= Measurement characterization of quantum fidelity
{title2=$F(\rho,\sigma)=\min_{\{E_i\}}\sum_i\sqrt{\operatorname{Tr}(\rho E_i)\operatorname{Tr}(\sigma E_i)}$}
A <POVM> $\{E_i\}$ gives outcome <probability distributions> $p_i=\operatorname{Tr}(\rho E_i)$ and $q_i=\operatorname{Tr}(\sigma E_i)$ by the <Born rule>. Their classical overlap, the <Bhattacharyya coefficient>, is $B(p,q)=\sum_i\sqrt{p_iq_i}$. Fuchs and Caves showed that minimizing this overlap over all <POVMs> gives the <unsquared quantum fidelity>:
$$
F(\rho,\sigma)=\min_{\{E_i\}}B(p,q)=\min_{\{E_i\}}\sum_i\sqrt{\operatorname{Tr}(\rho E_i)\operatorname{Tr}(\sigma E_i)}.
$$
The same <POVM> is applied to either possible input <quantum state>. Each <quantum measurement> has $B(p,q)\geq F$; some measurement attains equality, including when the <density operators> have zero <eigenvalues>. Thus <quantum fidelity> quantifies the smallest classical overlap that observation can reveal. For <fidelity of commuting mixed states>, measurement in a common <eigenbasis> attains the minimum. This operational interpretation compares statistics without assigning a probability that two <mixed states> are equal. https://arxiv.org/pdf/quant-ph/9604001v1[Fuchs and Caves, Mathematical Techniques for Quantum Communication Theory, equations 5–6 and 17; published 1995, archived 1996]{ref} https://jhwatrous.github.io/TQI.3.pdf#page=30[Watrous, The Theory of Quantum Information, Theorem 3.24, including singular density operators]{ref}
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