= Isometry parametrization of a density-matrix ensemble
{title2=$\sqrt{p_j}|\phi_j\rangle=\sum_{a=1}^r U_{ja}\sqrt{\lambda_a}|e_a\rangle,\quad U^\dagger U=I_r$}
For the nonzero spectral decomposition $\rho=\sum_{a=1}^r\lambda_a|e_a\rangle\langle e_a|$, every pure-state <quantum state ensemble> of $\rho$ is specified by the displayed column <linear isometry>. To prove this, write $w_j=\sqrt{p_j}\phi_j$. Positivity forces each $w_j$ into the support of $\rho$, and the coefficients $U_{ja}=\langle e_a|w_j\rangle/\sqrt{\lambda_a}$ have orthonormal columns because $\sum_j|w_j\rangle\langle w_j|=\rho$. Conversely this column identity reconstructs $\rho$; the squared vector norms are the <probabilities> and sum to one. Complete the columns to unitary bases after adding zero ensemble vectors to obtain the usual unitary-mixing formulation.
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