DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Dimensionless entropy of a finite density matrix

Definition

For a finite tensor density $ρ$ on $V_{N}$ , list its eigenvalues with multiplicity as $λ_{1}, \dots, λ_{q^{N}}$ . The finite Hermitian diagonalization in Finite Hermitian diagonalization and positivity of $ρ$ give $λ_{i} \geq 0$ and $\sum_{i} λ_{i} = 1$ . Define its dimensionless spectral entropy, measured in nats, by $s_{N} (ρ) : = - \sum_{i : λ_{i} > 0} λ_{i} \log λ_{i},$ with $0 \log 0 : = 0$ . This is independent of the ordering of the eigenvalues, including when eigenvalues are repeated. It contains no physical entropy unit; here $\log$ is the natural logarithm of The natural logarithm as the inverse of the exponential function, and a later physical postulate may assign $S_{N} = k_{B} s_{N}$ .

Dimensionless entropy of a finite density matrix

Definition

Depends on

Dependency tree · 2 levels

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