Monotonicity of quantum fidelity under partial trace ($F({\mathrm{Tr}}_B\rho ,{\mathrm{Tr}}_B\sigma )\ge F(\rho ,\sigma )$)

Discarding a subsystem cannot decrease unsquared quantum fidelity: $F({\rho }_A,{\sigma }_A)\ge F({\rho }_{AB},{\sigma }_{AB})$. 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.