ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Entanglement threshold versus thermodynamic singularity

Physical scope

The zero-field, antiferromagnetic finite Pauli dimer at its exact thermal separability boundary. It distinguishes a state-property threshold from a thermodynamic singularity and makes no experimental claim.

Example

For the zero-field dimer of Interacting Heisenberg dimer Gibbs model with $J > 0$ , the equality point $β J = (\log 3) / 4$ gives a separable Gibbs density matrix, and every larger $β J$ gives an entangled one. Nevertheless $Z = e^{3 β J} + 3 e^{- β J}$ and the free energy remain analytic at this threshold. The threshold changes the separability classification without producing a finite-temperature thermodynamic singularity.

Facts & Assumptions

Given: $J > 0$ , $h = 0$ , $β > 0$ , and the exact finite dimer result.

[F1]

Interacting Heisenberg dimer Gibbs model: at zero field the Gibbs state is separable for $β J \leq (\log 3) / 4$ and entangled above, while its finite partition and free energy are analytic for positive $β$ .

Verification

1.1givenF1algebra

Substituting $β J = (\log 3) / 4$ into [F1] gives the separable equality state; taking any $β J > (\log 3) / 4$ gives the entangled branch. These are conclusions about the defined two-qubit density matrix.

2.1givenF1step 1.1algebra∎

At $h = 0$ , $Z = e^{3 β J} + 3 e^{- β J}$ is a finite positive sum of exponentials, so $- β^{- 1} \log Z$ is analytic for $β > 0$ , including at the separability threshold. Therefore this state-property boundary is not a thermodynamic singularity of the finite dimer.

Depends on

Interacting Heisenberg dimer Gibbs model

Dependency tree · 2 levels

Sources

Damian Markham, Janet Anders, Vlatko Vedral, Mio Murao, and Akimasa Miyake, Survival of entanglement in thermal states, Example 1