Non-standard Generalizations of the Pauli Matrices that retain closedness, identity mapping & tracelessness?

Question

The Pauli matrices $\{\sigma_{0}=I,\sigma_{1}=X,\sigma_{2}=Y,\sigma_{3}=Z\}$ exhibit the following properties;

1. $$ \sigma_{i}\sigma_{j}=\pm \mathrm{i} \sigma_{k} $$
2. $$ \sigma_{i}^{2}=\sigma_{0}=I $$
3. $$ Tr(\sigma_{i})=0, i > 0$$.

This makes them a useful matrix basis for example in quantum mechanics, where we employ the Hilbert-Schmidt scalar product $$ \langle A,B\rangle = Tr(A^{\dagger}B) $$. The rationale here being that if $A,B$ are expressed in a Pauli basis the resulting sum contains only d non trivial terms due to (1)-(3) instead of up to d^2 in the case of a general matrix basis.

However, the Pauli matrices are only defined for dimension 2.

By 1) and 2) each summand will be equalt either to the identity or a single Pauli matrix. The trace of all single Pauli matrices is 0 by 3), thus the overall sum simlpifies tremendously. We would like to keep this nice behaviour in higher dimensions.

The most-well known higher dimensional generalizations are either (generalized) Gellmann matrices or the Heisenberg-Weyl matrices. However, they are not generalizing (1)-(3) but rather other importan properties of the Paulis (ie either Unitarity or Hermiticity). In fact, the (generalized) Gellmann matrices violate 1). For an example check 2 the product of two differing (anti-)symetric Gellmann matrices. It may either be the zero matrix or contain only a single element. Thus they the Gellmann matrices are not closed. The Heisenberg-Weyl matrices too violate 1) as well check for example the product of a shift and a clock operator. So both are no valid choice.

I am interested if there are any known matrix algebras that generalize these properties in the following (or some related) sense to higher dimensions:

The algebra should contain the identity $1$ and retain 1),2) and 3) in the following sense:

1) For closedness reasons, we would like to keep at least that any product of different operator is mapped to a single element of the Algebra. This can be generalized in several ways to higher dimensions. For example this mapping could or could not be bijective/injective/surjective. Any possible choice would be welcome. 2) should be kept strictly and not generalized to $\sigma_{i}^{d}=\sigma_{0}$. Otherwise there would be no big simplification. 3) should be kept or at least restrict the possible values of the trace to a (or only few) constant(s).

Answer 1

For a certain interpretation of your axioms there exist no interesting examples—they're all "tensor products" of Pauli matrices.

Let $\tau=\mathrm{i}\sigma$. Then

1. $\tau_i\tau_j=\pm\tau_k$ and
2. $\tau_i^2=\pm 1$.

Let $C_2=\{1,-1\}$. Let $A$ be an index set for the symbols $\tau_{a}$, $a\in A$, and assume that $\tau_a\notin C_2$ for $a\neq 0$. Then (1) and (2) imply that

• the set $G\stackrel{\text{def}}{=}C\sigma_A$ is a group,
• $C_2$ is the commutator subgroup of $G$,
• $A=G/C_2$ has a natural group structure, and
• $A$ has exponent $2$.

But every group of exponent $2$ is abelian; consequently, if $A$ is finitely generated, then the structure theorem for finitely generated abelian groups implies $A\cong (\mathbb{Z}/(2))^{\oplus r}$ for some $r$. Therefore $\lvert G\rvert =\lvert C_2\rvert\lvert A \rvert=2^{r+1}$.

Let's assume that $C_2$ is the center of $G$—i.e., $\tau_a\tau_b=\tau_b\tau_a$ for all $b\in A$ iff $a=0$. Then $G$ is an extra special $2$-group: $r=2s$ for some $s>0$, and $G$ is a central product of the groups $$\begin{align}D_{8}&=\langle \sigma_z,\sigma_x| \sigma_z^2=1,\sigma_x^2=1,\sigma_z\sigma_x=-\sigma_x\sigma_z\rangle \supseteq C_2\\ Q_8&=\langle \tau_x,\tau_y |\tau_x^2=-1,\tau_y^2=-1,\tau_y\tau_x=-\tau_x\tau_y\rangle\supseteq C_2\text{.} \end{align}$$

Each of these groups has an "obvious" representation by Pauli matrices—taking the tensor product of all of these gives a representation of $G$ on $\mathbb{C}^{2^s}$. Then $$\mathrm{Tr}\,g\neq 0 \Leftrightarrow g\in C_2\text{.}$$