Manuscript Hawk02-E@1 · Hawk02-E.tex · proposition · 4841592b

Proposition: Unique gapped full-residual vacuum criterion

Information endpoint and gravitational endpoint › A full-residual energy criterion and its limitations · lines 2009–2059 · proof included or linked

Hawk02-E@1#prop:residual-energy

Suppose the entire residual has a specified nonnegative operator $K_{B}$ satisfying

K_{B} \geq γ (I_{B} - ∣ Ω_{B} ⟩ ⟨ Ω_{B} ∣), γ > 0,

with one normalized vacuum $Ω_{B}$ . If a complete code isometry $V$ obeys

ρ_{C} sup Tr (K_{B} C (ρ_{C})) \leq u < γ,

then

δ_{full}, ε_{rec} \leq min {1, 2 (1 - 1 - u / γ)} .

Proof. Set $P = I_{R} \otimes ∣ Ω_{B} ⟩ ⟨ Ω_{B} ∣$ . The energy bound gives

V^{†} (I - P) V \leq (u / γ) I_{C} .

Thus $M = V^{†} P V$ is positive and invertible. The polar-normalized map

V_{0} = P V M^{- 1/2}

is an isometry of the form $A \otimes ∣ Ω_{B} ⟩$ . Its complementary channel is constant and its accessible channel is exactly recoverable.

Moreover $V^{†} V_{0} = M^{1/2}$ , so

∥ V - V_{0} ∥^{2} = 2 I - 2 M^{1/2} \leq 2 (1 - 1 - u / γ) .

Apply Lemma lem:isometry and decode the isometry $A$ on its range with a complete CPTP extension.

Displayed formulas in this passage

unlabeled · line 2013
$K_{B} \geq γ (I_{B} - ∣ Ω_{B} ⟩ ⟨ Ω_{B} ∣), γ > 0,$
unlabeled · line 2020
$ρ_{C} sup Tr (K_{B} C (ρ_{C})) \leq u < γ,$
unlabeled · line 2025
$δ_{full}, ε_{rec} \leq min {1, 2 (1 - 1 - u / γ)} .$
unlabeled · line 2037
$V^{†} (I - P) V \leq (u / γ) I_{C} .$
unlabeled · line 2042
$V_{0} = P V M^{- 1/2}$
unlabeled · line 2050
$∥ V - V_{0} ∥^{2} = 2 I - 2 M^{1/2} \leq 2 (1 - 1 - u / γ) .$