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

Proposition: Selected entropy does not identify recovery

Holographic non-identification · lines 1605–1623 · proof included or linked

En07-A@1#prop:entropy

There are complete isometries with identical zero radiation entropy on every pure code input but radiation recovery errors $0$ and $a_{d}$ , respectively.

Proof. Take

V_{good} ∣ ψ ⟩ = ∣ ψ ⟩_{R} ∣ 0 ⟩_{B}, V_{bad} ∣ ψ ⟩ = ∣ 0 ⟩_{R} ∣ ψ ⟩_{B} .

Both radiation marginals are pure for every pure input. The first channel is the identity; the second is constant. Their optimal recovery errors are therefore $0$ and $a_{d}$ .

Displayed formulas in this passage

unlabeled · line 1614
$V_{good} ∣ ψ ⟩ = ∣ ψ ⟩_{R} ∣ 0 ⟩_{B}, V_{bad} ∣ ψ ⟩ = ∣ 0 ⟩_{R} ∣ ψ ⟩_{B} .$

Nearby in this section

remark: Remark: The exact criterion is not a geometric dictionary
section: Holographic non-identification
corollary: Corollary: A dimension constraint, not an area law
section: Holographic non-identification