A Parameterization

Parameterize a traceless hermitian matrix \( A\) in an orthonormal basis of \( su(d)\). The (complex) traceless hermitian matrix \( A\) is written as

\[ A = \sum_{j=1}^{d^2-1} a_j A_j\ , \]

where the \( A_j\) are the normalized version of the generalized Gell-Mann matrices, i.e. \( A_j = \lambda_j/\sqrt2\) where \(\lambda_j\) are defined as in Refs. [1-3].

Whenever we talk about the A parameterization of a matrix which is not traceless, we imply the A parameterization of its traceless part, i.e. \( A - \mathrm{tr}(A)\mathbb{I}/d\).

1. Wolfram MathWorld: Generalized Gell-Mann Matrix;
2. Brüning et al., “Parametrizations of density matrices,” Journal of Modern Optics 59:1 1 (2012), doi:10.1080/09500340.2011.632097, arXiv:1103.4542;
3. Bertlmann & Krammer, “Bloch vectors for qudits,” Journal of Physics A 41:23 235303 (2008) doi:10.1088/1751-8113/41/23/235303, arXiv:0806.1174.