\backslash nu\backslash bigl(x\_\backslash mu\; \backslash sigma^\backslash mu\; \backslash bigr)\; \backslash Bigr)\; .$$

Similarly to the Pauli 3-vector case, we can find a matrix group that acts as isometries on

R ^{1,3} ; in this case the matrix group is SL ( 2 , C ) , and this shows SL ( 2 , C ) \cong Spin ( 1 , 3 ). Similarly to above, this can be explicitly realized for S \in SL ( 2 , C ) with componentsΛ (S) ^{\mu{}} _{\nu} = \tfrac{ 1 }{ 2 }\operatorname{tr} \left( \bar\sigma _{\nu} S \sigma ^{\mu} S ^{\dagger} \right).

In fact, the determinant property follows abstractly from trace properties of the

\sigma ^{\mu.} For 2 x 2 matrices, the following identity holds:\det(A + B) = \det(A) + \det(B) + \operatorname{tr}(A)\operatorname{tr}(B) - \operatorname{tr}(AB). That is, the 'cross-terms' can be written as traces. When A,B are chosen to be different \sigma ^{\mu ,} the cross-terms vanish. It then follows, now showing summation explicitly,$\backslash det\backslash left(\backslash sum\_\backslash mu\; x\_\backslash mu\; \backslash sigma^\backslash mu\backslash right)\; =\; \backslash sum\_\backslash mu\; \backslash det\backslash left(x\_\backslash mu\backslash sigma^\backslash mu\backslash right).$ Since the matrices are 2 x 2 , this is equal to $\backslash sum\_\backslash mu\; x\_\backslash mu^2\; \backslash det(\backslash sigma^\backslash mu)\; =\; \backslash eta(x,x).$

Relation to dot and cross product

Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives

\begin{align} \left[ \sigma _{j,} \sigma _{k\right]} + \{\sigma _{j,} \sigma _{k\}} & = (\sigma _{j} \sigma _{k} - \sigma _{k} \sigma _{j} ) + (\sigma _{j} \sigma _{k} + \sigma _{k} \sigma _{j)} \\ 2 i\varepsilon _{j} \sigma _{\ell} + 2 \delta _{j} I & = 2 \sigma _{j} \sigma _{k} \end{align} so that,

Contracting each side of the equation with components of two -vectors and (which commute with the Pauli matrices, i.e., for each matrix and vector component (and likewise with) yields

~~ \begin{align} a _{j} b _{k} \sigma _{j} \sigma _{k} & = a _{j} b _{k} \left(i\varepsilon _{jk\ell} \sigma _{\ell} + \delta _{jk} I\right) \\ a _{j} \sigma _{j} b _{k} \sigma _{k} & = i\varepsilon _{jk\ell} a _{j} b _{k} \sigma _{\ell} + a _{j} b _{k} \delta _{jk} I \end{align}.~

Finally, translating the index notation for the dot product and cross product results in

If is identified with the pseudoscalar then the right hand side becomes

a ⋅ b + a \wedge b , which is also the definition for the product of two vectors in geometric algebra.

If we define the spin operator as, then satisfies the commutation relation:$$\backslash mathbf\; \backslash times\; \backslash mathbf\; =\; i\backslash hbar\; \backslash mathbf$$Or equivalently, the Pauli vector satisfies:$$\backslash frac\; \backslash times\; \backslash frac\; =\; i\backslash frac$$

Some trace relations

The following traces can be derived using the commutation and anticommutation relations.

\begin{align} \operatorname{tr}\left(\sigma _{j} \right) & = 0 \\ \operatorname{tr}\left(\sigma _{j} \sigma _{k} \right) & = 2 \delta _{jk} \\ \operatorname{tr}\left(\sigma _{j} \sigma _{k} \sigma _{\ell} \right) & = 2 i\varepsilon _{jk\ell} \\ \operatorname{tr}\left(\sigma _{j} \sigma _{k} \sigma _{\ell} \sigma _{m} \right) & = 2 \left(\delta _{jk} \delta _{\ell} - \delta _{j\ell} \delta _{km} + \delta _{jm} \delta _{k\ell} \right) \end{align}

If the matrix is also considered, these relationships become

where Greek indices and assume values from and the notation $\backslash sum\_$ is used to denote the sum over the cyclic permutation of the included indices.

Exponential of a Pauli vector

For

\vec{a} = a\hat{n}, |\hat{n}| = 1 ,

one has, for even powers,

(\hat{n} ⋅ \vec{\sigma}) ^{2p} = I ,

which can be shown first for the case using the anticommutation relations. For convenience, the case is taken to be by convention.

For odd powers,

\left(\hat{n} ⋅ \vec{\sigma}\right) ^{2q+1} = \hat{n} ⋅ \vec{\sigma} .

Matrix exponentiating, and using the Taylor series for sine and cosine,

\begin{align} e ^{i} ⋅ \vec{\sigma}\right)} & = \sum ^{inf ty{ (i ^{k} \left[a \left(\hat{n)/(⋅)} _{k =0} \vec{\sigma}\right)\right] ^{k}{k!}}} \\ & = \sum ^{inf ty{ (( -1 ) ^{p} (a\hat{n)/(⋅)} _{p =0} \vec{\sigma}) ^{2p} }} + i\sum_^\infty \\ &= I\sum_^\infty + i (\hat\cdot \vec) \sum_^\infty\\\end.

In the last line, the first sum is the cosine, while the second sum is the sine; so, finally,

which is analogous to Euler's formula, extended to quaternions.

Note that

\det[i a(\hat{n} ⋅ \vec{\sigma})] = a ^{2} ,

while the determinant of the exponential itself is just, which makes it the generic group element of SU(2).

A more abstract version of formula for a general matrix can be found in the article on matrix exponentials. A general version of for an analytic (at a and −a) function is provided by application of Sylvester's formula,^[3]

f(a(\hat{n} ⋅ \vec{\sigma})) = I (f(a) + f( - a))/(2) + \hat{n} ⋅ \vec{\sigma} (f(a) - f( - a))/(2) .

The group composition law of

A straightforward application of formula provides a parameterization of the composition law of the group . One may directly solve for in

which specifies the generic group multiplication, where, manifestly, $$\backslash cos\; c\; =\; \backslash cos\; a\; \backslash cos\; b\; -\; \backslash hat\; \backslash cdot\; \backslash hat\; \backslash sin\; a\; \backslash sin\; b~,$$the spherical law of cosines. Given, then, $$\backslash hat\; =\; \backslash frac\backslash left(\backslash hat\; \backslash sin\; a\; \backslash cos\; b\; +\; \backslash hat\; \backslash sin\; b\; \backslash cos\; a\; -\; \backslash hat\backslash times\backslash hat\; \backslash sin\; a\; \backslash sin\; b\backslash right).$$

Consequently, the composite rotation parameters in this group element (a closed form of the respective BCH expansion in this case) simply amount to^[4]

$$e^\; =\; \backslash exp\; \backslash left(i\backslash frac\; \backslash left(\backslash hat\; \backslash sin\; a\; \backslash cos\; b\; +\; \backslash hat\; \backslash sin\; b\; \backslash cos\; a\; -\; \backslash hat\backslash times\backslash hat\; ~\; \backslash sin\; a\; \backslash sin\; b\backslash right)\; \backslash cdot\; \backslash vec\backslash right).$$

(Of course, when

\hat{n} is parallel to \hat{m} , so is \hat{k} , and .)

Adjoint action

It is also straightforward to likewise work out the adjoint action on the Pauli vector, namely rotation of any angle

a along any axis \hat n :$$R\_n(-a)\; ~\; \backslash vec\; ~\; R\_n(a)\; =\; e^\; ~\; \backslash vec\; ~\; e^\; =\; \backslash vec\backslash cos\; (a)\; +\; \backslash hat\; \backslash times\; \backslash vec\; ~\; \backslash sin(a)\; +\; \backslash hat\; ~\; \backslash hat\; \backslash cdot\; \backslash vec\; ~\; (1\; -\; \backslash cos(a))\; ~\; .$$

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 $R\_y\backslash mathord\backslash left(-\backslash frac\backslash right)\backslash ,\; \backslash sigma\_x\backslash ,\; R\_y\backslash mathord\backslash left(\backslash frac\backslash right)\; =\; \backslash hat\; \backslash cdot\; \backslash left(\backslash hat\; \backslash times\; \backslash vec\backslash right)\; =\; \backslash sigma\_z$.

See also: Rodrigues' rotation formula.

Completeness relation

An alternative notation that is commonly used for the Pauli matrices is to write the vector index in the superscript, and the matrix indices as subscripts, so that the element in row and column of the -th Pauli matrix is

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

\vec{\sigma} _{\alpha\beta} ⋅ \vec{\sigma} _{\gamma\delta} \equiv \sum ^{3} _{k =1} \sigma ^{k} _{\alpha\beta} \sigma ^{k} _{\gamma\delta} = 2 \delta _{\alpha\delta} \delta _{\beta\gamma} - \delta _{\alpha\beta} \delta _{\gamma\delta} .

As noted above, it is common to denote the 2 × 2 unit matrix by so The completeness relation can alternatively be expressed as$$\backslash sum\_^3\; \backslash sigma^k\_\backslash ,\backslash sigma^k\_\; =\; 2\backslash ,\backslash delta\_\backslash ,\backslash delta\_\; ~\; .$$

The fact that any Hermitian complex 2 × 2 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’ density matrix, (positive semidefinite 2 × 2 matrices with unit trace. This can be seen by first expressing an arbitrary Hermitian matrix as a real linear combination of

Notes and References

1. Gull, S. F. . Lasenby, A. N. . Doran, C. J. L. . January 1993 . Imaginary numbers are not Real – the geometric algebra of spacetime . geometry.mrao.cam.ac.uk . Found. Phys. . 23 . 9 . 1175–1201 . 10.1007/BF01883676 . 1993FoPh...23.1175G . 14670523 . 2023-05-05 . dmy-all.
2. See the spinor map.
3. Book: Quantum Computation and Quantum Information . Nielsen . Michael A. . Michael Nielsen . Chuang . Isaac L. . Isaac Chuang . 2000 . Cambridge University Press . Cambridge, UK . 978-0-521-63235-5 . 43641333.
4. Book: Gibbs, J.W. . 1884 . Elements of Vector Analysis . New Haven, CT . 67. In fact, however, the formula goes back to Olinde Rodrigues (1840), replete with half-angle: Olinde . Rodrigues . Olinde Rodrigues . 1840 . 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 . . 5 . 380–440 .