# Is there a relation between some kind of distance and the Schmidt basis?

Consider two bipartite quantum states $$|\phi\rangle^{AB}$$ and $$|\psi\rangle^{AB}$$ (in a finite dimensional Hilbert space $$\mathcal H_A\otimes \mathcal H_B$$), such that $$\| |\phi\rangle\langle\phi|^{AB} - |\psi\rangle\langle\psi|^{AB} \|_p \leq \varepsilon\ .$$ Does this imply that there exist Schmidt decompositions \begin{align} |\phi\rangle^{AB}&=\sum_i\sqrt{p_i}|e_i\rangle^A|\tilde{e}_i\rangle^B\ ,\\ |\psi\rangle^{AB}&=\sum_i\sqrt{q_i}|f_i\rangle^A|\tilde{f}_i\rangle^B \end{align} which are also close to each other, i.e., for which \begin{align} 1-|\langle e_i|f_i\rangle|^2 & \leq g(\varepsilon)\ ,\\ 1-|\langle \tilde e_i|\tilde f_i\rangle|^2 & \leq g(\varepsilon)\ ,\\ |p_i-q_i|&\le h(\varepsilon)\ , \end{align} where $$g(\varepsilon),h(\varepsilon)\to0$$ as $$\varepsilon\to0$$?

Cross-posted on QC.SE

• Hint: First, prove a relation between p-norm distance and overlap. Second, use this to show a relation between the Schmidt decompositions. Sep 18, 2023 at 7:45
• In the end, this boils down to understand whether closeby matrices have closeby SVDs, cf. math.stackexchange.com/questions/3389899/… Sep 19, 2023 at 16:43

To this end, consider $$\lvert\phi\rangle = a\lvert0\rangle\lvert0\rangle + b \lvert1\rangle\lvert1\rangle\ ,$$ and $$\lvert\psi\rangle = a\lvert+\rangle\lvert+\rangle + b \lvert-\rangle\lvert-\rangle\ ,$$ where $$a=\sqrt{\tfrac12-\varepsilon}$$, $$b=\sqrt{\tfrac12+\varepsilon}$$ [and with $$\lvert \pm\rangle = \tfrac12(\lvert0\rangle\pm\lvert1\rangle)$$].
$$\lvert\phi\rangle$$ and $$\lvert\psi\rangle$$ are in their Schmidt decomposition, and it is unique (as long as $$\varepsilon\ne 0$$). Moreover, $$\|\lvert\phi\rangle\langle\phi\rvert-\lvert\phi\rangle\langle\phi\rvert\|_p \to 0$$ as $$\varepsilon\to 0$$.
Yet, their Schmidt vectors do not become close to each other; in fact, they are completely independent of $$\varepsilon$$.
Thus, the only way in which this can be made to work is if you insist that you are sufficiently far (as comapred to $$\varepsilon$$) from a state with degenerate Schmidt coefficients.