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We first establish the purification-overlap formula for quantum fidelity, rather than assume the desired monotonicity. Write a canonical purification of a density operator asTracing the second factor gives $\rho$. The Schmidt decomposition shows that any other purification on a sufficiently large fixed ancillary space differs by a unitary on that space: its orthonormal ancillary vectors can be mapped to those of the canonical purification and the map extended to a unitary operator. Zero eigenvalues are handled by padding the ancillary basis.

The overlap of two canonical purifications, with a relative ancillary unitary $U$, is $\mathrm{Tr}(\sqrt{\rho }\sqrt{\sigma }{U}^T)$, with zero padding when the ancillary dimension is larger. The unitary variational formula for the trace norm isIndeed if $X=PD{Q}^{†}$ is a singular value decomposition, the trace is $\mathrm{Tr}(DW)$ with $W=Q^{†}VP$ unitary. Its modulus is at most ${\sum }_j{D}_{jj}$ because $|W_{jj}|\le 1$, and $V=Q{P}^{†}$ attains that bound. Thus the maximal absolute overlap of purifications equals $F(\rho ,\sigma )$; this also proves the needed form of Uhlmann's theorem.

Choose purifications $|\psi {⟩}_{ABE}$ and $|\varphi {⟩}_{ABE}$ attaining $F({\rho }_{AB},{\sigma }_{AB})$. Regard the very same vectors as purifications of ${\rho }_A,{\sigma }_A$ with ancillary system $BE$. Their overlap is among those maximized in the marginal purification formula, soThis proves the monotonicity of quantum fidelity under partial trace.