I want to know that there is a relation between the distance of two vectors and the corresponding elements of the Schmidt bases.
We assume that two bipartite vectors $|\phi\rangle^{AB}$ and $|\psi\rangle^{AB}$ can be decomposed such that
$$|\phi\rangle^{AB}=\sum_i\sqrt{p_i}|e_i\rangle^A|\tilde{e}_i\rangle^B\ \qquad(p_1\geq p_2 \geq ...), $$$$|\psi\rangle^{AB}=\sum_i\sqrt{q_i}|f_i\rangle^A|\tilde{f}_i\rangle^B\ \qquad (q_1 \geq q_2 \geq ...).$$
These Schomidt coefficients are arranged in decreasing order, and $|e_i\rangle^{A}|\tilde{e}_i\rangle^{B}$ and $|f_i\rangle^A|\tilde{f}_i\rangle^B$ are Schmidt basis corresponding to coefficient $\sqrt{p_i}$, $\sqrt{q_i}$ respectively.
If $$\| |\phi\rangle\langle\phi|^{AB} - |\psi\rangle\langle\psi|^{AB} \|_p \leq \varepsilon$$
then is there relation that $$|\langle e_i|^A\langle \tilde{e}_i|^B|f_i\rangle^A|\tilde{f}_i\rangle^B| \geq 1- g(\varepsilon)\ ,$$ where $g(\varepsilon)$ is some kind of function of $\varepsilon$ ?
Certainly, when it comes to degenerate cases, Schmidt bases are not uniquely determined. The question arises whether it's possible to pair up "Schmidt bases which satisfy that relation for any $i$."
Does a relation like this exist or not?