ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Pure-state entropy and entropy of a dephased average

Example

Use $H = C^{2}$ with $⟨ z, w ⟩ = z_{1} \overline{w_{1}} + z_{2} \overline{w_{2}}$ , linear in the first argument, and $u_{1} = (1, 0)$ , $u_{2} = (0, 1)$ . Choose real $E_{1} \neq E_{2}$ and define the self-adjoint energy operator $A (z_{1}, z_{2}) = (E_{1} z_{1}, E_{2} z_{2})$ . Fix $ℏ > 0$ . For $t \in R$ set $U_{t} u_{j} = e^{- i E_{j} t / ℏ} u_{j}$ , and define $P_{j} z = ⟨ z, u_{j} ⟩ u_{j}$ , $ρ_{t} = U_{t} ρ U_{t}^{*}$ , and for $T \in (0, \infty)$ define ${\overset{ˉ}{ρ}}_{T} = T^{- 1} \int_{0}^{T} ρ_{t} d t$ by entrywise integration. All limits below are as $T \to + \infty$ . Let $ψ = (u_{1} + u_{2}) / \sqrt{2}$ , $ρ = ∣ ψ ⟩ ⟨ ψ ∣$ . Define the finite-dimensional entropy of a density matrix with eigenvalues $p_{i}$ as $S (ρ) = - k_{B} \sum_{i} p_{i} \log p_{i}$ , with $0 \log 0 = 0$ and fixed $k_{B} > 0$ . Every instantaneous unitary state $ρ_{t}$ is pure and has entropy zero. The Cesàro states converge to $ρ_{d i a g} = (P_{1} + P_{2}) / 2$ , whose entropy is $k_{B} \log 2$ . In fact $S ({\overset{ˉ}{ρ}}_{T}) \to k_{B} \log 2$ , whereas $T^{- 1} \int_{0}^{T} S (ρ_{t}) d t = 0$ for every $T > 0$ . Entropy of the averaged state and average instantaneous entropy are distinct quantities.

Facts & Assumptions

Given: The two-level discrete-spectrum model, pure initial density, and entropy convention in the statement.

[A1]

$u_{1}, u_{2}$ are orthonormal, $E_{1} \neq E_{2}$ , $ℏ > 0$ , $t \in R$ , $T > 0$ , and $ρ = ∣ ψ ⟩ ⟨ ψ ∣$ for $ψ = (u_{1} + u_{2}) / \sqrt{2}$ .

[F1]

$\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The complex exponential is entire and its complex derivative is itself, and The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ : imaginary exponential phases have modulus one, their derivatives give the explicit antiderivative, and the real and imaginary integrals are evaluated componentwise. The finite-dimensional Cesàro limit is computed here and does not invoke an uninstantiated infinite-dimensional Gibbs-spectrum theorem.

[F2]

Real and complex inner-product spaces and their induced length gives the orthonormal-coordinate inner product; The complex exponential by its power series defines the complex exponential; $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ gives $∣ e^{i y} ∣ = e^{0} = 1$ for real $y$ .

[F3]

The natural logarithm as the inverse of the exponential function and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is continuous on positive reals and $\log 2$ is finite.

Verification

Given: [A1] and the entropy convention.

1.1A1F2algebra

The unitary evolution sends $ψ$ to $(e^{- i E_{1} t / ℏ} u_{1} + e^{- i E_{2} t / ℏ} u_{2}) / \sqrt{2}$ , which remains a unit vector by [F2]. Its rank-one density has eigenvalues $1, 0$ , so its entropy is $- k_{B} (1 \log 1 + 0 \log 0) = 0$ for every $t$ .

2.1A1F1F3step 1.1

The two energy eigenspaces are one-dimensional. The matrix entry between them is an exponential with nonzero frequency; by [F1] its average has magnitude at most $ℏ / (T ∣ E_{1} - E_{2} ∣)$ and tends to zero. The diagonal entries remain $1 / 2$ , so the direct finite-dimensional average yields $ρ_{d i a g} = \frac{1}{2} P_{1} + \frac{1}{2} P_{2}$ . Its eigenvalues are $(1 / 2, 1 / 2)$ , so $S (ρ_{d i a g}) = - k_{B} (\frac{1}{2} \log \frac{1}{2} + \frac{1}{2} \log \frac{1}{2}) = k_{B} \log 2.$ Since every $S (ρ_{t}) = 0$ , the time average of instantaneous entropy is exactly zero.

3.1A1F2F3step 2.1algebra∎

In the basis $(u_{1}, u_{2})$ , ${\overset{ˉ}{ρ}}_{T}$ has diagonal entries $1 / 2$ and off-diagonal entry $c_{T} = \frac{1}{2 T} \int_{0}^{T} e^{- i (E_{1} - E_{2}) t / ℏ} d t,$ whose magnitude is at most $ℏ / (T ∣ E_{1} - E_{2} ∣)$ by direct integration. Its eigenvalues are $1 / 2 \pm ∣ c_{T} ∣$ , so both tend to $1 / 2$ . Continuity of $x \mapsto - x \log x$ at $1 / 2$ gives $S ({\overset{ˉ}{ρ}}_{T}) \to k_{B} \log 2$ . Thus entropy of the averaged state has a nonzero limit even though the averaged instantaneous entropy remains zero.

Pure-state entropy and entropy of a dephased average

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources