We adopt the concept of the composite parameterization of the unitary group |$\mathcal {U}(d)$|$U(d)$ to the special unitary group |$\mathcal {SU}(d)$|$SU(d)$. Furthermore, we also consider the Haar measure in terms of the introduced parameters. We show that the well-defined structure of the parameterization leads to a concise formula for the normalized Haar measure on |$\mathcal {U}(d)$|$U(d)$ and |$\mathcal {SU}(d)$|$SU(d)$. With regard to possible applications of our results, we consider the computation of high-order integrals over unitary groups.

1.
These can be considered as generalizations of the Pauli matrix σy.
2.
The order of the product is
$\prod _{i=1}^{N}A_i=A_1 \cdot A_{2} \cdots A_N$
$∏i=1NAi=A1·A2⋯AN$
.
3.
Note that the indices m and n again run from 1 to d except that there is no λd, d.
4.
Intuitively: When changing from the orthogonal coordinates {uk} to the non-orthogonal coordinates {αl} the infinitesimal volume element transforms as
$\prod _{k=1}^{d^2}du_k=\left|\det \frac{\partial (u_1,\ldots ,u_{d^2})}{\partial (\alpha _1,\ldots ,\alpha _{d^2})} \right|\prod _{l=1}^{d^2} d\alpha _l$
$∏k=1d2duk=det∂(u1,...,ud2)∂(α1,...,αd2)∏l=1d2dαl$
according to the Jacobian determinant.
5.
The independence of
$\left|\det M_1 \right|$
$detM1$
on λm, n with min {m, n} ⩾ 2 can also be confirmed by exploiting again the left and right invariance of the Haar measure.
6.
A basis for the vector space of operators
$\mathbb {C}^d \times \mathbb {C}^d$
$Cd×Cd$
has d2 elements. The constraint
$\det U=1$
$detU=1$
on special unitary operators implies that the trace of any derivative ∂U/∂αl with respect to an arbitrary parameter is always zero. Traceless operators form a d2 − 1 dimensional subspace of
$\mathbb {C}^d \times \mathbb {C}^d$
$Cd×Cd$
. As we have d2 − 1 parameters {αl} it is required to express the derivatives ∂U/∂αl in a basis of this subspace to obtain d2 − 1 linearly independent column vectors.
7.
Note that these are the generalized diagonal Gell-Mann matrices (Ref. 38) in reversed order.
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