quantum mechanics - Explanation of why this derivation of Schmidt decomposition works

I'm following Preskill's notes and he derives the Schmidt decomposition in the following way:

Let a bipartite state be $\psi_{AB} = \sum_{i,j}\lambda_{ij}\vert i\rangle\vert j\rangle = \sum_{i} \vert i\rangle\vert \tilde{i}\rangle$, where I simply choose $\sum_j \lambda_{ij}\vert j\rangle = \vert \tilde{i}\rangle$.

I choose a set of basis vectors $\vert i\rangle$ such that the partial state is diagonal, that is $\rho_A = \sum_i p_i\vert i\rangle\langle i\vert$. But I can also obtain $\rho_A = Tr_B(\rho_{AB}) = Tr_B\sum_{i,j} \vert i\rangle\langle j\vert \otimes \vert \tilde{i}\rangle\langle \tilde{j}\vert = \sum_{ij} \langle \tilde{j}\vert\tilde{i}\rangle \vert i\rangle\langle j\vert$. The last part can be computed by explicitly writing out the trace over $B$ and using the properties of an orthonormal basis.

Thus, we have $\rho_{A} = \sum_i p_i\vert i\rangle\langle i\vert = \sum_{ij} \langle \tilde{j}\vert\tilde{i}\rangle \vert i\rangle\langle j\vert$. That is $\langle \tilde{j}\vert \tilde{i}\rangle = p_i\delta_{ij}$. Suddenly, the $\vert\tilde{i}\rangle$ are all orthogonal to each other.

Why does choosing the basis where $\rho_A$ is diagonal also give you orthogonal vectors in $B$? This seemed to drop out of the sky for me although the math is clear. What is the physical meaning of this?

Answer

Let us start from the Schmidt decomposition $|\psi\rangle = \sum s_i |a_i\rangle |b_i\rangle$.

Now consider the reduced state of $A$: $\rho_A=\sum s_i^2 |a_i\rangle\langle a_i|$. This is, the eigenbasis of A is exactly the basis you need for the Schmidt decomposition!

Thus, if you write your state using that eigenbasis of Alice, $$ |\psi\rangle = \sum_i |a_i\rangle \Big(\sum_j \lambda_{ij}|j\rangle\Big)\ , $$ the part $|\tilde b_i\rangle=\sum_j \lambda_{ij}|j\rangle$ must be equal to $s_i|b_i\rangle$, since the Schmidt decomposition is unique (modulo degeneracies).