Free input-independent preparation of prior entanglement

For every natural number $N$, every ownership assignment $o:\{0,\ldots,N-1\}\to\{\text{Alice},\text{Bob}\}$, and every complex vector $\psi$ indexed by $N$-bit strings with Euclidean norm $1$, there exists a unitary matrix $U$ such that $U|0^N\rangle=\psi$, where $|0^N\rangle$ has coefficient $1$ at the all-false bit string and $0$ elsewhere. Moreover, the single round in which Alice applies $U$ and transmits exactly the wires $q$ with $o(q)=\text{Bob}$ is legal when all wires initially belong to Alice: every transmitted wire initially belongs to her, and the applied matrix acts within her initially owned wires. Explicitly, the latter condition means that there exist a natural number $k$, an injection $w:\{0,\ldots,k-1\}\hookrightarrow\{0,\ldots,N-1\}$, and a complex matrix $M$ indexed by $k$-bit strings, with the image of $w$ contained in Alice’s initially owned wires, such that $U_{y,z}=M_{y\circ w,z\circ w}$ when $y$ and $z$ agree outside that image, and $U_{y,z}=0$ otherwise. Updating ownership after the round gives exactly $o$: transmitted wires become Bob’s, and the remaining wires stay Alice’s. The statement includes $N=0$, when there are no wires to transmit and the state space has one basis vector.