Corollary: Global retention versus marginal loss

For every logical state,

The nonzero eigenvalues of $V_{\eta }{\rho }_M{V}_{\eta }^{†}$ are those of ${\rho }_M$. In particular,

For every purified input, the joint referenced output is pure. Thus the declared isometry does not globally destroy information.

Proof. The occupation statement follows from $\eta +t=1$. An isometry preserves inner products and embeds the spectral decomposition of its input with the same nonzero eigenvalues. Tensoring with a reference preserves the norm and purity of a purified input.