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Suppose $|W⟩=(U_A\otimes U_B\otimes U_C)|GHZ⟩$. Measure C in the orthonormal basis $\{U_C|0⟩,U_C|1⟩\}$. Its two AB conditional vectors would be proportional torespectively. Both are product states, because measuring the original GHZ state's C factor in the computational basis selects $|00⟩$ or $|11⟩$. This contradicts part (i), which applies to every basic measurement basis of the W state's C factor. Therefore no local unitary transformation maps GHZ to W.

An additional invariant makes the distinction visible without measurement records. Across the partition AB versus C, the GHZ reduced density matrix on C has eigenvalues $1/2,1/2$, whereas the W state's are $2/3,1/3$. By local-unitary invariance of reduced-state spectra, these spectra cannot be changed by local unitaries. This supplies the same impossibility and emphasizes that the proposed transformation is a product of unitaries, not arbitrary postselected local processing.

Introduce the orthonormal local vectors $|+⟩=(|1⟩+|2⟩)/\sqrt{2}$ and $|-⟩=(|1⟩-|2⟩)/\sqrt{2}$. Reexpressing the second state's factors in this basis gives its Schmidt decompositionThe squared Schmidt coefficients are therefore $1/3,2/3$, whereas those of the first state are $1/2,1/2$.

To see directly why these distinguish the states, a local unitary operation $U_A\otimes U_B$ changes the reduced density matrix toThis is local-unitary invariance of reduced-state spectra: the unitary on the traced subsystem cancels, and conjugation on $A$ leaves its eigenvalues unchanged. Here the reductions areTheir spectra differ. Hence no pair of local unitary operations carries the first state to the second. More generally, normalized bipartite pure states are locally unitarily equivalent exactly when their squared Schmidt coefficients agree, including multiplicities; sufficiency follows by mapping one pair of Schmidt bases to the other.

Nielsen's pure-state conversion theorem concerns exact deterministic transformations of bipartite pure states by local operations and classical communication. Let $\lambda (\psi )$ be the decreasing vector of squared Schmidt coefficients, equivalently the eigenvalues of either party's reduced density matrix, padded with zeros when necessary. ThenThe majorization relation means ${\sum }_{j=1}^k{\lambda }_j(\psi )\le {\sum }_{j=1}^k{\lambda }_j(\chi )$ for every $k$, with equality for the total sum. The direction allows a maximally entangled state to become a product state, not the reverse. Local-unitary invariance of reduced-state spectra alone would require equality of all the weights; measurements and communicated outcomes permit this larger class of conversions. A comparison of entanglement entropies alone is not a sufficient test in higher dimension.^[ref]

For the supplied states, the weight vectors and cumulative sums areAt $k=2$, $0.8>0.75$, violating the required inequality. The desired conversion is impossible with certainty using LOCC alone. The reverse conversion also fails, already at $k=1$, so these states are incomparable under this deterministic ordering.

Taking a tensor product with the catalyst multiplies every original weight by each catalyst weight. Regrouping the two local systems on Alice's side and the two on Bob's side leaves a Schmidt decomposition across the Alice–Bob partition. After sorting, the resulting weight vectors areThe actual nonnegative Schmidt amplitudes are $\sqrt{x_j}$ and $\sqrt{y_j}$; the displayed numbers are their squares. The cumulative sums areEvery component obeys $X_k\le Y_k$, so $x\prec y$. HenceThe final catalyst is exactly its initial state and is uncorrelated with the desired output. It can therefore be returned and reused. This is entanglement catalysis; it is stronger than merely retaining the same catalyst marginal while leaving correlations with the output.^[ref]

For the maximally entangled two-level auxiliary state, the sorted input weights are $(0.2,0.2,0.2,0.2,0.05,0.05,0.05,0.05)$, while the output weights are $(0.25,0.25,0.125,0.125,0.125,0.125,0,0)$. Their first four sums are $0.8$ and $0.75$. This maximally entangled auxiliary state cannot catalyze the conversion while being returned unchanged.

Nevertheless, $(0.5,0.5)\prec (0.6,0.4)$, so it can first be converted deterministically into the nonmaximal catalyst. The desired conversion can then proceed, after which Alice and Bob can return $|\varphi ⟩$, rather than the borrowed maximally entangled state. Turning that returned pair alone back into a maximally entangled pair would increase its entanglement entropy and is forbidden by LOCC. No joint restoration procedure can both keep the exact desired $|{\psi }_2⟩$ and return the original maximally entangled auxiliary state deterministically: its composition with the preceding steps would contradict the failed four-weight inequality above.