Theorem: Exact delayed retention and its contraction coefficient

For a fixed sector and $T$ steps, set

Then

Phase-compensated projective readout has optimal success

The coherent-versus-node-dephased trace-distance witness for the plus input is $A_T/2$.

The exact carrier contraction coefficient of one ${\Lambda }_{x,s}$ is ${\eta }_s$, not generally ${\eta }_s{\lambda }_s$. For ${\eta }_s<1$, it therefore provides strict global carrier contraction even when ${\lambda }_s=1$. For ${\eta }_s=1,{\lambda }_s<1$, phase information decays, but global strict contraction is absent.

Proof. In Bloch coordinates, depolarization scales all three coordinates by ${\eta }_s$; dephasing scales $x,y$ by ${\lambda }_s$ and leaves $z$ unchanged. The $Z$-rotation acts orthogonally in the $x,y$ plane. The maps commute with that rotation, giving the displayed states.

Their trace distance is $A_T$. The phase-compensated projectors attain the binary optimal formula in Theorem thm:qforget. The dephased early input is $I/2$, giving witness $A_T/2$.

Qubit trace distance is half Euclidean Bloch distance. The largest singular value of the Bloch map is ${\eta }_s$, attained by the two opposite $Z$-axis states.