Manuscript Hawk02-D@1 · Hawk02-D.tex · proposition · e6e4b72f

Proposition: Gap–area-law nonimplication

Debugging the proposed imports and physical interpretation › A spectral gap and an area law do not imply message independence · lines 1680–1709 · proof included or linked

Hawk02-D@1#prop:area-law

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A positive spectral gap and zero spatial-cut entanglement for all code states can coexist with perfect retention of the code in an inaccessible subsystem.

Proof. On finitely many spins, let

H_{gap} = j = 2 \sum^{m} ∣ 1 ⟩ ⟨ 1 ∣_{j}, m \geq 2.

The first spin is free. The ground space consists of

∣ ψ ⟩_{1} ∣ 0 ⟩_{2} \dots ∣ 0 ⟩_{m},

and the gap above this space is one. Every such state is a product across spatial cuts. Encode the message in the first spin and declare it inaccessible. The residual channel is the identity on the message, with leakage $1 - d^{- 2}$ .

If an area-law theorem instead assumes a unique ground state, its hypotheses do not cover this nontrivial ground-space code. They also do not automatically cover the nonequilibrium evaporating preparation.

Displayed formulas in this passage

unlabeled · line 1689
$H_{gap} = j = 2 \sum^{m} ∣ 1 ⟩ ⟨ 1 ∣_{j}, m \geq 2.$
unlabeled · line 1696
$∣ ψ ⟩_{1} ∣ 0 ⟩_{2} \dots ∣ 0 ⟩_{m},$

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section: A spectral gap and an area law do not imply message independence