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
I
{\displaystyle \mathbb {I} }
(sometimes considered as the zeroth Pauli matrix
σ
0
{\displaystyle \sigma _{0}}
), the Pauli matrices form a basis of the vector space of
2
×
2
{\displaystyle 2\times 2}
Hermitian matrices over the real numbers, under addition. This means that any
2
×
2
{\displaystyle 2\times 2}
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:
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,
σ
k
{\displaystyle \sigma _{k}}
represents the observable corresponding to spin along the
k
{\displaystyle k}
th coordinate axis in three-dimensional Euclidean space
R
3
{\displaystyle \mathbb {R} ^{3}}
.
The Pauli matrices (after multiplication by
i
{\displaystyle i}
to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: The matrices
i
σ
1
{\displaystyle i\sigma _{1}}
,
i
σ
2
{\displaystyle i\sigma _{2}}
, and
i
σ
3
{\displaystyle i\sigma _{3}}
form a basis for the real Lie algebra
s
u
(
2
)
{\displaystyle {\mathfrak {su}}(2)}
, which exponentiates to the special unitary group SU(2). The algebra generated by the three Pauli matrices is isomorphic to the Clifford algebra of
R
3
{\displaystyle \ \mathbb {R} ^{3}}
and the (unital) associative algebra generated by
i
σ
1
{\displaystyle i\sigma _{1}}
,
i
σ
2
{\displaystyle i\sigma _{2}}
, and
i
σ
3
{\displaystyle i\sigma _{3}}
functions identically (is isomorphic) to that of quaternions (
H
{\displaystyle \mathbb {H} }
).
Contents
Algebraic properties
All three of the Pauli matrices can be compacted into a single expression:
Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products. Adding the commutator to the anticommutator gives
[
σ
j
,
σ
k
]
+
{
σ
j
,
σ
k
}
=
(
σ
j
σ
k
−
σ
k
σ
j
)
+
(
σ
j
σ
k
+
σ
Some trace relations
The following traces can be derived using the commutation and anticommutation relations.
which can be shown first for the p = 1 case using the anticommutation relations. For convenience, the case p = 0 is taken to be I by convention.
Completeness relation
An alternative notation that is commonly used for the Pauli matrices is to write the vector index k in the superscript, and the matrix indices as subscripts, so that the element in row α and column β of the k-th Pauli matrix is σ k αβ .
In this notation, the completeness relation for the Pauli matrices can be written
σ
→
α
β
⋅
σ
→
γ
δ
≡
∑
k
=
1
3
σ
α
β
k
σ
γ
δ
k
=
2
δ
α
δ
δ
β
γ
Relation with the permutation operator
Let Pjk be the transposition (also known as a permutation) between two spins σj and σk living in the tensor product space
This operator can also be written more explicitly as Dirac's spin exchange operator,
P
j
k
SU(2)
The group SU(2) is the Lie group of unitary 2 × 2 matrices with unit determinant; its Lie algebra is the set of all 2 × 2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra
s
u
2
{\displaystyle {\mathfrak {su}}_{2}}
is the three-dimensional real algebra spanned by the set {iσk}. In compact notation,
As a result, each iσj can be seen as an infinitesimal generator of SU(2). The elements of SU(2) are exponentials of linear combinations of these three generators, and multiply as indicated above in discussing the Pauli vector. Although this suffices to generate SU(2), it is not a proper representation of su(2), as the Pauli eigenvalues are scaled unconventionally. The conventional normalization is λ = 1/2 , so that
SO(3)
The Lie algebra
s
u
(
2
)
{\displaystyle \ {\mathfrak {su}}(2)\ }
is isomorphic to the Lie algebra
s
o
(
3
)
{\displaystyle {\mathfrak {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 the i σj are a realization (and, in fact, the lowest-dimensional realization) of infinitesimal rotations in three-dimensional space. However, even though
s
u
(
2
)
{\displaystyle \ {\mathfrak {su}}(2)\ }
and
s
o
(
3
)
{\displaystyle {\mathfrak {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 group homomorphism from SU(2) ↦ SO(3) , see relationship between SO(3) and SU(2).
Quaternions
The real linear span of {I, iσ1, i σ2, i σ3} is isomorphic to the real algebra of quaternions,
In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a spin 1⁄2 particle, in each of the three spatial directions. As an immediate consequence of the Cartan decomposition mentioned above,
i
σ
j
{\displaystyle i\sigma _{j}}
are the generators of a projective representation (spin representation) of the rotation group SO(3) acting on non-relativistic particles with spin 1⁄2. The states of the particles are represented as two-component spinors. In the same way, the Pauli matrices are related to the isospin operator.
An interesting property of spin 1⁄2 particles is that they must be rotated by an angle of
4
π
{\displaystyle 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
{\displaystyle S^{2}}
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
J
=
ℏ
Relativistic quantum mechanics
In relativistic quantum mechanics, the spinors in four dimensions are 4 × 1 (or 1 × 4) matrices. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices. They are defined in terms of 2 × 2 Pauli matrices as
matrices have the same algebraic properties as the σk matrices.
However, relativistic angular momentum is not a three-vector, but a second order four-tensor. Hence
Σ
k
{\displaystyle \ {\mathsf {\Sigma }}_{k}\ }
needs to be replaced by Σμν, the generator of Lorentz transformations on spinors. By the antisymmetry of angular momentum, the Σμν are also antisymmetric. Hence there are only six independent matrices.
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 ".
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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
×
2
{\displaystyle \ 2\times 2\ }
Hermitian traceless matrix-valued dual vector, that is, an element of
A standard result in linear algebra (a linear map that satisfies a polynomial equation written in distinct linear factors is diagonalizable) means this implies
Contracting each side of the equation with components of two 3-vectors ap and bq (which commute with the Pauli matrices, i.e., apσq = σqap) for each matrix σq and vector component ap (and likewise with bq) yields
while the determinant of the exponential itself is just 1, which makes it the generic group element of SU(2).
A more abstract version of formula (2) for a general 2 × 2 matrix can be found in the article on matrix exponentials. A general version of (2) for an analytic (at a and −a) function is provided by application of Sylvester's formula,
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
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 {σ0, σ1, σ2, σ3} as above, and then imposing the positive-semidefinite and trace 1 conditions.
Its eigenvalues are therefore 1 or −1. It may thus be utilized as an interaction term in a Hamiltonian, splitting the energy eigenvalues of its symmetric versus antisymmetric eigenstates.
forms a group isomorphic to SU(2), U gives yet another way of describing SU(2). The two-to-one homomorphism from SU(2) to SO(3) may be given in terms of the Pauli matrices in this formulation.
for rotations about the
x
{\displaystyle x}
-axis through an angle
θ
{\displaystyle \theta }
may be written in terms of Pauli matrices and the unit matrix as
, the fundamental representation of SU(2). By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations. That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using this spin operator and ladder operators. They can be found in Rotation group SO(3) § A note on Lie algebras. The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple.
Also useful in the quantum mechanics of multiparticle systems, the general Pauli group
