Universal resource for measurement based quantum computation

Question

Consider universal resources for measurement based quantum computation, as defined here:

We are now ready to formulate the following definition. A family $\Psi$ is called a universal resource for MQC if for each state $|\phi\rangle$ on $n$ qubits there exists a state $|\psi\rangle \in \Psi$ on m qubits, with $m \geq n$, such that the transformation $$|\psi\rangle 􏰆\rightarrow |\phi\rangle|+\rangle ^{m−n}$$ is possible deterministically (with probability 1) by LOCC.

However, note that $|\phi\rangle|+\rangle ^{m−n}$ is a product state. Its Schmidt coefficient is $1$. Thus, no matter what state $|\psi\rangle$ is, its Schmidt coefficients will always be majorized by $1$. Hence, by Nielsen's theorem, the transformation

$$|\psi\rangle 􏰆\rightarrow |\phi\rangle|+\rangle ^{m−n}$$

will always be possible by LOCC (it may be an inefficient LOCC protocol, but the authors explicitly remark they do not care about efficiency and just care about universality.)

Doesn't it make this definition trivial?

Answer 1

That only works if you consider the bipartition of the bit that's going to be $|\phi\rangle$, versus the other $m-n$ qubits. Thus, to make that work, you have to be able to do untaries over the entirety of those two systems, which are almost certain to be entangling unitaries, and therefore often cannot be implemented by LOCC (except by consuming some additional entanglement resource).

So no, it's not trivial.