Pauli matrices

In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. They are usually denoted by the Greek letter , and occasionally by 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, and circular polarization.
Each Pauli matrix is Hermitian, and together with the identity matrix , the Pauli matrices form a basis of the vector space of Hermitian matrices over the real numbers, under addition. 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.
The Pauli matrices satisfy the useful product relation:
where is the Kronecker delta, which equals if otherwise, and the Levi-Civita symbol is used.
Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the complex two-dimensional Hilbert space. In the context of Pauli's work, represents the observable corresponding to spin along the th coordinate axis in three-dimensional Euclidean space.
The Pauli matrices also generate transformations in the sense of Lie algebras: The matrices ,, and form a basis for the real Lie algebra, which exponentiates to the special unitary group SU. The algebra generated by the three Pauli matrices is isomorphic to the Clifford algebra of and the associative algebra generated by ,, and functions identically to that of quaternions.

Algebraic properties

×

All three of the Pauli matrices can be compacted into a single expression:
This expression is useful for "selecting" any one of the matrices numerically by substituting values of in turn useful when any of the matrices is to be used in algebraic manipulations.
The matrices are involutory:
where is the identity matrix.
The determinants and traces of the Pauli matrices are
from which we can deduce that each matrix has eigenvalues.
With the inclusion of the identity matrix , the Pauli matrices form an orthogonal basis of the Hilbert space of Hermitian matrices over and the Hilbert space of all complex matrices over.

Commutation and anti-commutation relations

Commutation relations

The Pauli matrices obey the following commutation relations:
These commutation relations make the Pauli matrices the generators of a representation of the Lie algebra

Anticommutation relations

They also satisfy the anticommutation relations:
where is defined as 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 denoted
The usual construction of generators of 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 Pauli matrices has two eigenvalues:. The corresponding normalized eigenvectors are

Pauli vectors

The Pauli vector is defined by
where,, and are an equivalent notation for the more familiar,, and.
The Pauli vector provides a mapping mechanism from a vector basis to a Pauli matrix basis as follows:
More formally, this defines a map from to the vector space of traceless Hermitian matrices. This map encodes structures of as a normed vector space and as a Lie algebra 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 Hermitian traceless matrix-valued dual vector, that is, an element of that maps

Completeness relation

Each component of can be recovered from the matrix
This constitutes an inverse to the map, making it manifest that the map is a bijection.

Determinant

The norm is given by the determinant
Then, considering the conjugation action of an matrix on this space of matrices,
we find and that is Hermitian and traceless. It then makes sense to define where has the same norm as and therefore interpret as a rotation of three-dimensional space. In fact, it turns out that the special restriction on implies that the rotation is orientation preserving. This allows the definition of a map given by
where This map is the concrete realization of the double cover of by and therefore shows that The components of can be recovered using the tracing process above:

Cross-product

The cross-product is given by the matrix commutator
In fact, the existence of a norm follows from the fact that is a Lie algebra.
This cross-product can be used to prove the orientation-preserving property of the map above.

Eigenvalues and eigenvectors

The eigenvalues of are 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 since this can be factorised into A standard result in linear algebra means this implies is diagonalizable with possible eigenvalues The tracelessness of means it has exactly one of each eigenvalue.
Its normalized eigenvectors are
These expressions become singular for They can be rescued by letting and taking the limit which yields the correct eigenvectors and of
Alternatively, one may use spherical coordinates to obtain the eigenvectors and

Pauli 4-vector

The Pauli 4-vector, used in spinor theory, is written with components
This defines a map from to the vector space of Hermitian matrices,
which also encodes the Minkowski metric in its determinant:
This 4-vector also has a [|completeness relation]. It is convenient to define a second Pauli 4-vector
and allow raising and lowering using the Minkowski metric tensor. The relation can then be written
Similarly to the Pauli 3-vector case, we can find a matrix group that acts as isometries on in this case the matrix group is and this shows Similarly to above, this can be explicitly realized for with components
In fact, the determinant property follows abstractly from trace properties of the For matrices, the following identity holds:
That is, the 'cross-terms' can be written as traces. When are chosen to be different the cross-terms vanish. It then follows, now showing summation explicitly,
Since the matrices are this is equal to

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
so that,
Contracting each side of the equation with components of two -vectors and yields
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 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: Or equivalently, the Pauli vector satisfies:

Some trace relations

The following traces can be derived using the commutation and anticommutation relations.
If the matrix is also considered, these relationships become
where Greek indices and assume values from and the notation is used to denote the sum over the cyclic permutation of the included indices.

Exponential of a Pauli vector

For
one has, for even powers,
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,
Matrix exponentiating, and using the Taylor series for sine and cosine,
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. In particular,
Note that
while the determinant of the exponential itself is just, which makes it the generic group element of SU.
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 function is provided by application of Sylvester's formula,

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,
the spherical law of cosines. Given, then,
Consequently, the composite rotation parameters in this group element simply amount to
(Of course, when is parallel to so are and

Adjoint action

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

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
As noted above, it is common to denote the 2 × 2 unit matrix by so The completeness relation can alternatively be expressed as
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 as above, and then imposing the positive-semidefinite and trace conditions.
For a pure state, in polar coordinates, the idempotent density matrix
acts on the state eigenvector with eigenvalue +1, hence it acts like a projection operator.