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Finite modular energy and entropy balance — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)Open item page →

Two-level entropy first variation

Example

On $H = C^{2}$ with its standard inner product and standard basis, fix $p \in (0, 1)$ and set $σ : = diag (p, 1 - p)$ . For $∣ ε ∣ < \min {p, 1 - p}$ define $ρ (ε) : = diag (p + ε, 1 - p - ε)$ and $s (ρ (ε)) : = - tr (ρ (ε) \log ρ (ε))$ . Then every $ρ (ε)$ is a positive-definite density matrix, $ρ (0) = σ$ , and

${\frac{d}{d ε} s (ρ (ε)) ∣}_{ε = 0} = \log ⁣ (\frac{1 - p}{p}) .$

Example references

Finite-dimensional entropy first variation at a faithful state

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm

The natural logarithm as the inverse of the exponential function

John Watrous, The Theory of Quantum Information, §5.2.1

Mark M. Wilde, From Classical to Quantum Shannon Theory, §11.1

Facts & Assumptions

Given: $H = C^{2}$ with its standard inner product and basis, $p \in (0, 1)$ , and $∣ ε ∣ < \min {p, 1 - p}$ .

[F1]

Finite-dimensional entropy first variation at a faithful state: for a differentiable density path through a faithful finite-dimensional reference, $s^{'} (0) = tr (ρ^{'} (0) (- \log σ))$ .

[F2]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is differentiable on $(0, \infty)$ with $\log^{'} (x) = 1 / x$ .

[F3]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: for $a, b > 0$ , $\log (a) - \log (b) = \log (a / b)$ .

[F4]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : sums, scalar multiples, and products of differentiable real functions have the usual sum, scalar, and product derivatives.

[F5]

The natural logarithm as the inverse of the exponential function: $\log$ is the real inverse of the exponential on $(0, \infty)$ .

Verification

1.1givenF1algebra

The bounds on $ε$ imply $p + ε > 0$ and $1 - p - ε > 0$ , and their sum is $1$ . Thus $ρ (ε)$ is a positive diagonal trace-one matrix, $σ = ρ (0)$ is strictly positive, and the path satisfies the finite-dimensional hypotheses in [F1].

1.2givenF2F3F4F5algebra

The eigenvalues give $s (ρ (ε)) = - (p + ε) \log (p + ε) - (1 - p - ε) \log (1 - p - ε)$ . By the difference-quotient formula in [F2], $\frac{d}{d ε} \log (p + ε) = 1 / (p + ε)$ and, after replacing the increment by its negative, $\frac{d}{d ε} \log (1 - p - ε) = - 1 / (1 - p - ε)$ . The product rule [F4] therefore gives $\frac{d}{d ε} s (ρ (ε)) = \log (1 - p - ε) - \log (p + ε)$ ; at $0$ , [F3] turns this into $\log ((1 - p) / p)$ .

2.1F1step 1.2algebra∎

Directly, $ρ^{'} (0) = diag (1, - 1)$ and $- \log σ = diag (- \log p, - \log (1 - p))$ , so $tr (ρ^{'} (0) (- \log σ)) = \log (1 - p) - \log p = \log ((1 - p) / p)$ . This agrees with the local calculation and the general result [F1].