Proposition: Sharp record-only recovery lower bound

If a $d$-dimensional unknown input, $d\ge 2$, reaches a decoder only through a classical record, the decoded channel $\Lambda$ obeys

Here $d_{\diamond }(\Phi ,\Psi )=\frac{1}{2}{\Vert \Phi -\Psi \Vert }_{\diamond }$. The bound is attained by basis measurement and basis preparation.

Proof. The decoded channel is measure-and-prepare. On one half of a maximally entangled state its output is separable. Every product state has overlap at most $1/d$ with that maximally entangled state, by Cauchy–Schwarz; the same holds for mixtures. Testing its rank-one projector gives the lower bound.

Computational dephasing is $d^{-1}{{\sum }_{k=0}}^{d-1}{\mathrm{Ad}}_{{Z_d}^k}$. The triangle inequality bounds its half-diamond distance from identity by $1-1/d$, giving equality.