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The finite-temperature trace entropy is continuous at occupation endpoints

Statement

For $ϕ (x) = x \log x + (1 - x) \log (1 - x)$ with $0 \log 0 = 0$ , the map $D \mapsto tr ϕ (D)$ is continuous on the full domain $D_{r}$ , including matrices with eigenvalues $0$ or $1$ .

Facts & Assumptions

Given: The continuous scalar extension $ϕ$ on $[0, 1]$ and Hermitian matrices $D \in D_{r}$ .

[F1]

The entropy integrand and endpoint convention are as in the finite-temperature model (Fixed-particle finite-temperature density-matrix free energy).

[F2]

Ordered eigenvalues of complex Hermitian matrices change by at most the operator norm of the matrix perturbation (Ordered eigenvalues of Hermitian matrices are operator-norm Lipschitz).

[F3]

The natural logarithm satisfies $\log^{'} (x) = 1 / x$ and $\log x = \int_{1}^{x} d s / s$ for $x > 0$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t).

[F4]

For real or complex matrices, the induced operator norm is bounded by the Frobenius norm, $∥ A ∥_{2} \leq ∥ A ∥_{F}$ (Spectral and Frobenius norms are unitarily invariant, are given by singular values, and satisfy the sharp rank comparison).

Proof

1.1F1F3algebra

For $0 < x < 1$ , put $t = 1 / x$ . By [F3], $0 \leq \log t = \int_{1}^{t} d s / s \leq \int_{1}^{t} d s / \sqrt{s} = 2 (\sqrt{t} - 1)$ , so $∣ x \log x ∣ = (\log t) / t \leq 2 / \sqrt{t} \to 0$ . The other term $(1 - x) \log (1 - x)$ tends to $0$ as $x ↓ 0$ by continuity of $y \mapsto y \log y$ at $y = 1$ . Since $ϕ (1 - x) = ϕ (x)$ , the same limits hold at $1$ . Thus the stipulated endpoint values make $ϕ$ continuous on $[0, 1]$ .

1.2F1F2F4given

By [F2], if $D_{k} \to D$ in Frobenius norm, then every ordered eigenvalue $d_{k, j}$ tends to the corresponding $d_{j}$ ; indeed [F4] gives $∥ D_{k} - D ∥_{2} \leq ∥ D_{k} - D ∥_{F} \to 0$ . All these eigenvalues lie in $[0, 1]$ by the domain definition.

2.1F1step 1.1step 1.2algebra∎

Spectral functional calculus gives $tr ϕ (D_{k}) = \sum_{j = 1}^{n} ϕ (d_{k, j})$ and likewise for $D$ . Scalar continuity from step 1.1 makes each of the finitely many summands converge, so $tr ϕ (D_{k}) \to tr ϕ (D)$ .

Depends on

Used by

Cited to discharge well-definedness by Fixed-particle finite-temperature density-matrix free energy.

Dependency tree · two levels

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Sources

Daniel Arovas, Thermodynamics and Statistical Mechanics, §7.4 Variational Density Matrix Method, UC San Diego / Physics LibreTexts (standard reference, not scraped)
Christoph Helmberg, Guilherme Porto, Guilherme Torres, and Vilmar Trevisan, An interlacing property of the signless Laplacian of threshold graphs, §2, Theorem 2 (2023) (standard reference, not scraped)