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Does proximity of two bipartite states in a norm force high overlap between the elements of the Schmidt bases?

No. Here is an example without small Schmidt coefficients. To this end, consider $$ \lvert\phi\rangle = a\lvert0\rangle\lvert0\rangle + b \lvert1\rangle\lvert1\rangle\ , $$ and $$ \lvert\psi\rangle = ...
Norbert Schuch's user avatar
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Does proximity of two bipartite states in a norm force high overlap between the elements of the Schmidt bases?

TL;DR: No such relation exists, because the upper bound on the norm fails to impose any constraints whatsoever on the basis elements corresponding to very small Schmidt coefficients $\sqrt{p_i}$ and $\...
Adam Zalcman's user avatar
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Is fidelity of mixed $\sigma$ and pure $|\psi\rangle$ equal to $\||\psi\rangle\langle\psi|\sigma\|_1$?

No, in general $\langle\psi|\sigma|\psi\rangle\ne\||\psi\rangle\langle\psi|\sigma\|_1$. Singular value decomposition Recall that if \begin{equation} A=\sum_i s_i|a_i\rangle\langle b_i|\tag1 \end{...
Adam Zalcman's user avatar

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