= Solution
<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. Then
$$
\boxed{|\psi\rangle\longrightarrow|\chi\rangle\text{ by deterministic LOCC}
\quad\Longleftrightarrow\quad\lambda(\psi)\prec\lambda(\chi).}
$$
The <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.<Conditions for a class of entanglement transformations by Michael A. Nielsen>[Necessary and sufficient majorization criterion for deterministic bipartite pure-state conversion]{ref}
For the supplied states, the weight vectors and cumulative sums are
$$
\begin{array}{c|rrrr}
\lambda(\psi_1)&0.4&0.4&0.1&0.1\\
\lambda(\psi_2)&0.5&0.25&0.25&0
\end{array}\qquad
\begin{array}{c|rrrr}
\text{input sums}&0.4&0.8&0.9&1\\
\text{output sums}&0.5&0.75&1&1
\end{array}
$$
At $k=2$, $0.8>0.75$, violating the required inequality. \b[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 are
$$
\begin{aligned}
x&=(0.24,0.24,0.16,0.16,0.06,0.06,0.04,0.04),\\
y&=(0.30,0.20,0.15,0.15,0.10,0.10,0,0).
\end{aligned}
$$
The actual nonnegative Schmidt amplitudes are $\sqrt{x_j}$ and $\sqrt{y_j}$; the displayed numbers are their squares. The cumulative sums are
$$
\begin{aligned}
X&=(0.24,0.48,0.64,0.80,0.86,0.92,0.96,1),\\
Y&=(0.30,0.50,0.65,0.80,0.90,1,1,1).
\end{aligned}
$$
Every component obeys $X_k\le Y_k$, so $x\prec y$. Hence
$$
\boxed{|\psi_1\rangle\otimes|\phi\rangle
\xrightarrow{\rm LOCC}|\psi_2\rangle\otimes|\phi\rangle.}
$$
The 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.<Entanglement-assisted local manipulation of pure quantum states by Daniel Jonathan and Martin B. Plenio>[Two-qubit catalyst example and unchanged product-state return]{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 $0.75$. \b[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 \b[$|\phi\rangle$, 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 $|\psi_2\rangle$ and return the original maximally entangled auxiliary state deterministically: its composition with the preceding steps would contradict the failed four-weight inequality above.
\Image[/past-exam-of-the-quantum-information-course-of-the-university-of-cambridge/2014/paper-1-catalytic-majorization.png]
{title=The catalyst repairs every failed majorization inequality, while a maximally entangled pair does not}
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