Generalized Pauli matrices

In mathematics and physics, in particular quantum information, the term **generalized Pauli matrices** refers to families of matrices which generalize the (linear algebraic) properties of the Pauli matrices. Here, a few classes of such matrices are summarized.

## Pauli matrices

σ
1

=

σ
x

=

(

0

1

1

0

)

σ
2

=

σ
y

=

(

0

−
i

i

0

)

σ
3

=

σ
z

=

(

1

0

0

−
1

)

.

{\displaystyle
\begin{align}
 \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} \,.
\end{align}}

## Generalized Gell-Mann matrices (Hermitian)

### Construction

Let *E**jk* be the matrix with 1 in the *jk*-th entry and 0 elsewhere. Consider the space of *d*×*d* complex matrices, ℂ*d*×*d*, for a fixed *d*.

Define the following matrices,

 *f**k,j**d* =
*Ekj* + *Ejk*, for *k* < *j* .
−*i* (*Ejk* − *Ekj*), for *k* > *j* .
*h**k**d* =
*I**d*, the identity matrix, for *k* = 1,.
*h**k**d*−1 ⊕ 0, for 1 < *k* < *d* .

2

d
(
d
−
1
)

(

h
1

d
−
1

⊕
(
1
−
d
)

)

=

2

d
(
d
−
1
)

(

I

d
−
1

⊕
(
1
−
d
)

)

,

{\displaystyle \sqrt{\tfrac{2}{d(d-1)}} \left( h\_1^{d-1} \oplus (1 - d)\right) = \sqrt{\tfrac{2}{d(d-1)}} \left( I\_{d-1} \oplus (1 - d)\right),}

 for *k* = *d*.
The collection of matrices defined above without the identity matrix are called the *generalized Gell-Mann matrices*, in dimension d.
The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum.

The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices. One can also check that they are orthogonal in the Hilbert–Schmidt inner product on ℂ*d*×*d*. By dimension count, one sees that they span the vector space of *d* × *d* complex matrices,

g
l

{\displaystyle \mathfrak{gl}}

(d,ℂ). They then provide a Lie-algebra-generator basis acting on the fundamental representation of

s
u

{\displaystyle \mathfrak{su}}

(d ).

In dimensions d = 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.

λ
1

=

(

0

1

0

1

0

0

0

0

0

)

{\displaystyle \lambda\_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}}

λ
2

=

(

0

−
i

0

i

0

0

0

0

0

)

{\displaystyle \lambda\_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}}

λ
3

=

(

1

0

0

0

−
1

0

0

0

0

)

{\displaystyle \lambda\_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}}

λ
4

=

(

0

0

1

0

0

0

1

0

0

)

{\displaystyle \lambda\_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}}

λ
5

=

(

0

0

−
i

0

0

0

i

0

0

)

{\displaystyle \lambda\_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix}}

λ
6

=

(

0

0

0

0

0

1

0

1

0

)

{\displaystyle \lambda\_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}}

λ
7

=

(

0

0

0

0

0

−
i

0

i

0

)

{\displaystyle \lambda\_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}}

λ
8

=

1

3

(

1

0

0

0

1

0

0

0

−
2

)

{\displaystyle \lambda\_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}}

and

g
i

=

λ
i

/

2

{\displaystyle g\_i = \lambda\_i/2}

.

## A non-Hermitian generalization of Pauli matrices

The Pauli matrices

σ
1

{\displaystyle \sigma\_1}

 and

σ
3

{\displaystyle \sigma\_3}

 satisfy the following:

σ
1
2

=

σ
3
2

=
I
,

σ
1

σ
3

=
−

σ
3

σ
1

=

e

π
i

σ
3

σ
1

.

{\displaystyle
\sigma \_1 ^2 = \sigma \_3 ^2 = I, \; \sigma \_1 \sigma \_3 = - \sigma \_3 \sigma \_1 = e^{\pi i} \sigma \_3 \sigma\_1.
}

The so-called Walsh–Hadamard conjugation matrix is

# W

1

2

[

1

1

1

−
1

]

.

{\displaystyle
W = \frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\ 1 & -1
\end{bmatrix}.
}

Like the Pauli matrices, *W* is both Hermitian and unitary.

σ
1

,

σ
3

{\displaystyle \sigma \_1, \; \sigma \_3}

 and *W* satisfy the relation

σ
1

=
W

σ
3

W
∗

.

{\displaystyle \; \sigma \_1 = W \sigma \_3 W^\* .}

The goal now is to extend the above to higher dimensions, *d*, a problem solved by J. J. Sylvester (1882).

H
1

=

[

1

]

,

H
2

=

[

1

1

1

−
1

]

,

H
4

=

[

1

1

1

1

1

−
1

1

−
1

1

1

−
1

−
1

1

−
1

−
1

1

]

,

{\displaystyle \begin{align}
 H\_1 &= \begin{bmatrix} 1 \end{bmatrix}, \\
 H\_2 &= \begin{bmatrix}
 1 & 1 \\
 1 & -1
 \end{bmatrix}, \\
 H\_4 &= \begin{bmatrix}
 1 & 1 & 1 & 1\\
 1 & -1 & 1 & -1\\
 1 & 1 & -1 & -1\\
 1 & -1 & -1 & 1
 \end{bmatrix},
\end{align}}

and

H

2
k

=

[

H

2

k
−
1

H

2

k
−
1

H

2

k
−
1

−

H

2

k
−
1

]

=

H
2

⊗

H

2

k
−
1

,

{\displaystyle
 H\_{2^k} = \begin{bmatrix}
 H\_{2^{k-1}} & H\_{2^{k-1}}\\
 H\_{2^{k-1}} & -H\_{2^{k-1}}
 \end{bmatrix} = H\_2 \otimes H\_{2^{k-1}},
}

Sylvester's matrices have a number of special properties. They are symmetric and, when *k* ≥ 1, have trace zero. The elements in the first column and the first row are all positive. The elements in all the other rows and columns are evenly divided between positive and negative. Sylvester matrices are closely connected with Walsh functions.

### Construction: The clock and shift matrices

Fix the dimension d as before. Let *ω* = exp(2*πi*/*d*), a root of unity. Since *ω**d* = 1 and *ω* ≠ 1, the sum of all roots annuls:

1
ω
⋯

ω

d
−
1

=
0
.

{\displaystyle 1 + \omega + \cdots + \omega ^{d-1} = 0 .}

Integer indices may then be cyclically identified mod d.

Now define, with Sylvester, the
**shift matrix**

Σ
1

=

[

0

0

0

⋯

0

1

1

0

0

⋯

0

0

0

1

0

⋯

0

0

0

0

1

⋯

0

0

⋮

⋮

⋮

⋱

⋮

⋮

0

0

0

⋯

1

0

]

{\displaystyle
\Sigma \_1 =
\begin{bmatrix}
0 & 0 & 0 & \cdots &0 & 1\\
1 & 0 & 0 & \cdots & 0 & 0\\
0 & 1 & 0 & \cdots & 0 & 0\\
0 & 0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots &\vdots &\vdots \\
0 & 0 &0 & \cdots & 1 & 0\\
\end{bmatrix}
}

and the **clock matrix**,

Σ
3

=

[

1

0

0

⋯

0

0

ω

0

⋯

0

0

0

ω
2

⋯

0

⋮

⋮

⋮

⋱

⋮

0

0

0

⋯

ω

d
−
1

]

.

{\displaystyle
\Sigma \_3 =
\begin{bmatrix}
1 & 0 & 0 & \cdots & 0\\
0 & \omega & 0 & \cdots & 0\\
0 & 0 &\omega ^2 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & \omega ^{d-1}
\end{bmatrix}.
}

These matrices generalize *σ*1 and *σ*3, respectively.

Note that the unitarity and tracelessness of the two Pauli matrices is preserved, but not Hermiticity in dimensions higher than two. Since Pauli matrices describe Quaternions, Sylvester dubbed the higher-dimensional analogs "nonions", "sedenions", etc.

These two matrices are also the cornerstone of **quantum mechanical dynamics in finite-dimensional vector spaces** as formulated by Hermann Weyl, and find routine applications in numerous areas of mathematical physics. The clock matrix amounts to the **exponential of position** in a "clock" of *d* hours, and the shift matrix is just the translation operator in that cyclic vector space, so the **exponential of the momentum**. They are (finite-dimensional) representations of the corresponding elements of the Weyl-Heisenberg on a *d*-dimensional Hilbert space.

The following relations echo and generalize those of the Pauli matrices:

Σ
1
d

=

Σ
3
d

=
I

{\displaystyle \Sigma \_ 1 ^d = \Sigma \_ 3 ^d = I}

and the braiding relation,

Σ
3

Σ
1

=
ω

Σ
1

Σ
3

=

e

2
π
i

/

d

Σ
1

Σ
3

,

{\displaystyle \; \Sigma\_3 \Sigma \_1 = \omega \Sigma\_1 \Sigma \_3 = e^{2 \pi i / d} \Sigma\_1 \Sigma \_3 ,}

the Weyl formulation of the CCR, and can be rewritten as

Σ
3

Σ
1

Σ
3

d
−
1

Σ
1

d
−
1

=
ω

.

{\displaystyle \; \Sigma\_3 \Sigma \_1 \Sigma \_3^{d-1} \Sigma\_1 ^{d-1} = \omega ~.}

On the other hand, to generalize the Walsh–Hadamard matrix *W*, note

# W

1

2

[

1

1

1

ω

2
−
1

]

=

1

2

[

1

1

1

ω

d
−
1

]

.

{\displaystyle
W = \frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\ 1 & \omega ^{2 -1}
# \end{bmatrix}
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\ 1 & \omega ^{d -1}
\end{bmatrix}.
}

Define, again with Sylvester, the following analog matrix, still denoted by *W* in a slight abuse of notation,

# W

1

d

[

1

1

1

⋯

1

1

ω

d
−
1

ω

2
(
d
−
1
)

⋯

ω

(
d
−
1

)
2

1

ω

d
−
2

ω

2
(
d
−
2
)

⋯

ω

(
d
−
1
)
(
d
−
2
)

⋮

⋮

⋮

⋱

⋮

1

ω

ω
2

⋯

ω

d
−
1

]

.

{\displaystyle
W =
\frac{1}{\sqrt{d}}
\begin{bmatrix}
1 & 1 & 1 & \cdots & 1\\
1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)^2}\\
1 & \omega^{d-2} & \omega^{2(d-2)} & \cdots & \omega^{(d-1)(d-2)}\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 &\omega &\omega ^2 & \cdots & \omega^{d-1}

\end{bmatrix}~.
}

It is evident that *W* is no longer Hermitian, but is still unitary. Direct calculation yields

Σ
1

=
W

Σ
3

W
∗

,

{\displaystyle \; \Sigma\_1 = W \Sigma\_3 W^\* ~,}

which is the desired analog result. Thus, W , a Vandermonde matrix, arrays the eigenvectors of Σ1, which has the same eigenvalues as Σ3.

When *d* = 2*k*, *W* \* is precisely the matrix of the discrete Fourier transform, converting position coordinates to momentum coordinates and vice versa.

The complete family of *d*2 unitary (but non-Hermitian) independent matrices

(

Σ
1

)
k

(

Σ
3

)
j

=

∑

# m
0

d
−
1

m
k
⟩

ω

j
m

⟨
m

,

{\displaystyle
(\Sigma\_1)^k (\Sigma\_3)^j =\sum\_{m=0}^{d-1} |m+k\rangle \omega^{jm} \langle m| ,
}

provides Sylvester's well-known trace-orthogonal basis for

g
l

{\displaystyle \mathfrak{gl}}

(*d*,ℂ), known as "nonions"

g
l

{\displaystyle \mathfrak{gl}}

(3,ℂ), "sedenions"

g
l

{\displaystyle \mathfrak{gl}}

(4,ℂ), etc...

This basis can be systematically connected to the above Hermitian basis. (For instance, the powers of Σ3, the Cartan subalgebra,
map to linear combinations of the *h**k**d*s.) It can further be used to identify

g
l

{\displaystyle \mathfrak{gl}}

(*d*,ℂ) , as *d* → ∞, with the algebra of Poisson brackets.