The Exercise

The operators on the state space of a $d$-level system form a vector space of their own. In a fixed orthonormal basis $|1\rangle, \dots, |d\rangle$ of the system they are the complex $d \times d$ matrices; we write $M_d(\mathbb{C})$ for this space, which has dimension $d^2$. The Hilbert-Schmidt inner product of two $d \times d$ matrices $A$ and $B$ is the trace of $A^\dagger B$: $$(A, B) = \operatorname{tr}\bigl(A^\dagger B\bigr).$$(1)^✦

The exercise asks for an orthonormal basis of $M_d(\mathbb{C})$ for this inner product whose elements are all Hermitian, each of them an observable of the system.

The Gell-mann Basis

The library answers with an explicit family of $d^2$ matrices, one for each ordered pair of indices, built from the matrix units. The matrix unit $E_{jk} = |j\rangle\langle k|$ is the $d \times d$ matrix with a $1$ in row $j$ and column $k$ and zeros elsewhere.

The generalised Gell-Mann matrix $G_{jk}$ of an ordered pair $(j, k)$ of indices in $\{1, \dots, d\}$ is the complex $d \times d$ matrix $$G_{jk} = \begin{cases} \dfrac{1}{\sqrt 2}\,\bigl(E_{jk} + E_{kj}\bigr), & j < k,\\[8pt] \dfrac{i}{\sqrt 2}\,\bigl(E_{kj} - E_{jk}\bigr), & j > k,\\[8pt] E_{jj}, & j = k. \end{cases}$$(2)^✦

The Gell-Mann basis $\mathcal{G}$ of $M_d(\mathbb{C})$ is the basis whose elements are the $d^2$ matrices $G_{jk}$: every complex $d \times d$ matrix $A$ has a unique expansion $$A = \sum_{j,k=1}^{d} c_{jk}\, G_{jk}, \qquad c_{jk} \in \mathbb{C}.$$(3)^✦

The Claim

What Is Checked

The claim is stated over the Gell-Mann basis $\mathcal{G}$ of (3), whose elements are the matrices (2), and over the Hilbert-Schmidt inner product (1), for a system of any finite number $d$ of levels.

About these constructions the library proves that the $d^2$ generalised Gell-Mann matrices $G_{jk}$ are linearly independent over $\mathbb{C}$.^✦