We establish an exact equivalence in distribution between D-dimensional random mixed states induced by partial traces over K-dimensional environments and the Mai-Alquier distribution, a mixture of K independent Haar-random pure states weighted b
We establish an exact equivalence in distribution between D-dimensional random mixed states induced by partial traces over K-dimensional environments and the Mai-Alquier distribution, a mixture of K independent Haar-random pure states weighted by a symmetric Dirichlet distribution. This identification recasts a class of expectation values for induced ensembles into calculations involving Dirichlet moments and low-order Haar averages on a single-system state space. As applications, we recover exact purity moments up to fourth order, derive the mean Hilbert–Schmidt distance between independent induced ensembles with possibly different environment dimensions, and obtain exact average determinants. These results provide a unified and constructive perspective on induced random mixed states and on the evaluation of quantities such as purity moments, overlaps, and determinants.