CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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A two-level generator fixes many states

Statement refuted

Let $d \geq 1$ and let $(T_{t})_{t \geq 0}$ be complex-linear maps on $M_{d} (C)$ that are completely positive (each finite ancilla amplification preserves positive matrices) and trace-preserving ( $Tr (T_{t} (X)) = Tr X$ for every $X$ ), with $T_{0} = id$ and $T_{t + s} = T_{t} \circ T_{s}$ for all $s, t \geq 0$ . A density matrix $ρ$ means $ρ = ρ^{*}$ , $v^{*} ρ v \geq 0$ for every $v \in C^{d}$ , and $Tr ρ = 1$ ; it is faithful when $v^{*} ρ v > 0$ for every nonzero $v \in C^{d}$ . For a Hermitian matrix $H \in M_{d} (C)$ and $β \in R$ , define $Z_{β} : = Tr (e^{- β H}) > 0$ and the Gibbs density $σ_{β} : = e^{- β H} / Z_{β}$ . A density matrix $σ$ is stationary for $(T_{t})$ when $T_{t} (σ) = σ$ for every $t \geq 0$ . For a density matrix $ρ$ and faithful density $σ$ , let $D (ρ ∥ σ) : = Tr (ρ \log ρ) - Tr (ρ \log σ)$ , using the natural logarithm, the finite spectral logarithm on the support of $ρ$ , and $0 \log 0 : = 0$ . The claim refuted here is that nonincrease of relative entropy to a faithful stationary Gibbs density is sufficient to guarantee that the Gibbs density is the unique stationary density and that every initial density converges to it.

Statement references

Stationary finite quantum semigroups decrease relative entropy

Finite completely positive trace-preserving maps

Finite GKSL generators and units

Finite GKSL generates CPTP semigroups

Facts & Assumptions

Units convention: In the refuted Gibbs formula, $H$ is an energy matrix with entries in joules and $β \in R$ is an inverse-energy parameter in joules $^{- 1}$ , so $β H$ is dimensionless. All real values of $β$ , including zero and negative values, are retained; no positive-temperature restriction is imposed.

Given: The space is $H = C^{2}$ with standard basis $e_{0} = (1, 0)^{T}$ , $e_{1} = (0, 1)^{T}$ , written $∣ j ⟩ = e_{j}$ ; $E_{j k}$ denotes the matrix unit defined by $E_{j k} e_{ℓ} = δ_{k ℓ} e_{j}$ . Let $Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ , $H = 0$ joules, $γ > 0$ have units seconds $^{- 1}$ , and $β \in R$ have units joules $^{- 1}$ . The generator acts on dimensionless matrices as $L (ρ) = γ (Z ρ Z - ρ)$ . Time $t$ is in seconds. An initial density has the form $ρ (0) = (\begin{array}{cc} p & z \\ \overline{z} & 1 - p \end{array})$ with $0 \leq p \leq 1$ and $∣ z ∣^{2} \leq p (1 - p)$ .

[A1]

Stationary finite quantum semigroups decrease relative entropy (this batch’s draft theorem; its current proof is presented here as a conditional input and remains subject to independent review): for a finite channel semigroup with a faithful stationary density, relative entropy to that density is nonincreasing.

[F1]

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

[F2]

Finite GKSL generators and units: a finite GKSL generator on dimensionless matrices has Hamiltonian term $- i [H, ρ] / ℏ$ and finite jump terms $J ρ J^{*} - \frac{1}{2} {J^{*} J, ρ}$ , with jump matrices in seconds $^{- 1 / 2}$ .

[F3]

Finite GKSL generates CPTP semigroups: a finite GKSL generator produces finite channel maps $T_{t} = e^{t L}$ for every $t \geq 0$ .

[F4]

Every finite-dimensional normed space is Banach: every finite-dimensional normed space is complete.

[F5]

The exponential series of a bounded operator: on a Banach space the operator exponential is given by its operator-norm convergent series.

Counterexample

1.1givenF1F2F3F4F5algebra

Let $J = \sqrt{γ} Z$ , which is Hermitian and has units seconds $^{- 1 / 2}$ , so $J^{*} J = γ I_{2}$ . Substitution of $H = 0$ and this jump into [F2] gives $J ρ J^{*} - \frac{1}{2} {J^{*} J, ρ} = γ (Z ρ Z - ρ) = L (ρ)$ . Use the max-entry norm on $M_{2} (C)$ ; the space is finite-dimensional and hence Banach by [F4]. Since $Z$ is diagonal with entries $1, - 1$ , $∥ L (X) ∥_{\max} \leq 2 γ ∥ X ∥_{\max}$ , so $L$ is bounded. Thus [F5] supplies the norm-convergent exponential series. By [F3] and [F1], this generator produces a finite channel semigroup.

1.2givenF5algebra

Since $H = 0$ , $e^{- β H} = I_{2}$ and $Z_{β} = Tr I_{2} = 2$ , so the faithful Gibbs density is $σ_{β} = I_{2} / 2 > 0$ . Direct multiplication gives $L (E_{00}) = L (E_{11}) = 0$ and $L (E_{01}) = - 2 γ E_{01}$ , $L (E_{10}) = - 2 γ E_{10}$ . Since $ρ (0) = p E_{00} + (1 - p) E_{11} + z E_{01} + \overline{z} E_{10}$ , applying the exponential series [F5] to these four eigenmatrices yields $T_{t} ρ (0) = p E_{00} + (1 - p) E_{11} + e^{- 2 γ t} z E_{01} + e^{- 2 γ t} \overline{z} E_{10} = (\begin{array}{cc} p & e^{- 2 γ t} z \\ e^{- 2 γ t} \overline{z} & 1 - p \end{array})$ . Every diagonal density is therefore stationary, including the faithful Gibbs density $I_{2} / 2$ and the distinct density $diag (1, 0)$ .

2.1givenA1step 1.1step 1.2∎

The hypotheses of [A1] hold by steps 1.1 and 1.2, so relative entropy to $I_{2} / 2$ is nonincreasing in this example. Yet $diag (1, 0)$ remains fixed and differs from $I_{2} / 2$ for every $t \geq 0$ . Hence the Gibbs density is not the unique stationary density, and this initial state does not converge to it, refuting both conclusions.

Remarks

The vertices are the chosen eigenvectors of $Z$ , and the generator leaves their population coordinates fixed. Because $H = 0$ , they belong to the same degenerate energy eigenspace. An arrow would represent a nonzero population-transfer rate; here no such arrow occurs, while the separate coherence mode decays exponentially.

\begin{tikzpicture}
\node (zero) at (0,0) {$|0\rangle$};
\node (one) at (3,0) {$|1\rangle$};
\node at (1.5,-0.7) {$\dot p_0=\dot p_1=0$};
\node at (1.5,0.7) {$\rho_{01}(t)=e^{-2\gamma t}\rho_{01}(0)$};
\end{tikzpicture}

The drawing records the population and coherence equations proved above; no proof step relies on it.

A two-level generator fixes many states

Statement refuted

Statement references

Facts & Assumptions

Counterexample

Remarks

Depends on

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