Entropy and complementary information

Entropy calculations use input $|0⟩$; recovery uses a Bell reference with unchanged dynamics.

For $t\le 3$, define

Let ${\mathsf{f}}_t$ contain $v_t$ and the nonempty flag weights. For even parity the rail entropy is one for a singleton and two for a pair or triple. For odd parity the exception is $s_L(\{1,3\})=1$. Then

This follows from the orthogonal-block entropy identity. For $a=1-w_0$,

Propagation preserves entropy of the complete emitted region.

For the Bell experiment let $K_t=C_t{F}_{\le t}$ and

A constant qubit channel has a complementary Bell state isometric to ${\Phi }_{RE}\otimes \tau$, so its distance is $3/4$. This follows from a dilation $|i⟩\mapsto {\sum }_j\sqrt{{\lambda }_j}|j⟩|i,j⟩$ of a fixed output ${\sum }_j{\lambda }_j|j⟩⟨j|$.

Consequently the even benchmark has

Complementary full-transmission sectors are constant; failure sectors complement constant accessible channels. The vacuum sector is retained coherently as one sector.

This equality is special to this channel, not a general identification of Bell-input decoupling with worst-case recovery.