= Monotonicity of quantum fidelity under partial trace
{title2=$F(\operatorname{Tr}_B\rho,\operatorname{Tr}_B\sigma)\geq F(\rho,\sigma)$}
Discarding a subsystem cannot decrease unsquared <quantum fidelity>: $F(\rho_A,\sigma_A)\geq F(\rho_{AB},\sigma_{AB})$. Maximizing purifications of the joint states are also candidates for purifications of the marginals, with the discarded subsystem included in the reference. <Uhlmann's theorem> then proves the inequality. Information loss makes the two states at least as similar.
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