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In Cleve et al.'s paper "How to share a quantum secret", two encodings of quantum states are mentioned. The first one encodes one qutrit into three qutrits,

$\alpha|0\rangle + \beta|1\rangle + \gamma|2\rangle \mapsto \\ \alpha(|000\rangle + |111\rangle + |222\rangle) \\ + \beta(|012\rangle + |120\rangle + |201\rangle) \\ + \gamma(|021\rangle + |102\rangle + |210\rangle).$

The second encoding is in fact a quantum error-correcting code and encodes one qubit into four qubits,

$\alpha|0\rangle + \beta|1\rangle \mapsto \alpha(|0000\rangle + |1111\rangle) + \beta(|0011\rangle + |1100\rangle)$.

Such succinct ways of encoding pure quantum states can often be found in the literature of quantum error correction. I am wondering how the above encodings can be implemented unitarily and if there are any general results that these brief descriptions draw on that I am missing here.

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Let's say you're given a transformation $$ \sum_i\alpha_i|\psi_i\rangle\mapsto\sum_i\alpha_i|\phi_i\rangle $$ where $\langle\psi_i|\psi_j\rangle=\langle\phi_i|\phi_j\rangle=\delta_{ij}$. What this really means is that you want a unitary $U$ that transforms $$ U|\psi_i\rangle=|\phi_i\rangle $$ for all $i$.

Now, in the cases you give, the Hilbert space dimensions don't match, so that's a problem. We get around this by introducing ancillas. Basically, we introduce extra quantum systems, in a known state (usually denoted $|0\rangle$) to make up the Hilbert space to the correct dimension. So, perhaps we really mean $$ U|\psi_i\rangle|00\rangle=|\phi_i\rangle, $$ for example.

Now, this set of conditions basically means that you can fill out several columns of the unitary matrix. Then, you are completely free to fill out the remaining columns however you want so long as the matrix at the end is unitary. Basically, that just means making every column length 1, and making it orthogonal to all other columns. That freedom of choice is typically helpful in making the circuit corresponding to the unitary as simple as possible. However, for the sorts of results you're talking about, they're not really worried about finding good circuits, they just need to know that a circuit exists (which is does due to universality results, provided you can write the matrix down, which you always can). Since it's a unitary of a finite size, the circuit to create it is never going to be too bad.

For example, if you're looking at the transformation $\alpha|0\rangle+\beta|1\rangle\mapsto \alpha|00\rangle+\beta|11\rangle$, we know that $$ U|00\rangle=|00\rangle,\qquad U|10\rangle=|11\rangle. $$ This tells us certain parts of the unitary: $$ U=\left[\begin{array}{cccc} 1 & ? & 0 & ? \\ 0 & ? & 0 & ? \\ 0 & ? & 0 & ? \\ 0 & ? & 1 & ?\end{array} \right]. $$ Orthogonality of the two remaining columns means we can fill in a bit more $$ U=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & ? & 0 & ? \\ 0 & ? & 0 & ? \\ 0 & 0 & 1 & 0\end{array} \right]. $$ The remaining $2\times 2$ submatrix can be any $2\times 2$ unitary that you want, but the choice $$ U=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{array} \right] $$ conveniently corresponds to the controlled-not gate.

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