Generalizations of 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. In this article we give a few classes of such matrices.

1 Generalized Gell-Mann matrices
- 1.1 Construction
- 1.2 Properties
2 A non-Hermitian generalization of Pauli matrices
- 2.1 Construction
3 A unitary generalization of the Pauli matrices
- 3.1 Construction
- 3.2 Properties
4 See also
5 Notes

Generalized Gell-Mann matrices

Construction

Let $\; E_{jk}$ be the matrix with 1 in the $jk$ -th entry and 0 elsewhere. Consider the space of $d \times d$ complex matrices, $\mathbb{C}^{d \times d}$ , for a fixed d. Define the following matrices

For $k < j$ , $f_{k,j} ^d = E_{kj} %2B E_{jk}$ .

For $k > j$ , $f_{k,j} ^d = - i ( E_{jk} - E_{kj} )$ .

Let $h_1 ^d = I_d$ , the identity matrix.

For $1 < k < d$ , $h_k ^d = h ^{d-1} _k \oplus 0$ .

For $k = d$ , $h_d ^d = \sqrt{\frac{2}{d(d-1)}} (h_1 ^{d-1} \oplus (1-d))$ .

The collection of matrices defined above are called the generalized Gell-Mann matrices, in dimension d. The symbol $\oplus$ above means matrix direct sum.

Properties

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 $\mathbb{C}^{d \times d}$ . By the dimension count, we see that they span the vector space of $d \times d$ complex matrices.

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

A non-Hermitian generalization of Pauli matrices

The Pauli matrices $\sigma _1$ and $\sigma _3$ satisfy the following:

$\sigma _1 ^2 = \sigma _3 ^2 = I, \; \sigma _1 \sigma _3 = - \sigma _3 \sigma _1 = e^{\pi i} \sigma _3 \sigma_1.$

The so-called Walsh-Hadamard matrix is

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

Like the Pauli matrices, W is both Hermitian and unitary. $\sigma _1, \; \sigma _3$ and W satisfy the relation

$\; \sigma _1 = W \sigma _3 W^* .$

The goal now is to extend above to higher dimensions.

Construction

Fix the dimension d as before. Let $\; \sigma = e^{2 \pi i / d}$ , a root of unity. Since $\sigma ^d = 1$ and $\sigma \neq 1$ , we have

${\hat \sigma} = 1 %2B \sigma %2B \cdots %2B \sigma ^{d-1} = 0 .$

Now define, with J. J. Sylvester (1882) the shift matrix^[1]

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

and the clock matrix,

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

These matrices generalize $\sigma _1$ and $\sigma _3$ respectively. Note that the 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^[2] ^[3] 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.

The following relations echo those of the Pauli matrices:

$\Sigma _ 1 ^d = \Sigma _ 3 ^d = I$

and

$\; \Sigma_3 \Sigma _1 = \sigma \Sigma_1 \Sigma _3 = e^{2 \pi i / d} \Sigma_1 \Sigma _3 .$

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

$W = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & \sigma ^{2 -1} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & \sigma ^{d -1} \end{bmatrix}.$

Define the following matrix, still denoted by W as a slight abuse of notation:

$W = \frac{1}{\sqrt{d}} \begin{bmatrix} 1 & 1 & 1 & \cdots & 1\\ 1 & \sigma^{d-1} & \sigma^{2(d-1)} & \cdots & \sigma^{(d-1)^2}\\ 1 & \sigma^{d-2} & \sigma^{2(d-2)} & \cdots & \sigma^{(d-1)(d-2)}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \sigma & \sigma^2 & \cdots & \sigma^{d-1} \end{bmatrix}.$

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

$\; \Sigma_1 = W \Sigma_3 W^* ~,$

which is the desired result.

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

A unitary generalization of the Pauli matrices

As noted above, the Pauli matrices are both Hermitian and unitary. The unitarity has not been extended by generalizations given so far. We now give a generalization which does so. (Although Hermiticity will no longer hold, in general.)

Construction

Again fix the dimension d. Let $\mathbb{Z}_d$ be the abelian ring of integers modulo d. All indices in the subsequent discussion will be considered elements of this group, that is, all operations are to be understood modulo d. The set

$\{v_k, k \in \mathbb{Z}_d \}$

denotes the standard orthonormal basis for the d-dimensional Hilbert space. Put $\; \xi = e^{2 \pi i / d}$ . The generalization we are interested in is defined by

$S_{j,k} = \sum _{m = 0} ^ {d-1} \xi ^{jm} v_m v_{m%2Bk} ^*.$

Clearly the family specified by above consists of unitary matrices.

To see that they indeed generalize the Pauli matrices, in some sense, we compute for $\; d=2$ , where $\; \xi = -1$ :

$S_{0,0} = v_0 v_0 ^* %2B v_1 v_1 ^* = I ,$

$S_{0,1} = v_0 v_1 ^* %2B v_1 v_0 ^* = \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} = \sigma _1 ,$

$S_{1,0} = v_0 v_0 ^* - v_1 v_1 ^* = \begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix} = \sigma _3 ,$

and

$S_{1,1} = v_0 v_1 ^* - v_1 v_0 ^* = \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} = i \sigma _2.$

Properties

The set $\; \{ S_{j,k} \}$ are called generalized spin matrices.^[4] Some properties of $\; \{ S_{j,k} \}$ are:

$\; \{ S_{j,k} \}$ form an orthogonal set in $\mathbb{C}^{d \times d}$ in the Hilbert-Schmidt sense. Therefore by the dimension count, they span the set of $d \times d$ matrices. (Recall this is also true for $\{I, \sigma _1, \sigma _2, \sigma _3\}$ in dimension 2.)

$\; S_{j,k} S_{s,t} = \xi ^{ks} S_{j%2Bs, k%2Bt}.$

$S_{j,k} = S_{1, 0} ^j S_{0,1} ^k$ .

Notes

^ Sylvester, J. J., (1882), Johns Hopkins University Circulars I: 241-242; ibid II (1883) 46; ibid III (1884) 7-9. Summarized in The Collected Mathematics Papers of James Joseph Sylvester (Cambridge University Press, 1909) v III
^ Weyl, H., "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1–46, doi:10.1007/BF02055756.
^ Weyl, H., The Theory of Groups and Quantum Mechanics (Dover, New York, 1931)
^ Pittenger, A. (2004). "Mutually unbiased bases, generalized spin matrices and separability". Linear Algebra and its Applications 390: 255–278. doi:10.1016/j.laa.2004.04.025. edit

Generalizations of Pauli matrices

Contents

Generalized Gell-Mann matrices

Construction

Properties

A non-Hermitian generalization of Pauli matrices

Construction

A unitary generalization of the Pauli matrices

Construction

Properties

See also

Notes