CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Amplitude damping can decrease system entropy

Statement refuted

Every finite quantum channel $Φ$ and every positive semidefinite trace-one matrix $ρ$ satisfy $S (Φ (ρ)) \geq S (ρ)$ .

Facts & Assumptions

Given: The finite channel and entropy definitions, $k_{B} > 0$ with units $J K^{- 1}$ , and the full Axiom of Choice as required by the KL supplier below.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. It is an explicit hypothesis of the published KL nonnegativity supplier used for the strict entropy comparison.

[F1]

Finite completely positive trace-preserving maps: a finite channel is a complex-linear completely positive trace-preserving map.

[F2]

Complete finite Choi-Kraus equivalence: a finite Kraus map $X \mapsto \sum_{a} K_{a} X K_{a}^{*}$ is completely positive, and it is trace-preserving when $\sum_{a} K_{a}^{*} K_{a} = I$ .

[F3]

Quantum spectral entropy and relative-entropy domains: for a positive trace-one matrix with eigenvalues $λ_{i}$ , $S (ρ) = - k_{B} \sum_{i} λ_{i} \ln λ_{i}$ , with $0 \ln 0 = 0$ .

[F4]

Kullback–Leibler divergence: for probability laws $P ≪ Q$ , $D (P ∥ Q) = \int \ln (d P / d Q) d P$ , using the natural logarithm.

[F5]

Kullback–Leibler divergence is nonnegative: under full AC, $D (P ∥ Q) \geq 0$ , with equality exactly when $P = Q$ .

[F6]

The real exponential function and the number $e$ by a power series: $\exp (x)$ is the real exponential function defined by its power series, with $\exp (0) = 1$ .

[F7]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (x) > 0$ for every real $x$ .

[F8]

The exponential function is strictly increasing: $x < y$ implies $\exp (x) < \exp (y)$ .

Counterexample

1.1givenF1F2F6F7F8algebra

Fix $γ > 0$ in $s^{- 1}$ and $t > 0$ in seconds, so $0 < e^{- γ t} < 1$ by [F6]–[F8]. On $C^{2}$ , define $K_{0} = diag (e^{- γ t / 2}, 1)$ , $K_{1} = \sqrt{1 - e^{- γ t}} ∣ 1 ⟩ ⟨ 0 ∣$ , and $Φ_{t} (X) = K_{0} X K_{0}^{*} + K_{1} X K_{1}^{*}$ . Since $K_{0}^{*} K_{0} + K_{1}^{*} K_{1} = I_{2}$ , [F2] and [F1] show that $Φ_{t}$ is a finite channel.

2.1givenA1F3F4F5F6F7F8step 1.1algebra∎

The matrix $ρ_{0} = I_{2} / 2$ is positive semidefinite with trace one. Direct multiplication gives $Φ_{t} (ρ_{0}) = diag (p, 1 - p)$ , where $p = e^{- γ t} / 2 \in (0, 1 / 2)$ by [F6]–[F8]. The entropy difference is $S (ρ_{0}) - S (Φ_{t} (ρ_{0})) = k_{B} (\ln 2 - h (p))$ , where $h (p) = - p \ln p - (1 - p) \ln (1 - p)$ . For the binary laws $P = (p, 1 - p)$ and $Q = (1 / 2, 1 / 2)$ , [F4] gives $D (P ∥ Q) = \ln 2 - h (p)$ ; since $P \neq Q$ , [F5] gives $D (P ∥ Q) > 0$ . With $k_{B} > 0$ , this yields $S (Φ_{t} (ρ_{0})) < S (ρ_{0})$ and refutes the universal claim.

Amplitude damping can decrease system entropy

Statement refuted

Facts & Assumptions

Counterexample

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