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

Proposition: Comparison of the recovery witnesses

Improved intermediate recovery control › A balanced decoder with an exact Pauli error · lines 1385–1428 · proof included or linked

Em05-B@1#prop:dominance

The optimized Bell lower bound dominates all the elementary lower bounds stated in Section sec:complementary:

B (η) \geq max {\frac{t ^{2}}{2}, \frac{t}{2}, \frac{( 1 - η ^{2} ) ^{2}}{4}} .

At $η = 1/2$ , the improved sandwich is

\frac{5}{16} \leq ε_{rec} (N_{1/2}) \leq \frac{3}{5} - \frac{1}{2 5} .

As $t ↓ 0$ ,

B (η) = \frac{t}{2} + \frac{t ^{2}}{4}, U (η) = \frac{t}{2} + \frac{3 t ^{2}}{4} + O (t^{3}) .

Proof. The first two comparisons follow directly from $B = t /2 + t^{2} /4$ . For the complementary term,

4 (B - \frac{t ^{2} ( 2 - t ) ^{2}}{4}) = t (2 - 3 t + 4 t^{2} - t^{3}) .

For $0 \leq t \leq 1$ ,

2 - 3 t + 4 t^{2} - t^{3} = \frac{5}{4} + 3 (t - \frac{1}{2})^{2} + t^{2} (1 - t) > 0.

Substitution gives the half-transfer values. Taylor expansion of (eq:balancedUpper) gives the endpoint asymptotics.

Displayed formulas in this passage

unlabeled · line 1389
$B (η) \geq max {\frac{t ^{2}}{2}, \frac{t}{2}, \frac{( 1 - η ^{2} ) ^{2}}{4}} .$
unlabeled · line 1397
$\frac{5}{16} \leq ε_{rec} (N_{1/2}) \leq \frac{3}{5} - \frac{1}{2 5} .$
unlabeled · line 1403
$B (η) = \frac{t}{2} + \frac{t ^{2}}{4}, U (η) = \frac{t}{2} + \frac{3 t ^{2}}{4} + O (t^{3}) .$
unlabeled · line 1414
$4 (B - \frac{t ^{2} ( 2 - t ) ^{2}}{4}) = t (2 - 3 t + 4 t^{2} - t^{3}) .$
unlabeled · line 1420
$2 - 3 t + 4 t^{2} - t^{3} = \frac{5}{4} + 3 (t - \frac{1}{2})^{2} + t^{2} (1 - t) > 0.$