Proposition: Classical readout obstruction

If the only accessible output is a finite classical measurement record, every quantum decoder produces an entanglement-breaking channel and has full-code recovery error at least

For a replacement accessible channel,

the optimal recovery error is

Proof. A decoded classical record has the form

On a referenced input its output is a sum of positive reference operators tensored with ${\sigma }_z$, hence is separable. A product state has overlap at most $1/d$ with a maximally entangled state; convexity gives the same bound for separable states. The Bell-projector effect yields (eq:classicalbound).

Every decoded replacement is a replacement with some state ${\tau }_C$. Its normalized Bell output is $I_Q/d\otimes {\tau }_C$, whose Bell overlap is $d^{-2}$, yielding the lower bound $1-d^{-2}$. Preparing $I_C/d$ attains it: the completely depolarizing channel is the uniform average of the $d^2$ Weyl conjugations. Separating identity and using the triangle inequality gives half-diamond distance at most $1-d^{-2}$.