Manuscript En07-A@1 · En07-A.tex · proposition · 44229ce9

Proposition: Uniformly featureless radiation

Thermodynamic access: three separate ledgers › Residual leakage is not heat · lines 1397–1430 · proof included or linked

En07-A@1#prop:featureless, En07-A@1#eq:featureless

If, for some radiation state $σ$ ,

d_{⋄} (N, T_{σ}) \leq η,

then

ε_{rec} (N) \geq max {0, a_{d} - η} = max {0, 1 - d^{- 2} - η} .

Proof. For any deterministic decoder,

D T_{σ} = T_{D (σ)} .

Contraction and the triangle inequality give

d_{⋄} (DN, id_{C}) \geq d_{⋄} (T_{D (σ)}, id_{C}) - d_{⋄} (DN, D T_{σ}) \geq a_{d} - η,

by Lemma lem:unitary-mixture. Take the decoder infimum and use nonnegativity.

Displayed formulas in this passage

En07-A@1#eq:featureless · line 1404
$ε_{rec} (N) \geq max {0, a_{d} - η} = max {0, 1 - d^{- 2} - η} .$
unlabeled · line 1400
$d_{⋄} (N, T_{σ}) \leq η,$
unlabeled · line 1415
$D T_{σ} = T_{D (σ)} .$
unlabeled · line 1419
$d_{⋄} (DN, id_{C}) \geq d_{⋄} (T_{D (σ)}, id_{C}) - d_{⋄} (DN, D T_{σ}) \geq a_{d} - η,$