 Solution

The Schrödinger–Jaynes–Hughston–Jozsa–Wootters theorem classifies all finite quantum state ensembles of a density matrix. Write its nonzero spectral decomposition as $\rho ={\sum }_{a=1}^r{\lambda }_a|e_a⟩⟨e_a|$, with ${\lambda }_a>0$ and orthonormal eigenvectors. An ensemble with subnormalized vectors $|w_j⟩=\sqrt{p_j}|{\varphi }_j⟩$, $j=1,\dots ,m$, represents $\rho$ exactly whenHere $U$ 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 $|v⟩$ lies in the kernel of $\rho$, thenEvery $w_j$ therefore lies in the support of $\rho$. Define $U_{ja}=⟨e_a|w_j⟩/\sqrt{{\lambda }_a}$. The ensemble identity givesso these columns are orthonormal and the displayed parametrization follows.

Conversely, start with any such isometry and define $w_j$ by the boxed equation. ThenThe weights are $p_j=\Vert w_j{\Vert }^2={\sum }_a{\lambda }_a|U_{ja}{|}^2\ge 0$ and sum to $\mathrm{Tr}\rho =1$. 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 $\tilde{U}$ and $\tilde{V}$. Then $W=\tilde{V}{\tilde{U}}^{†}$ is unitary and $V=WU$, so the second ensemble vectors satisfy $v_k={\sum }_j{W}_{kj}{w}_j$. Conversely this unitary mixing preserves ${\sum }_j|w_j⟩⟨w_j|$, because ${\sum }_k{W}_{ki}{W}_{kj}^{\ast }={\delta }_{ij}$. 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 $|\Psi ⟩={\sum }_a\sqrt{{\lambda }_a}|e_a⟩|a⟩$. On the ancillary system define $|{\eta }_j⟩={\sum }_a{U}_{ja}^{\ast }|a⟩$. The effects $E_j=|{\eta }_j⟩⟨{\eta }_j|$ sum to $I_r$, hence form a POVM. The conditional unnormalized system ket is $(I\otimes ⟨{\eta }_j|)|\Psi ⟩=w_j$, 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.