= Solution
The <Schrödinger–Jaynes–Hughston–Jozsa–Wootters theorem> classifies all finite <quantum state ensembles> of a <density matrix>. Write its nonzero spectral decomposition as $\rho=\sum_{a=1}^r\lambda_a|e_a\rangle\langle e_a|$, with $\lambda_a>0$ and orthonormal <eigenvectors>. An ensemble with subnormalized vectors $|w_j\rangle=\sqrt{p_j}|\phi_j\rangle$, $j=1,\ldots,m$, represents $\rho$ exactly when
$$
\boxed{|w_j\rangle=\sum_{a=1}^rU_{ja}\sqrt{\lambda_a}|e_a\rangle,\qquad U^\dagger U=I_r.}
$$
Here $U$ is an $m\times r$ column <linear isometry>. Zero-weight vectors may be included or discarded. Equivalently, any two ensembles represent the same <density matrix> if and only if, after padding the shorter list with zero vectors, their subnormalized vectors are related by a <unitary matrix> acting on the ensemble index. This is the <isometry parametrization of a density-matrix ensemble>.
For necessity, if $|v\rangle$ lies in the kernel of $\rho$, then
$$
0=\langle v|\rho|v\rangle=\sum_j|\langle v|w_j\rangle|^2.
$$
Every $w_j$ therefore lies in the support of $\rho$. Define $U_{ja}=\langle e_a|w_j\rangle/\sqrt{\lambda_a}$. The ensemble identity gives
$$
\sum_jU_{ja}^*U_{jb}
=\frac{\langle e_b|\rho|e_a\rangle}{\sqrt{\lambda_a\lambda_b}}
=\delta_{ab},
$$
so these columns are orthonormal and the displayed parametrization follows.
Conversely, start with any such isometry and define $w_j$ by the boxed equation. Then
$$
\sum_j|w_j\rangle\langle w_j|
=\sum_{a,b}\sqrt{\lambda_a\lambda_b}\left(\sum_jU_{ja}U_{jb}^*\right)|e_a\rangle\langle e_b|
=\rho.
$$
The weights are $p_j=\|w_j\|^2=\sum_a\lambda_a|U_{ja}|^2\geq0$ and sum to $\operatorname{Tr}\rho=1$. Normalize every nonzero $w_j$ to obtain its <pure state>. This proves both directions and includes degenerate <eigenvalues> without any preferred choice of basis in their eigenspaces.
To prove the unitary-mixing version, pad two ensembles to a common length $L$. Their matrices $U$ and $V$ have $r$ orthonormal columns in $\mathbb C^L$. Extend each column family to an <orthonormal basis>, obtaining square unitaries $\widetilde U$ and $\widetilde V$. Then $W=\widetilde V\widetilde U^\dagger$ is unitary and $V=WU$, so the second ensemble vectors satisfy $v_k=\sum_jW_{kj}w_j$. Conversely this unitary mixing preserves $\sum_j|w_j\rangle\langle w_j|$, because $\sum_kW_{ki}W_{kj}^*=\delta_{ij}$. Independent phase choices for ensemble kets are absorbed in the row phases of these matrices.
The classification also has a measurement interpretation. Purify $\rho$ as $|\Psi\rangle=\sum_a\sqrt{\lambda_a}|e_a\rangle|a\rangle$. On the ancillary system define $|\eta_j\rangle=\sum_aU_{ja}^*|a\rangle$. The effects $E_j=|\eta_j\rangle\langle\eta_j|$ sum to $I_r$, hence form a <POVM>. The conditional unnormalized system ket is $(I\otimes\langle\eta_j|)|\Psi\rangle=w_j$, with <probability> $p_j$. Thus every ensemble can be realized by measuring a purifying system; with a sufficiently enlarged ancilla the measurement can be projective. The <density matrix> specifies the common statistics without selecting one particular ensemble decomposition.
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