Manuscript Em05-B@1 · Em05-B-v1.tex · proposition · 6c050f5b

Proposition: Noisy Bell optimum and balanced recovery bound

Population-preserving dephasing and its full environment › Bounds with both extraction loss and dephasing · lines 1572–1636 · proof included or linked

Em05-B@1#prop:noisyBounds, Em05-B@1#eq:noisyBellOpt, Em05-B@1#eq:noisySandwich

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For the noisy channel,

F_{e}^{opt} (N_{η, λ}) = \frac{2 - t ^{2} + 2 λ η}{4} .

The decoder $D_{1}$ attains this Bell optimum. Define

B (η, λ) = \frac{2 + t ^{2} - 2 λ η}{4}, U (η, λ) = 1 - \frac{1}{2 ( 1 + t ^{2} )} - \frac{λ η}{2 1 + t ^{2}} .

Then

B (η, λ) \leq ε_{rec} (N_{η, λ}) \leq min {\frac{3}{4}, t + \frac{η ( 1 - λ )}{2}, U (η, λ)} .

The balanced decoder attains $U (η, λ)$ as its own exact error.

Proof. Repeat Theorem thm:optimalBell, replacing the coherence coefficient $η$ by $λ η$ . All coefficients remain nonnegative, and $D_{1}$ still attains the bound. The Bell witness gives $B (η, λ)$ .

The balanced decoded channel has the same population multiplier $s_{η}$ , and coherence multiplier $λ c_{η}$ . Its Pauli identity probability is

\frac{1}{2 ( 1 + t ^{2} )} + \frac{λ η}{2 1 + t ^{2}},

so Lemma lem:witnesses gives the exact decoder error $U (η, λ)$ .

Finally, the specified decoder yields

D_{⋆} N_{η, λ} = η Δ_{λ} + t R_{∣0 ⟩ ⟨ 0∣}, Δ_{λ} = a_{λ} id_{M} + b_{λ} Ad_{Z} .

The channel triangle inequality gives error at most $t + η b_{λ}$ . The constant decoder gives the remaining upper bound $3/4$ .

Displayed formulas in this passage

Em05-B@1#eq:noisyBellOpt · line 1575
$F_{e}^{opt} (N_{η, λ}) = \frac{2 - t ^{2} + 2 λ η}{4} .$
Em05-B@1#eq:noisySandwich · line 1594
$B (η, λ) \leq ε_{rec} (N_{η, λ}) \leq min {\frac{3}{4}, t + \frac{η ( 1 - λ )}{2}, U (η, λ)} .$
unlabeled · line 1583
$B (η, λ) = \frac{2 + t ^{2} - 2 λ η}{4}, U (η, λ) = 1 - \frac{1}{2 ( 1 + t ^{2} )} - \frac{λ η}{2 1 + t ^{2}} .$
unlabeled · line 1618
$\frac{1}{2 ( 1 + t ^{2} )} + \frac{λ η}{2 1 + t ^{2}},$
unlabeled · line 1626
$D_{⋆} N_{η, λ} = η Δ_{λ} + t R_{∣0 ⟩ ⟨ 0∣}, Δ_{λ} = a_{λ} id_{M} + b_{λ} Ad_{Z} .$