# What's up with PauliMatrix with a string argument?

Looking at the definition of PauliMatrix (not the documentation), I realized that it is defined for arguments 0, 1, 2, 3, 4, "A", "a", "B", "b", "C", "c". The first five are reasonable: there are three Pauli matrices (with indices 1, 2, 3) and the identity, which depending on the convention can be denoted with index 0 or 4. The next six are weird: as shown by the following code, "A" and "a" give the identity, while "B", "b", "C", "c" give the second Pauli matrix.

Association@@Table[x -> MatrixForm[PauliMatrix[x]],
{x, {0, 1, 2, 3, 4, "A", "B", "C", "a", "b", "c"}}] // PositionIndex


Is there any rationale for these definitions?

• It seems that there might be another naming convention here; see e.g. this article. May 19, 2016 at 16:54

Thanks to MarcoB's pointer, I realized the following properties of conjugating one of the numbered Pauli matrices by $A$, $B$, or $C$.

Table[ConjugateTranspose[PauliMatrix[i]] ==
PauliMatrix["A"].PauliMatrix[i].MatrixPower[PauliMatrix["A"], -1],
{i, 3}]
Table[-Transpose[PauliMatrix[i]] ==
PauliMatrix["B"].PauliMatrix[i].MatrixPower[PauliMatrix["B"], -1],
{i, 3}]
Table[-Conjugate[PauliMatrix[i]] ==
PauliMatrix["C"].PauliMatrix[i].MatrixPower[PauliMatrix["C"], -1],
{i, 3}]


To understand the use of $A$, $B$, $C$, one should remember that Pauli matrices are a representation of the $\text{su}(2)$ Lie algebra $[\sigma_i,\sigma_j]\colon=\sigma_i\sigma_j-\sigma_j\sigma_i=2i\epsilon_{ijk}\sigma_k$ with $1\leq i,j,k\leq 3$, implicit summation over $k$, and where $\epsilon_{ijk}$ is the completely antisymmetric symbol. This abstract algebra has a unique representation of dimension 2 (up to isomorphism), namely Pauli matrices.

The commutation relations are invariant under Hermitian conjugation (conjugate transpose) so the matrices $\sigma_i^\dagger$ form another representation of the $\text{su}(2)$ lie algebra, which must be isomorphic: this implies the existence of $A$ such that $\sigma_i^\dagger=A\sigma_i A^{-1}$ for all $i$. For the concrete case of Pauli matrices, the matrices are Hermitian, so $A=\text{id}$ works.

The commutation relations are also invariant under "minus-transposition" or under "minus-conjugation" so the matrices $-\sigma_i^T$ and $-\sigma_i^*$ also form a representation, which proves the existence of $B$ and $C$, equal since Pauli matrices are Hermitian and $-\sigma_i^T$ coincides with $-\sigma_i^*$.