Manuscript Oai265-A@2 · Oai265-A.tex · proposition · c0a34a45

Proposition: The same boundary size can have different entropies

Stage 4: hypothesis tests on explicit models › An upper bound is not a strict equality · lines 1056–1084 · proof included or linked

Oai265-A@2#prop:strict

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There is a unique, nearest-neighbour, unit-gap ground state satisfying

S (A) \leq (lo g 2) ∣ \partial_{Λ} A ∣

for every $A$ , but no constants $α, γ$ satisfy

S (A) = α ∣ \partial_{Λ} A ∣ - γ

for all its interior one-site regions.

Proof. Choose a sufficiently large open square, two neighbouring interior sites $a, b$ , and another interior site $c \in / {a, b}$ . Let the ground state be a Bell pair on $a, b$ and $∣0 ⟩$ elsewhere, with Hamiltonian

I - ∣ Ω_{2} ⟩ ⟨ Ω_{2} ∣_{ab} + x \neq = a, b \sum (I - ∣0 ⟩ ⟨ 0 ∣_{x}) .

The disjoint projectors give a unique ground state and gap one. Only cuts separating $a, b$ have nonzero entropy, at most $lo g 2$ , and every such cut crosses their edge.

Both ${a}$ and ${c}$ have boundary size four, but their entropies are $lo g 2$ and zero.

Displayed formulas in this passage

unlabeled · line 1059
$S (A) \leq (lo g 2) ∣ \partial_{Λ} A ∣$
unlabeled · line 1063
$S (A) = α ∣ \partial_{Λ} A ∣ - γ$
unlabeled · line 1074
$I - ∣ Ω_{2} ⟩ ⟨ Ω_{2} ∣_{ab} + x \neq = a, b \sum (I - ∣0 ⟩ ⟨ 0 ∣_{x}) .$

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section: An upper bound is not a strict equality