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.

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

Define the following matrices,

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 $ℂ d \times d$ . By dimension count, one sees that they span the vector space of $d \times d$ complex matrices, ${\mathfrak {gl}}$ ( $d$ ,ℂ). They then provide a Lie-algebra-generator basis acting on the fundamental representation of ${\mathfrak {su}}$ ( $d$ ).

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