Theorem: Ideal five-qubit access table

The matrix $U_B$ is unitary and

Consequently

recovers the qubit exactly, including every external reference.

For an arbitrary accessible set $A$,

The infimum is over all CPTP decoders on the declared accessible set.

Proof. Privacy on $E$ gives

The eight columns in (eq:UB) are therefore orthonormal. Summing the four environment components proves (eq:block-channel), first on matrix units and hence on all operators. Tensoring this equality with a reference proves the reference statement. For larger access, discard to a triple.

For $|A|\le 2$, every decoder output is a replacement state $\tau$. On a Bell input, $\tau \otimes I_2/2$ has Bell overlap $1/4$. The Bell-projector measurement gives error at least $3/4$. The replacement $I_2/2$ attains this value because

so its half-diamond distance from ${\mathrm{id}}_2$ is at most $3/4$.