I have two states, $|\psi\rangle$ and $|\phi\rangle$. I have in mind that they live on a length $L$ spin chain with finite local Hilbert space dimension.
I know that for every Schmidt decomposition bipartitioning the system into a region $A$ and a region $B$, the following is true:
$$|\psi\rangle = \sum_{i=1}^n \lambda_i|a_i\rangle|b_i\rangle $$
$$|\phi\rangle = \sum_{i=1}^n \lambda_i|a'_i\rangle|b'_i\rangle $$
That is, while the states are not necessarily equal, their Schmidt coefficients are equal across every cut in physical space. The Schmidt coefficients may depend on the choice of cut. I imagine that all of my cuts are in physical space, but I do allow regions $A$ and $B$ to contain sites that are not contiguous; for example, $A$ could contain all even sites and $B$ can contain all odd sites.
Given this, am I guaranteed that there exists a unitary that is a tensor product of single-site unitaries, $U = \otimes_{i=1}^L U_i$, such that $ U|\psi\rangle = |\phi\rangle$?
Here are my thoughts. If I had the weaker statement that the Schmidt coefficients were equal $$|\psi\rangle = \sum_{i=1}^n \lambda_i|a_i\rangle|b_i\rangle $$
$$|\phi\rangle = \sum_{i=1}^n \lambda_i|a'_i\rangle|b'_i\rangle $$ for some specific regions $A$ and $B$, then I know I can make a unitary $U = U_A \otimes U_B$ with $U_A = \sum_i |a'_i\rangle \langle a_i|$ and $U_B = \sum_i |b'_i\rangle \langle b_i|$ that takes $|\psi\rangle$ to $|\phi\rangle$.
However, it's less clear to me how to use the information from all of the bipartite cuts together. I was thinking I could consider all $L$ cuts where $A$ contains a single site, and then attempt to argue that $U$ can be written as a product of single-site operators, but I'm not sure that will work. In particular, I'm getting suspicious that perhaps all of the bipartite cuts aren't enough, and that I'll need to know things about multipartite decompositions of the state.
Best Answer
Your claim already fails in the simplest non-trivial case, namely for three qubits.
This follows from a (the?) classic result in the theory of multipartite entanglement -- Three qubits can be entangled in two inequivalent ways by Dür, Vidal, and Cirac.
They show that for three qubits, there are two classes of genuine tripartite entangled states, the W class and the GHZ class. Given states $\lvert \phi_\mathrm{W}\rangle$ and $\lvert \phi_\mathrm{GHZ}\rangle$, from the two classes, it is impossible to convert between them using SLOCC, i.e., it is impossible to write $$ \lvert\phi_\mathrm{W}\rangle\propto (A\otimes B\otimes C)\lvert\phi_\mathrm{GHZ}\rangle $$ (or vice versa) for any $A$, $B$, and $C$. In particular, this rules out the possibility of unitaries doing the job, as you are asking.
What remains is to show that there are states $\lvert\phi_\mathrm{W}\rangle$ and $\lvert \phi_\mathrm{GHZ}\rangle$ with identical Schmidt coefficients in every bipartition. This is equivalent to demanding that their single-qubit reduced states have the same spectra.
Using the results of the paper of Dür, Vidal, and Cirac, it is now easy to find such states (and indeed, they really should exist -- it would be rather disappointing if the two classes were distinguished by the spectra of their reduced density matrices). For instance, you can choose $$ \lvert \phi_\mathrm{W}\rangle = \sqrt{\gamma}\lvert 001\rangle + \sqrt{\gamma}\lvert 010\rangle + \sqrt{\gamma}\lvert 100\rangle + \sqrt{1-3\gamma}\lvert 000\rangle $$ (cf. Eq. (20) in the paper), and $$ \lvert \phi_\mathrm{GHZ}\rangle \propto \lvert 0\rangle \lvert 0\rangle \lvert 0\rangle + \lvert \theta\rangle\lvert \theta\rangle\lvert \theta\rangle $$ (cf. Eq. (15)), with $\lvert\theta\rangle=\sqrt{\mu}\lvert 0\rangle + \sqrt{1-\mu}\lvert 1\rangle$. You can now easily check that for the spectra $(\lambda,1-\lambda)$ ($\lambda\le1/2$) of the single-qubit reduced states (which are all equal by symmetry)
Thus, you can easily find values $\mu$ and $\gamma$ where the reduced states have identical spectra, and thus the Schmidt spectra in all bipartitions are equal, yet the states cannnot be converted into each other by local unitaries (or even SLOCC).