Corollary: Global retention

For every logical state,

The nonzero eigenvalues of $V_{\eta }{\rho }_M{V}_{\eta }^{†}$ are those of ${\rho }_M$. Every purified input has a pure joint referenced output.

Proof. The first equality uses $\eta +t=1$. An isometry embeds a spectral decomposition without changing its nonzero eigenvalues and maps pure vectors to pure vectors.