Manuscript Oai276-A@1 · Oai276-A.tex · section · 215068af

Explicit output matrices

Stages 3 and 6: the explicit finite maximum certificate › Explicit output matrices · lines 1220–1258

Oai276-A@1#eq:outputs

The output states of the finite pair are

ρ^{0}_{\pm} = 2561/4000 \pm \frac{996279}{8000000} \pm \frac{996279}{8000000} 1439/4000 .

Their average is diagonal, and

det ρ^{0}_{\pm} = \frac{1692721}{16000000}, 1 - 4 det ρ^{0}_{\pm} = \frac{2307279}{4000000} = q_{0} .

These identities are direct substitutions into (eq:channel); they also independently check the scalar arguments in the certificate.

The exact optimal outputs are obtained by replacing $p_{0}$ by the uniquely specified $p^{⋆}$ :

ρ^{⋆}_{\pm} = 7/8 - p^{⋆} /2 \pm \frac{p ^{⋆} ( 1 - p ^{⋆} )}{2} \pm \frac{p ^{⋆} ( 1 - p ^{⋆} )}{2} 1/8 + p^{⋆} /2 .

This writes the binary ensemble explicitly without supplying an asymptotic decoding circuit.

Displayed formulas in this passage

Oai276-A@1#eq:outputs · line 1223
$ρ^{0}_{\pm} = 2561/4000 \pm \frac{996279}{8000000} \pm \frac{996279}{8000000} 1439/4000 .$
unlabeled · line 1235
$det ρ^{0}_{\pm} = \frac{1692721}{16000000}, 1 - 4 det ρ^{0}_{\pm} = \frac{2307279}{4000000} = q_{0} .$
unlabeled · line 1246
$ρ^{⋆}_{\pm} = 7/8 - p^{⋆} /2 \pm \frac{p ^{⋆} ( 1 - p ^{⋆} )}{2} \pm \frac{p ^{⋆} ( 1 - p ^{⋆} )}{2} 1/8 + p^{⋆} /2 .$