General programs and selective distance

Define

For $J=[a_0-d,a_0+d]$, equation (eq:variation) gives

where

This follows by integrating the derivative block by block. The trace norm of a block-diagonal Hermitian record is the sum of its block trace norms. For classical laws, replace the trace norms in $L_P(J)$ by $\left|{\beta }_y{s}^2-{\gamma }_{y}st\right|$.

There is also a program-independent estimate:

Indeed, a positive trace-preserving map contracts the trace norm on Hermitian differences. For $Z=Z_{+}-Z_{-}$, positivity, the triangle inequality, and trace preservation give

This standard contraction principle is also recorded in Watrous (Corollary 3.40). Direct differentiation of (eq:state) gives

Integration proves (eq:contractive).

For arbitrary encodings, constancy on a bounded segment need not imply constancy outside it. For the retained amplitude family, however, exact selective constancy on any nontrivial segment still forces $B_y=C_y=0$. After dividing its zero derivative by $s(a{)}^2>0$, one obtains

throughout the segment; two different values of $a$ force both operators to vanish. The useful alternative here is approximate closure, not an exact loophole in Option A.