Petz and physical implementation comparisons

For a full-rank input state $\sigma$, write $s=\mathcal{N}(\sigma )$, $P=\mathrm{supp}s$, and invert $s$ only on $P$. A completed Petz candidate is

It is CP, and the traces of its two terms are $\mathrm{Tr}(PY)$ and $\mathrm{Tr}((I-P)Y)$, so it is TP on the entire accessible space. This alone does not establish (eq:decerror). Petz and Yoshida–Kitaev decoders have additional theorems under their own hypotheses ChenPetz,YoshidaKitaev.

Any specified candidate can instead be tested by

which gives half-diamond error at most $d{r}_{\mathrm{dec}}$ by Lemma lem:Choi.