# 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. Here, a few classes of such matrices are summarized.

## 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,

- For
*k*<*j*,*f*_{k,j}^{d}=*E*+_{kj}*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*_{k}^{d−1}⊕ 0 .

The collection of matrices defined above without the identity matrix are called the *generalized Gell-Mann matrices*, in dimension Template:Mvar.^{[1]}
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, (Template:Mvar,ℂ). They then provide a Lie-algebra-generator basis acting on the fundamental representation of (Template:Mvar ).

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

## A non-Hermitian generalization of Pauli matrices

The Pauli matrices and satisfy the following:

The so-called Walsh–Hadamard conjugation matrix is

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

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

### Construction: The clock and shift matrices

Fix the dimension Template:Mvar 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:

Integer indices may then be cyclically identified mod Template:Mvar.

Now define, with Sylvester, the
**shift matrix**^{[2]}

and the **clock matrix**,

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**^{[3]}^{[4]}^{[5]} as formulated by Hermann Weyl, and find routine applications in numerous areas of mathematical physics.^{[6]} 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 Heisenberg group on a *d*-dimensional Hilbert space.

The following relations echo those of the Pauli matrices:

and the braiding relation,

the Weyl formulation of the CCR, or

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

Define, again with Sylvester, the following analog matrix,^{[7]} still denoted by *W* in a slight abuse of notation,

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

which is the desired analog result. Thus, Template:Mvar , a Vandermonde matrix, arrays the eigenvectors of Σ_{1}, which has the same eigenvalues as Σ_{3}.

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

The family of *d*^{2} unitary (but non-Hermitian) independent matrices

provides Sylvester's well-known basis for (*d*,ℂ), known as "nonions" (3,ℂ), "sedenions" (4,ℂ), etc...^{[8]}

This basis can be systematically connected to the above Hermitian basis.^{[9]} (For instance, the powers of Σ_{3}, the Cartan subalgebra,
map to linear combinations of the *h*_{k}^{d}s.) It can further be used to identify (*d*,ℂ) , as *d* → ∞, with the algebra of Poisson brackets.

## See also

- Hermitian matrix
- Bloch sphere
- Discrete Fourier transform
- Generalized Clifford algebra
- Weyl–Brauer matrices
- Circulant matrix
- Shift operator

## Notes

- ↑ Template:Cite doi, {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ 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**. online and further. - ↑ Weyl, H., "Quantenmechanik und Gruppentheorie",
*Zeitschrift für Physik*,**46**(1927) pp. 1–46, Template:Hide in printTemplate:Only in print. - ↑ Weyl, H.,
*The Theory of Groups and Quantum Mechanics*(Dover, New York, 1931) - ↑ Template:Cite doi
- ↑ For a serviceable review,
see Vourdas A. (2004), "Quantum systems with finite Hilbert space",
*Rep. Prog. Phys.***67**267. doi: 10.1088/0034-4885/67/3/R03. - ↑ J.J. Sylvester, J. J. (1867) .
*Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers.*Philosophical Magazine, 34:461–475. online - ↑ Template:Cite doi
- ↑ Template:Cite doi