Theorem: Exact control accounting

The complete code vectors, up to a common phase, are

They give

and the exact formulas

In particular, ${{\epsilon }_{\mathrm{rec}}}^R\ge 1/2$ and ${\delta }_{\mathrm{full}}\ge 1/2$. The complete output is exactly reversible.

Proof. On source eigenvalue $a_i$, let $q_i=g{a}_i{h_B}^{-1}f$. The Weyl identity gives

where $\mathcal{W}(q)=e^{a^{†}(q)-a(q)}$. The vacuum therefore evolves to the coherent amplitude $(e^{-\mathrm{i}t{h}_B}-I)q_i={\eta }_i(t)$. The phase is common for $a_i=\pm 1$. Orthogonal packet decomposition factors the coherent state. Tracing the orthogonal source labels gives the radiation channel.

Its two pure output states have trace distance $z_R$. Their optimal equal-prior discrimination probability is $(1+z_R)/2$, obtained by the positive part of their difference. For any radiation decoder, its Bell overlap is

The two diagonal decoder effects constitute a discrimination POVM. The Bell-projector test gives error at least $(3-z_R)/4$.

Bath parity interchanges the two states, so an optimal discrimination POVM can be symmetrized to have equal conditional errors. Measurement followed by preparation of the inferred basis state gives

Its Pauli probabilities are

Orthogonal Pauli Bell outputs and the triangle inequality give half-diamond error $1-p_I$, proving the radiation formula.

Define the isometry

Then

Retracting the range of $L_t$ gives the dephasing recovery error $(1-c_R)/2$, by the same upper bound and $|+⟩$, $|-⟩$ lower bound used in Theorem thm:rotation.

Isometric embedding and CPTP retraction preserve leakage. For ${\Delta }_c$, Pauli covariance and convexity show that twirling any candidate replacement state to $I_2/2$ cannot increase its distance. Relative to that replacement, the Pauli coefficients, in order $I,Z,X,Y$, are

The half-diamond norm is half the sum of their absolute values: a Bell input gives the lower bound and the triangle inequality gives the upper bound. Hence

The complete columns are orthonormal, so global recovery follows from Theorem thm:global.