What is meant by "local isometry" in this paper?

Question

I apologize for such a basic question here. I've been reading this paper, and I am wondering what is the usage of the term "local isometry" in the paper? The paper is open access, so you can access it and search with CTRL+F for "isometry" and "isometries" for context. In particular, they say,

Recall that the set of Schmidt coefficients of a state is preserved under local isometries.

By "local isometries" do they just mean bijective linear maps $\mathcal{H}\rightarrow\mathcal{H}'$ that preserve the inner-product? If so, why do we say they are "local?" Why not just call them isometries?

I can understand that unitary isn't the right word, because unitary transformations have the same domain and codomain. Is my understanding here correct, or am I way off?

Answer 1

If you're interested in Schmidt coefficients it means you have a bipartite structure to your Hilbert space. Then local just means that the isometry is of the form $V_1\otimes V_2$ where $V_1$ is an isometry acting locally on the first system in the bipartition and similarly $V_2$ acts only on the second system.

Operationally the isometey can be performed by two separated parties who only have access to the their respective local systems.

Answer 2

Given a bipartite pure state $|\psi\rangle\in\mathcal H_A\otimes\mathcal H_B$, a "local isometry" is an operator of the form $(V\otimes I)$ with $V$ an isometry in $\mathcal H_A$ (and analogous definition for $\mathcal H_B$).

Clearly, not all isometries are local. A general isometry would act nontrivially on the full space rather than only on $\mathcal H_A$ or $\mathcal H_B$. For example, a unitary such as the QFT matrix on the pair of qubits is a nonlocal isometry.