Manuscript Hawk04-A@1 · Hawk04-A.tex · section · 526092ba

Entropy and source energy are insufficient

Endpoints and the missing gravitational implications › Entropy and source energy are insufficient · lines 1937–1957

The two isometries

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

give radiation entropy zero for every pure code input. Their optimal radiation recovery errors are respectively $0$ and $a_{d}$ . Thus a selected radiation entropy does not characterize the channel.

Likewise, $V_{bad}$ can have a source in a fixed zero-energy vacuum while an inaccessible degenerate memory retains the code. Vanishing source energy is not complete residual independence. These are the same logical distinctions emphasized in Hawk01-B and Hawk02-E CheungHawk01B,CheungHawk02E.

Displayed formulas in this passage

unlabeled · line 1940
$V_{good} ∣ ψ ⟩ = ∣ ψ ⟩_{R} ∣ 0 ⟩_{B}, V_{bad} ∣ ψ ⟩ = ∣ 0 ⟩_{R} ∣ ψ ⟩_{B}$