Manuscript En07-B@1 · En07-B.tex · proposition · 45dff4a1

Proposition: Featureless accessible output obstructs recovery

Featureless output, entropy, and availability · lines 788–838 · proof included or linked

En07-B@1#prop:featureless

If, for one density operator $ω$ on $R$ ,

d_{⋄} (N, T_{ω}) \leq η,

then every deterministic decoder satisfies

ε (D; N) \geq max {0, 1 - \frac{1}{d ^{2}} - η} .

In particular, a constant accessible channel has optimal recovery error $1 - d^{- 2}$ .

Proof. For any density operator $σ$ on $C$ ,

(id_{Q} \otimes T_{σ}) (P_{Φ_{d}}) = I_{Q} / d \otimes σ,

whose Bell overlap is $d^{- 2}$ . The Bell-projector test gives

d_{⋄} (T_{σ}, id_{C}) \geq 1 - d^{- 2} .

CPTP contraction gives

d_{⋄} (DN, T_{D (ω)}) \leq η .

The reverse triangle inequality proves the asserted lower bound.

When $N$ is constant, a decoder can ignore its input and prepare $I_{C} / d$ . The resulting channel is

T_{I_{C} / d} = \frac{1}{d ^{2}} a, b = 0 \sum^{d - 1} Ad_{X^{a} Z^{b}} .

The same triangle argument as in Theorem thm:classical gives

id_{C} - T_{I_{C} / d}_{⋄} \leq 2 (1 - d^{- 2}) .

The Bell witness gives the matching lower bound.

Displayed formulas in this passage

unlabeled · line 791
$d_{⋄} (N, T_{ω}) \leq η,$
unlabeled · line 795
$ε (D; N) \geq max {0, 1 - \frac{1}{d ^{2}} - η} .$
unlabeled · line 806
$(id_{Q} \otimes T_{σ}) (P_{Φ_{d}}) = I_{Q} / d \otimes σ,$
unlabeled · line 813
$d_{⋄} (T_{σ}, id_{C}) \geq 1 - d^{- 2} .$
unlabeled · line 817
$d_{⋄} (DN, T_{D (ω)}) \leq η .$
unlabeled · line 826
$T_{I_{C} / d} = \frac{1}{d ^{2}} a, b = 0 \sum^{d - 1} Ad_{X^{a} Z^{b}} .$
unlabeled · line 833
$id_{C} - T_{I_{C} / d}_{⋄} \leq 2 (1 - d^{- 2}) .$