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Specified finite GNS detailed balance

1 · Prerequisites

2 · Summary

This page fixes one finite GNS detailed-balance convention: the Hamiltonian part commutes with the faithful Gibbs state, while the dissipator is self-adjoint in the specified weighted matrix inner product. A scalar dissipative kernel gives an explicit gap, centered correlation integrability, and trace-norm relaxation. Connected Gibbs jump rates supply a concrete finite model; the commuting relative entropy production and monitored-record variance are proved for that model.

The bath state law and energy-resolved instrument are adopted effective assumptions with physical applicability only on the finite resolved interval $[0, t_{r e s}]$ specified by the postulate. All-time relaxation estimates and long-time coefficients are mathematical consequences of continuing the same finite generator and selected chain; they carry no physical-applicability claim beyond $t_{r e s}$ . The instrument specifies one record law; the one-time GKSL semigroup alone does not determine a unique monitoring record. All results are finite-dimensional conditional model statements. No microscopic bath limit, measured rate, or experimental observation is asserted.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: AI-generatedjudge pass (gpt-6-luna)Open item page →

Specified finite GNS detailed balance

Definition

Let $d \geq 1$ be finite, let $H = C^{d}$ with its standard inner product, and write $M_{d} (C)$ for its matrices. Let $H_{s} = H_{s}^{*} \in M_{d} (C)$ be a Hamiltonian in joules, let $ℏ > 0$ have units joule-seconds, and let $m \geq 0$ be an integer with $L_{1}, \dots, L_{m} \in M_{d} (C)$ a finite family of jump matrices in seconds $^{- 1 / 2}$ . Matrices on which the generators act are dimensionless. The Schrödinger generator $L$ is the finite GKSL generator of Finite GKSL generators and units.

Its trace-dual observable generator is

$G (A) = \frac{i}{ℏ} [H_{s}, A] + D (A), D (A) = \sum_{j = 1}^{m} (L_{j}^{*} A L_{j} - \frac{1}{2} {L_{j}^{*} L_{j}, A}) .$

Here $[U, V] = U V - V U$ , ${U, V} = U V + V U$ , and $A^{*}$ is the conjugate transpose. The units of $G$ , $D$ , and $L$ are seconds $^{- 1}$ . The generator preserves the identity, $D (I) = 0$ , so $e^{t D} (I) = I$ .

Let $σ = σ^{*} > 0$ be a dimensionless density matrix on $H$ , so $Tr σ = 1$ . Define

$⟨ A, B ⟩_{σ} : = Tr (σ B^{*} A), ∥ A ∥_{σ} : = ⟨ A, A ⟩_{σ}^{1 / 2} .$

This inner product is linear in its first variable. It is positive definite because $σ > 0$ . The data satisfy the specified finite GNS detailed-balance convention when $[H_{s}, σ] = 0$ and $D$ is self-adjoint for this inner product, meaning $⟨ D (A), B ⟩_{σ} = ⟨ A, D (B) ⟩_{σ}$ for all $A, B \in M_{d} (C)$ . The Hamiltonian part of $G$ is kept separate and is anti-self-adjoint. Quantum detailed-balance conventions vary; in particular, this definition does not identify GNS symmetry with KMS or Kubo–Mori symmetry.

Facts & Assumptions

Given: The finite GKSL data $d, H_{s}, ℏ, (L_{j})_{j = 1}^{m}$ , a faithful density matrix $σ$ , and the finite trace.

[F1]

Finite GKSL generators and units: $L (X) = - \frac{i}{ℏ} [H_{s}, X] + \sum_{j} (L_{j} X L_{j}^{*} - \frac{1}{2} {L_{j}^{*} L_{j}, X})$ .

[F2]

Finite matrix spectral and state calculus: finite Hermitian matrices have basis-independent spectral calculus; a finite density is positive semidefinite with trace one.

[F3]

Finite matrix trace cyclicity and nondegeneracy: the finite trace cycles rectangular and square matrix products.

Proof

1.1givenF2F3algebra

The square root $σ^{1 / 2}$ exists by [F2] and is invertible because $σ > 0$ . Thus $⟨ A, A ⟩_{σ} = Tr ((A σ^{1 / 2})^{*} (A σ^{1 / 2})) > 0$ for every nonzero $A$ . Linearity in the first variable is immediate, and cyclicity [F3] gives Hermitian symmetry; hence the displayed form is a positive-definite inner product.

2.1givenF1F3step 1.1algebra

For the Hamiltonian term, cyclicity in [F3] gives $Tr (- i [H_{s}, X] A / ℏ) = Tr (X i [H_{s}, A] / ℏ)$ . Indeed, $Tr (H_{s} X A) = Tr (X A H_{s})$ and $Tr (X H_{s} A)$ is already in the desired input-first order.

3.1givenF1F3step 2.1algebra∎

For each jump matrix, $Tr (L_{j} X L_{j}^{*} A) = Tr (X L_{j}^{*} A L_{j})$ . The first anticommutator term satisfies $Tr (L_{j}^{*} L_{j} X A) = Tr (X A L_{j}^{*} L_{j})$ by cyclicity, and the second is already $Tr (X L_{j}^{*} L_{j} A)$ . Thus the trace-dual dissipator is exactly the displayed $D$ . Substituting $A = I$ cancels $L_{j}^{*} L_{j} - \frac{1}{2} (L_{j}^{*} L_{j} + L_{j}^{*} L_{j}) = 0$ term by term, so $D (I) = 0$ .

Remarks

The cited Alicki–Lendi convention uses the same dissipative self-adjointness condition with the Hamiltonian part separated. Its inner product is written conjugate-linear in the first variable; swapping the two slots and conjugating gives the first-linear convention used here, without changing self-adjointness. Carlen–Maas explicitly distinguish the GNS inner product from the KMS inner product. No equivalence with another detailed-balance convention is assumed.

TheoremStatement: Literature-sourcedProof: AI-generatedOpen item page →

Faithful scalar-kernel detailed balance gives a relaxation gap

Statement

Let $d \geq 1$ , and let $G = K + D$ and the faithful density matrix $σ$ satisfy Specified finite GNS detailed balance, with $\ker D = C I$ . Write $M_{0} = {A \in M_{d} (C) : Tr (σ A) = 0}$ . If $d > 1$ , the restriction $D ∣_{M_{0}}$ has only strictly negative eigenvalues. Define

$γ : = \min {- λ : λ \in spec (D ∣_{M_{0}})} > 0.$

Then for every $A \in M_{0}$ and $t \geq 0$ ,

$∥ e^{t G} A ∥_{σ} \leq e^{- γ t} ∥ A ∥_{σ} .$

For every density matrix $ρ$ , let $ρ_{t} = e^{t L} ρ$ , where $L$ is the Schrödinger trace-dual generator. Then

$∥ ρ_{t} - σ ∥_{1} \leq ∥ σ^{- 1 / 2} (ρ - σ) ∥_{2} e^{- γ t} .$

Here $∥ X ∥_{1} = Tr \sqrt{X^{*} X}$ and $∥ X ∥_{2} = (Tr X^{*} X)^{1 / 2}$ . For centered $A, B \in M_{0}$ , the correlation $t \mapsto ⟨ e^{t G} A, B ⟩_{σ}$ is absolutely integrable. Also

$Γ (A) : = Re \int_{0}^{\infty} ⟨ e^{t G} A, A ⟩_{σ} d t = Re ⟨ (- G ∣_{M_{0}})^{- 1} A, A ⟩_{σ} \geq 0.$

For $d = 1$ , $M_{0} = {0}$ , $ρ = σ$ , and all centered and relaxation claims are trivial; no $γ$ is defined. The units of $γ$ are seconds $^{- 1}$ . These are finite-dimensional conditional model results; they assert no infinite-dimensional bath limit.

Facts & Assumptions

Given: The finite observable generator $G = K + D$ , faithful density $σ$ , self-adjoint dissipator $D$ , $[H_{s}, σ] = 0$ , and $\ker D = C I$ .

[F1]

Specified finite GNS detailed balance: $D (I) = 0$ , $D$ is self-adjoint for $⟨ A, B ⟩_{σ} = Tr (σ B^{*} A)$ , and $K (A) = i [H_{s}, A] / ℏ$ with $[H_{s}, σ] = 0$ .

[F2]

Finite GKSL generates CPTP semigroups: the finite Schrödinger semigroup $e^{t L}$ is completely positive and trace-preserving for $t \geq 0$ .

[F3]

Channel duals and norm closure: the trace-dual of a finite channel is unital and completely positive.

[F4]

Schwarz and trace-norm channel contraction: a unital completely positive map satisfies the Schwarz inequality; its proof also establishes $∥ X ∥_{1} = \sup_{∥ A ∥_{\infty} \leq 1} ∣ Tr (A X) ∣$ .

[F5]

Finite matrix spectral and state calculus: finite self-adjoint endomorphisms have an orthonormal eigenbasis; the finite weighted inner product satisfies Cauchy–Schwarz.

[F6]

Finite matrix trace cyclicity and nondegeneracy: finite matrix traces are cyclic.

Proof

1.1F1F6givenalgebra

If $d = 1$ , then $M_{0} = {0}$ and the only density matrix is $σ$ , so all assertions are immediate. For the remainder assume $d > 1$ . By [F1], self-adjointness and $D (I) = 0$ give $Tr (σ D (A)) = ⟨ D (A), I ⟩_{σ} = ⟨ A, D (I) ⟩_{σ} = 0$ . For the Hamiltonian part, cyclicity [F6] and $[H_{s}, σ] = 0$ give $Tr (σ K (A)) = 0$ . Since $G$ is the trace dual of $L$ , differentiating $Tr (σ e^{t G} A)$ now shows it is constant. Applying the same calculation with $K = 0$ shows $Tr (σ e^{t D} A) = Tr (σ A)$ . Thus $σ$ is stationary for both dual semigroups.

2.1F1F2F3F4F6step 1.1algebra

Let $L_{D} (X) = \sum_{j} (L_{j} X L_{j}^{*} - \frac{1}{2} {L_{j}^{*} L_{j}, X})$ , the Schrödinger GKSL generator with zero Hamiltonian and the same jump matrices. The finite trace calculation in [F1], [F6] shows its trace dual is $D$ . Applying [F2] to $L_{D}$ and [F3] to its channel semigroup makes $e^{t D}$ unital and completely positive. The Schwarz inequality [F4] and stationarity from step 1.1 give

∥ e^{t D} A ∥_{σ}^{2} = Tr (σ (e^{t D} A)^{*} e^{t D} A) \leq Tr (σ e^{t D} (A^{*} A)) = ∥ A ∥_{σ}^{2} .

Differentiating at $t = 0$ yields $Re ⟨ D (A), A ⟩_{σ} \leq 0$ ; self-adjointness in [F1] makes this quantity real.

3.1givenF1F5step 2.1algebra

The subspace $M_{0}$ is the orthogonal complement of $C I$ , and [F1] gives $D (M_{0}) \subseteq M_{0}$ . The finite-dimensional spectral theorem [F5] applies to the self-adjoint endomorphism $D$ on $(M_{d}, ⟨ \cdot, \cdot ⟩_{σ})$ . Step 2.1 makes all eigenvalues nonpositive; $\ker D = C I$ excludes zero on $M_{0}$ . Because $d > 1$ , $M_{0} \neq {0}$ , so its finite nonempty list of eigenvalues has maximum $- γ < 0$ , and $⟨ D (A), A ⟩_{σ} \leq - γ ∥ A ∥_{σ}^{2}$ .

4.1givenF1F5F6step 1.1step 3.1algebra

By [F6] and $[H_{s}, σ] = 0$ , $K$ is anti-self-adjoint for $⟨ \cdot, \cdot ⟩_{σ}$ . Both $K$ and $D$ preserve $M_{0}$ : for $D$ this follows from step 1.1, and for $K$ from the same trace-cyclicity calculation. For $A_{t} = e^{t G} A$ with centered $A$ , differentiating gives

\frac{d}{d t} ∥ A_{t} ∥_{σ}^{2} = 2 Re ⟨ G (A_{t}), A_{t} ⟩_{σ} = 2 ⟨ D (A_{t}), A_{t} ⟩_{σ} \leq - 2 γ ∥ A_{t} ∥_{σ}^{2} .

Thus $e^{2 γ t} ∥ A_{t} ∥_{σ}^{2}$ is nonincreasing, proving the asserted decay. Cauchy–Schwarz [F5] then bounds each centered correlation by $e^{- γ t} ∥ A ∥_{σ} ∥ B ∥_{σ}$ , an integrable function.

5.1step 4.1algebra

The decay in step 4.1 makes $Y : = \int_{0}^{\infty} e^{t G} A d t$ converge in the finite-dimensional space $M_{0}$ . Differentiating $e^{t G} A$ and integrating gives $- G Y = A$ . If $G Z = 0$ for centered $Z$ , then step 4.1 gives $∥ Z ∥_{σ} = ∥ e^{t G} Z ∥_{σ} \leq e^{- γ t} ∥ Z ∥_{σ}$ , so $Z = 0$ ; hence $- G ∣_{M_{0}}$ is invertible and $Y = (- G ∣_{M_{0}})^{- 1} A$ .

6.1F1step 3.1step 5.1algebra

Let $Y$ be as in step 5.1. Then $Γ (A) = Re ⟨ Y, - G (Y) ⟩_{σ}$ . The anti-self-adjoint $K$ contributes zero real part, while step 3.1 gives $⟨ Y, - D (Y) ⟩_{σ} \geq γ ∥ Y ∥_{σ}^{2} \geq 0$ . Therefore $Γ (A) \geq 0$ .

7.1F1F4F5F6step 1.1step 4.1algebra∎

Let $X = ρ - σ = X^{*}$ , and take any $A$ with $∥ A ∥_{\infty} \leq 1$ . Set $A_{0} = A - Tr (σ A) I$ . Stationarity and trace duality give $Tr ((ρ_{t} - σ) A) = Tr (X e^{t G} A_{0})$ . By cyclicity [F6], the trace-norm dual formula [F4] applies to this product order. Put $C = e^{t G} A_{0}$ . Since $Tr (X C) = Tr (C X) = Tr ((C σ^{1 / 2}) (σ^{- 1 / 2} X))$ , the Hilbert–Schmidt trace inequality $∣ Tr (U V) ∣ \leq ∥ U ∥_{2} ∥ V ∥_{2}$ yields

∣ Tr (X e^{t G} A_{0}) ∣ \leq ∥ σ^{- 1 / 2} X ∥_{2} ∥ e^{t G} A_{0} ∥_{σ} \leq ∥ σ^{- 1 / 2} X ∥_{2} e^{- γ t} ∥ A_{0} ∥_{σ} .

The identity $∥ A_{0} ∥_{σ}^{2} = Tr (σ A^{*} A) - ∣ Tr (σ A) ∣^{2} \leq ∥ A ∥_{\infty}^{2} \leq 1$ gives a bound independent of $A$ . Taking the supremum in [F4] proves the stated trace-norm relaxation estimate.

Remarks

The proof uses only finite matrices. In particular, the decay of $e^{t G}$ follows from the dissipator's gap even when $K$ and $D$ do not commute. The correlation integral is a model coefficient, not a measured transport quantity.

TheoremStatement: Literature-sourcedProof: AI-generatedjudge pass (gpt-6-luna)Open item page →

Connected Gibbs jump generators satisfy detailed balance

Statement

Let $d \geq 2$ , let $E = {1, \dots, d}$ , let $E_{a} \in R$ be finite energies in joules, and let $β > 0$ have units joule $^{- 1}$ . Define

$Z_{β} : = \sum_{a \in E} e^{- β E_{a}}, π_{a} : = Z_{β}^{- 1} e^{- β E_{a}} .$

For each $a \neq b$ , let $w_{a b} \geq 0$ be a rate in seconds $^{- 1}$ satisfying Gibbs detailed balance $π_{a} w_{a b} = π_{b} w_{b a}$ , and assume the undirected graph with edge ${a, b}$ whenever $w_{a b} > 0$ is connected. On $H = C^{d}$ , fix an orthonormal energy basis $(∣ a ⟩)$ , set $H_{s} = \sum_{a} E_{a} ∣ a ⟩ ⟨ a ∣$ , and define $L_{a b} = \sqrt{w_{a b}} ∣ b ⟩ ⟨ a ∣$ for $a \neq b$ . Fix $ℏ > 0$ in joule-seconds. For real functions $f, g : E \to R$ , define $⟨ f, g ⟩_{π} : = \sum_{a} π_{a} f_{a} g_{a}$ , and define the row-generator action on column observables by $(W f)_{a} : = \sum_{b \neq a} w_{a b} (f_{b} - f_{a})$ . Let

$σ : = \sum_{a} π_{a} ∣ a ⟩ ⟨ a ∣, r_{a} : = \sum_{b \neq a} w_{a b} .$

Use the finite GKSL generator with Hamiltonian $H_{s}$ and the jump matrices $L_{a b}$ . Its observable dissipator $D$ is self-adjoint for the $σ$ -GNS inner product of Specified finite GNS detailed balance, $H_{s}$ commutes with $σ$ , and $\ker D = C I$ . On diagonal observables, $W$ is self-adjoint on $L^{2} (E, π)$ with constants as its kernel; for $d \geq 2$ its restriction to $π^{⊥} : = {f : E \to R : \sum_{a} π_{a} f_{a} = 0}$ has a positive spectral gap $γ_{c l}$ . Thus the generator satisfies the specified finite GNS detailed-balance convention and the generic gap conclusion of Faithful scalar-kernel detailed balance gives a relaxation gap applies. This claim does not assert a microscopic bath limit.

Facts & Assumptions

Given: $d \geq 2$ , the energy basis, $E_{a}$ , $β > 0$ , and rates $w_{a b}$ satisfying the finite connected Gibbs detailed-balance hypotheses above.

[F1]

Specified finite GNS detailed balance: the observable dissipator is $D (A) = \sum_{j} (L_{j}^{*} A L_{j} - \frac{1}{2} {L_{j}^{*} L_{j}, A})$ ; GNS detailed balance means $D$ is self-adjoint for $Tr (σ B^{*} A)$ and the Hamiltonian commutes with $σ$ .

[F2]

Finite GKSL generates CPTP semigroups: the finite GKSL generator determined by $H_{s}$ and the finite family $L_{a b}$ generates CPTP maps.

[F3]

Faithful scalar-kernel detailed balance gives a relaxation gap: if a faithful finite GNS detailed-balance model has $\ker D = C I$ , then $D ∣_{M_{0}}$ has a strictly positive gap and its semigroup has the stated exponential relaxation and correlation bounds.

[F4]

Finite matrix spectral and state calculus: finite Hermitian matrices have an orthonormal eigenbasis and the weighted state form is an inner product when the density is faithful.

[F5]

Finite matrix trace cyclicity and nondegeneracy: finite traces are cyclic.

Proof

1.1givenF2algebra

The finite sum $Z_{β}$ is strictly positive, so each $π_{a} > 0$ and $\sum_{a} π_{a} = 1$ . Hence $σ > 0$ , $Tr σ = 1$ , and $[H_{s}, σ] = 0$ . The finite GKSL semigroup exists by [F2].

1.2givenF1F5algebra

Put $R = \sum_{x \neq y} L_{x y}^{*} L_{x y} = \sum_{a} r_{a} ∣ a ⟩ ⟨ a ∣$ . For a diagonal density $ρ = \sum_{a} p_{a} ∣ a ⟩ ⟨ a ∣$ , $L_{x y} ρ L_{x y}^{*} = w_{x y} p_{x} ∣ y ⟩ ⟨ y ∣$ , while the anticommutator contributes $- r_{a} p_{a} ∣ a ⟩ ⟨ a ∣$ . Summing entries gives the Schrödinger population equation

{\dot{p}}_{a} = \sum_{b \neq a} (p_{b} w_{b a} - p_{a} w_{a b}),

For a diagonal observable $f = \sum_{a} f_{a} ∣ a ⟩ ⟨ a ∣$ , direct multiplication gives

D (f) = \sum_{a} (\sum_{b \neq a} w_{a b} (f_{b} - f_{a})) ∣ a ⟩ ⟨ a ∣,

so its diagonal action is exactly $W$ . For $a \neq b$ , every gain term $L_{x y} ∣ a ⟩ ⟨ b ∣ L_{x y}^{*}$ vanishes; the $R$ anticommutator contributes $- (r_{a} + r_{b}) ∣ a ⟩ ⟨ b ∣ / 2$ , and the Hamiltonian commutator contributes $- i (E_{a} - E_{b}) ∣ a ⟩ ⟨ b ∣ / ℏ$ . Thus the coherence equation is

{\dot{ρ}}_{a b} = (- \frac{i (E_{a} - E_{b})}{ℏ} - \frac{r_{a} + r_{b}}{2}) ρ_{a b} .

For $c_{a b} : = π_{a} w_{a b} = c_{b a}$ and sums over $a \neq b$ , the cross terms satisfy $\sum_{a \neq b} c_{a b} f_{b} g_{a} = \sum_{a \neq b} c_{a b} f_{a} g_{b}$ after swapping $a, b$ , so $⟨ W f, g ⟩_{π} = ⟨ f, W g ⟩_{π}$ . Symmetrizing the quadratic form gives

- ⟨ W f, f ⟩_{π} = \frac{1}{2} \sum_{a} \sum_{b \neq a} π_{a} w_{a b} ∣ f_{b} - f_{a} ∣^{2},

so $W$ is self-adjoint on the diagonal subspace.

2.1F1F4step 1.2algebra

For $a \neq b$ , direct multiplication with diagonal $σ$ gives $⟨ ∣ a ⟩ ⟨ b ∣, ∣ c ⟩ ⟨ d ∣ ⟩_{σ} = 0$ unless $(a, b) = (c, d)$ , and $∥ ∣ a ⟩ ⟨ b ∣ ∥_{σ}^{2} = π_{b}$ . Also each jump gain $L_{x y}^{*} ∣ a ⟩ ⟨ b ∣ L_{x y}$ is zero when $a \neq b$ , and the anticommutator with $R$ gives $D (∣ a ⟩ ⟨ b ∣) = - (r_{a} + r_{b}) ∣ a ⟩ ⟨ b ∣ / 2$ . Diagonal and off-diagonal subspaces are orthogonal. Together with the diagonal self-adjointness from step 1.2, these real eigenvalues prove that $D$ is self-adjoint on all of $M_{d} (C)$ .

3.1givenF1F4step 1.2step 2.1algebra

Connectivity and $d \geq 2$ imply $r_{a} > 0$ for each $a$ , so every off-diagonal matrix unit has a strictly negative $D$ -eigenvalue. The Dirichlet identity in step 1.2 vanishes only when $f_{a} = f_{b}$ along every positive-rate edge; connectedness then forces $f$ to be constant. Therefore the diagonal kernel is $C I$ , the off-diagonal kernel is zero, and $\ker D = C I$ . Applying the finite spectral theorem [F4] to the self-adjoint $W ∣_{π^{⊥}}$ shows its finite list of eigenvalues is strictly negative; their minimum absolute value is $γ_{c l} > 0$ .

4.1F1F3step 1.1step 1.2step 2.1step 3.1∎

Steps 1.1, 1.2, 2.1, and 3.1 verify every hypothesis of Specified finite GNS detailed balance and [F3]. Applying [F3] yields the positive gap, exponential relaxation, trace-norm bound, and absolutely integrable centered correlations for this Gibbs jump generator.

Remarks

This is a finite mathematical rate model. Gibbs detailed balance specifies a stationary state and a symmetric generator; it does not derive the rates from a microscopic reservoir or establish that a physical system realizes this model.

DefinitionDefinition: Literature-sourcedProof: AI-generatedOpen item page →

Specified energy-jump instrument realizes a finite chain

Definition

Use the finite energy basis, energies $E_{a}$ , rates $w_{a b}$ , Hamiltonian $H_{s}$ , and jump matrices $L_{a b} = \sqrt{w_{a b}} ∣ b ⟩ ⟨ a ∣$ of Connected Gibbs jump generators satisfy detailed balance. Define $R : = \sum_{a \neq b} L_{a b}^{*} L_{a b} = \sum_{a} r_{a} ∣ a ⟩ ⟨ a ∣$ and the no-click matrix $K : = - i H_{s} / ℏ - R / 2$ . The specified energy-resolved instrument has the following outcomes:

If the current state is the energy eigenstate $∣ a ⟩$ , then during a no-click interval of length $t$ the unnormalized state is $e^{t K} ∣ a ⟩$ .
In an infinitesimal interval $d t$ , the outcome labelled $a \to b$ has jump operation $\sqrt{d t} L_{a b}$ and probability $d t ⟨ a ∣ L_{a b}^{*} L_{a b} ∣ a ⟩ + o (d t)$ . For $w_{a b} > 0$ , its normalized post-click state is $L_{a b} ∣ a ⟩ / ∥ L_{a b} ∣ a ⟩ ∥ = ∣ b ⟩$ . If $w_{a b} = 0$ , this branch is the zero operation and has exactly zero probability; no conditional state is assigned.
The no-click operation over $d t$ is $M_{0} (d t) = I + K d t + o (d t)$ . Since $K^{*} + K = - R$ and $\sum_{a \neq b} L_{a b}^{*} L_{a b} = R$ , the no-click and jump effects sum to $I + o (d t)$ .
After each recorded jump, repeat the same rule from the new energy label.

The output path records the current label $a (t) \in E$ . This is one selected instrument model; the one-time GKSL semigroup alone does not specify a unique record process.

Facts & Assumptions

Given: The Axiom of Choice, the connected finite Gibbs jump data, the jump operators $L_{a b}$ , and the finite-state chain supplier.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. The full axiom is retained because the supplied chain construction uses Ionescu–Tulcea.

[F1]

Connected Gibbs jump generators satisfy detailed balance: $E$ is finite, the positive-rate graph is connected, $r_{a} = \sum_{b \neq a} w_{a b} > 0$ , and the energy-resolved jump matrix is $\sqrt{w_{a b}} ∣ b ⟩ ⟨ a ∣$ .

[F2]

Finite GKSL generators and units: the Schrödinger generator for Hamiltonian $H_{s}$ and jumps $L_{a b}$ is $L (X) = - i [H_{s}, X] / ℏ + \sum_{a \neq b} (L_{a b} X L_{a b}^{*} - \frac{1}{2} {L_{a b}^{*} L_{a b}, X})$ .

[F3]

Nonexplosive finite continuous-time chain and reversible spectral gap: under full AC, the finite connected reversible rates construct a nonexplosive continuous-time Markov chain with waiting rate $r_{a}$ , destination probabilities $w_{a b} / r_{a}$ , and transition matrix $e^{t W}$ .

Proof

1.1givenF1constructalgebra

The jump matrices give $L_{a b}^{*} L_{a b} = w_{a b} ∣ a ⟩ ⟨ a ∣$ , so $R = \sum_{a} r_{a} ∣ a ⟩ ⟨ a ∣$ and $K^{*} + K = - R$ . Hence $M_{0} (d t)^{*} M_{0} (d t) = I - R d t + o (d t)$ , while the sum of jump effects is $R d t$ ; the total effect is $I + o (d t)$ as required for a normalized infinitesimal instrument. Also $K ∣ a ⟩ = (- i E_{a} / ℏ - r_{a} / 2) ∣ a ⟩$ , so $∥ e^{t K} ∣ a ⟩ ∥^{2} = e^{- r_{a} t}$ . The no-click survival probability is exactly the exponential holding-time law of rate $r_{a}$ .

1.2givenF1algebra

For a short interval $d t$ , the instrument outcome $a \to b$ has probability $d t ⟨ a ∣ L_{a b}^{*} L_{a b} ∣ a ⟩ + o (d t) = w_{a b} d t + o (d t)$ . Summing over $b \neq a$ gives total jump rate $r_{a}$ , and conditioning on a jump gives destination probability $w_{a b} / r_{a}$ . For $w_{a b} > 0$ the post-jump normalized state is $∣ b ⟩$ ; for $w_{a b} = 0$ the jump operation is zero, so that outcome never occurs and is not normalized.

2.1F1F2step 1.1algebra

For an arbitrary matrix $X$ , the no-click branch contributes $M_{0} (d t) X M_{0} (d t)^{*} = X + d t (K X + X K^{*}) + o (d t)$ , and the jump branches contribute $d t \sum_{a \neq b} L_{a b} X L_{a b}^{*} + o (d t)$ . Since $K = - i H_{s} / ℏ - R / 2$ and $R = \sum_{a \neq b} L_{a b}^{*} L_{a b}$ , their sum is $X + d t L (X) + o (d t)$ with exactly the generator in [F2]. Thus the selected infinitesimal outcomes have the specified GKSL generator.

3.1A1F3step 1.1step 1.2∎

The holding-time and destination laws in steps 1.1–1.2 are exactly those of the chain in [F3]. Full AC [A1] and [F3] supply its joint path law and nonexplosion; hence the specified instrument's recorded path is the finite continuous-time chain with generator $W$ , not a law inferred from noncommuting one-time correlations.

Remarks

The record path is a mathematical model of a chosen energy-resolved monitoring scheme. Other unravelings of the same GKSL evolution can produce different records; no unique trajectory follows from the one-time channel family.

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Gibbs jump commuting relative entropy production

Statement

Use the finite connected Gibbs jump model of Connected Gibbs jump generators satisfy detailed balance. For a diagonal density $ρ_{t} = \sum_{a} p_{a} (t) ∣ a ⟩ ⟨ a ∣$ , the populations satisfy $p_{a}^{'} = \sum_{b \neq a} (p_{b} w_{b a} - p_{a} w_{a b})$ . Define the dimensionless classical relative entropy

$K (p ∣ π) : = \sum_{a \in E} p_{a} \log \frac{p_{a}}{π_{a}}, 0 \log (0 / π_{a}) : = 0,$

where $\log$ is the natural logarithm. For every $t > 0$ , $p_{a} (t) > 0$ for all $a$ , and

$\frac{d}{d t} K (p_{t} ∣ π) = - \frac{1}{2} \sum_{\begin{array}{c} a, b \in E \\ a \neq b \end{array}} π_{a} w_{a b} (u_{a} - u_{b}) (\log u_{a} - \log u_{b}) \leq 0, u_{a} : = p_{a} / π_{a} .$

Equality in this derivative holds exactly when $p_{t} = π$ . For any initial probability vector, including one with zero entries, $t \mapsto K (p_{t} ∣ π)$ is continuous at $t = 0$ and nonincreasing on $[0, \infty)$ . This is a finite commuting-state statement; it asserts no entropy-production theorem for noncommuting densities.

Facts & Assumptions

Given: The finite state set $E$ , positive Gibbs weights $π_{a}$ , connected rates $w_{a b}$ , and the population equation in Connected Gibbs jump generators satisfy detailed balance.

[F1]

Connected Gibbs jump generators satisfy detailed balance: $π_{a} w_{a b} = π_{b} w_{b a}$ , the positive-rate graph is connected, and diagonal populations obey $p_{a}^{'} = \sum_{b \neq a} (p_{b} w_{b a} - p_{a} w_{a b})$ .

Proof

1.1givenF1chooseconstructalgebra

The population equation in [F1] is $p_{t}^{'} = p_{t} W$ for row probability vectors, so $p_{t} = p_{0} e^{t W}$ . Let $λ > \max_{a} r_{a}$ and $P : = I + W / λ$ . Then $P$ is a finite stochastic matrix with $P_{a a} > 0$ , and the matrix exponential satisfies

e^{t W} = e^{- λ t} \sum_{n = 0}^{\infty} \frac{(λ t)^{n}}{n!} P^{n} .

For any initial probability vector, choose $a$ with $p_{a} (0) > 0$ . For every $b$ , graph connectivity gives a finite positive-rate path from $a$ to $b$ ; the corresponding power $P^{n}$ has $(P^{n})_{a b} > 0$ . Its coefficient in the series is positive for $t > 0$ , so $p_{b} (t) > 0$ .

2.1givenF1step 1.1algebra

For $t > 0$ , put $u_{a} = p_{a} (t) / π_{a} > 0$ and, for $a \neq b$ , put $c_{a b} : = π_{a} w_{a b} = c_{b a}$ . Differentiating $K$ and using $\sum_{a} p_{a}^{'} = 0$ gives

\frac{d}{d t} K (p_{t} ∣ π) = \sum_{a} p_{a}^{'} \log u_{a} = \sum_{a \neq b} c_{a b} (u_{b} - u_{a}) \log u_{a} .

Pairing the $(a, b)$ and $(b, a)$ terms yields the displayed symmetric formula. Since $\log$ is increasing, each $(u_{a} - u_{b}) (\log u_{a} - \log u_{b}) \geq 0$ .

3.1givenF1step 2.1algebra∎

Equality in step 2.1 forces $u_{a} = u_{b}$ on every positive-rate edge. Connectedness makes all $u_{a}$ equal, and $\sum_{a} p_{a} = \sum_{a} π_{a} = 1$ then gives $u_{a} = 1$ , so $p_{t} = π$ . Conversely $p_{t} = π$ makes every term zero. Finally $x \log (x / π_{a}) \to 0$ as $x ↓ 0$ , so $K (p_{t} ∣ π)$ is continuous at $t = 0$ ; applying the derivative inequality on $[ε, t]$ and taking $ε ↓ 0$ proves nonincrease from any initial probability vector.

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Gap-controlled correlation resolvent and positive coefficients

Statement

Assume the finite GNS detailed-balance hypotheses, $d > 1$ , and the positive gap $γ$ of Faithful scalar-kernel detailed balance gives a relaxation gap. For centered $A, B \in M_{0}$ , define $C_{A, B} (t) : = ⟨ e^{t G} A, B ⟩_{σ}$ . Then

$∣ C_{A, B} (t) ∣ \leq e^{- γ t} ∥ A ∥_{σ} ∥ B ∥_{σ}, t \geq 0,$

so the correlation is absolutely integrable and

$\int_{0}^{\infty} C_{A, B} (t) d t = {⟨ (- G ∣_{M_{0}})^{- 1} A, B ⟩}_{σ} .$

For centered $A$ , the real coefficient

$Γ (A) : = Re \int_{0}^{\infty} ⟨ e^{t G} A, A ⟩_{σ} d t$

is nonnegative. Its real polarization on any finite list of Hermitian centered observables is a positive-semidefinite real symmetric correlation-integral matrix. No mixing, central-limit theorem, or infinite-volume transport claim follows.

Facts & Assumptions

Given: The finite observable generator $G = K + D$ , the faithful state $σ$ , the centered subspace $M_{0}$ , and the gap $γ > 0$ .

[F1]

Faithful scalar-kernel detailed balance gives a relaxation gap: $M_{0}$ is invariant, $∥ e^{t G} A ∥_{σ} \leq e^{- γ t} ∥ A ∥_{σ}$ for centered $A$ , and $D ∣_{M_{0}} \leq - γ I$ .

[F2]

Specified finite GNS detailed balance: $G = K + D$ , $D$ is self-adjoint for $⟨ \cdot, \cdot ⟩_{σ}$ , and $K$ is anti-self-adjoint.

Proof

1.1givenF1algebra

Cauchy–Schwarz in the finite weighted inner product and [F1] give $∣ C_{A, B} (t) ∣ \leq e^{- γ t} ∥ A ∥_{σ} ∥ B ∥_{σ}$ . The right side is integrable on $[0, \infty)$ , so the correlation integral converges absolutely.

1.2givenF1algebra

The integral $Y : = \int_{0}^{\infty} e^{t G} A d t$ converges in the finite-dimensional invariant space $M_{0}$ . Since $\frac{d}{d t} e^{t G} A = G (e^{t G} A)$ and the decay in [F1] sends the endpoint to zero, integrating gives $G (Y) = - A$ . If $G Z = 0$ for centered $Z$ , then [F1] gives $∥ Z ∥_{σ} = ∥ e^{t G} Z ∥_{σ} \leq e^{- γ t} ∥ Z ∥_{σ}$ , hence $Z = 0$ . Thus $- G ∣_{M_{0}}$ is invertible and $Y = (- G ∣_{M_{0}})^{- 1} A$ , which proves the resolvent identity.

2.1F1F2step 1.2algebra∎

For $Y$ from step 1.2, $Γ (A) = Re ⟨ Y, - G (Y) ⟩_{σ}$ . The term $Re ⟨ Y, - K (Y) ⟩_{σ}$ vanishes because $K$ is anti-self-adjoint. The remaining term is $⟨ Y, - D (Y) ⟩_{σ} \geq γ ∥ Y ∥_{σ}^{2} \geq 0$ by [F1]–[F2]. Therefore $Γ$ is a nonnegative real quadratic form on the real vector space of Hermitian centered observables; real polarization gives a positive-semidefinite symmetric matrix.

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Fixed-dissipator finite response and integrable kernel

Statement

Let $L_{0}$ be the finite Schrödinger generator trace-dual to $G$ in Specified finite GNS detailed balance, and assume the gap hypotheses of Faithful scalar-kernel detailed balance gives a relaxation gap with $d > 1$ . Let $A \in M_{d} (C)$ , $B = B^{*} \in M_{d} (C)$ be dimensionless observables, and let $h : [0, T] \to R$ be continuous with units joules. For dimensionless $ϵ \in R$ , perturb only the Hamiltonian by $H_{ϵ} (t) = H_{s} - ϵ h (t) B$ ; hold the dissipator fixed. Let $ρ_{t}^{ϵ}$ be the unique state solution with $ρ_{0}^{ϵ} = σ$ , supplied by Continuous finite GKSL channel propagators.

Define $A_{0} : = A - Tr (σ A) I$ and, for $τ \geq 0$ ,

$R_{A B} (τ) : = \frac{i}{ℏ} Tr ⁣ (σ [e^{τ G} A_{0}, B]) .$

Then, on every finite interval $[0, T]$ , the derivative at $ϵ = 0$ exists and

${\frac{d}{d ϵ} Tr (ρ_{t}^{ϵ} A) ∣}_{ϵ = 0} = \int_{0}^{t} h (s) R_{A B} (t - s) d s .$

The kernel is absolutely integrable on $[0, \infty)$ . If $μ$ is the smallest eigenvalue of $σ$ , then

$∣ R_{A B} (τ) ∣ \leq \frac{2 ∥ B ∥_{\infty}}{ℏ \sqrt{μ}} e^{- γ τ} ∥ A_{0} ∥_{σ} .$

This is an open-system model response with a fixed dissipator. It does not assert that the driven stationary state is the Gibbs state of $H_{s} - ϵ h B$ .

Facts & Assumptions

Given: The finite GKSL dynamics of $L_{0}$ , its trace-dual $G$ , faithful stationary density $σ$ , positive gap $γ$ , Hermitian $B$ , continuous field $h$ , and initial preparation $σ$ .

[F1]

Specified finite GNS detailed balance: $G$ is the trace-dual observable generator $i [H_{s}, \cdot] / ℏ + D$ , with $D (I) = 0$ and $[H_{s}, σ] = 0$ .

[F2]

Continuous finite GKSL channel propagators: a continuous finite GKSL family has a unique CPTP propagator on every compact time interval.

[F3]

Faithful scalar-kernel detailed balance gives a relaxation gap: $σ$ is stationary, centered observables decay in $∥ \cdot ∥_{σ}$ at least as $e^{- γ t}$ , and $σ$ is faithful.

[F4]

Finite matrix trace cyclicity and nondegeneracy: the finite trace is cyclic.

Proof

1.1givenF2F3algebra

Put $V (X) : = i [B, X] / ℏ$ . The perturbed Schrödinger generator is $L_{ϵ} (t) = L_{0} + ϵ h (t) V$ . Since $L_{0}$ leaves $σ$ stationary by [F3], variation of constants gives

ρ_{t}^{ϵ} = σ + ϵ \int_{0}^{t} h (s) e^{(t - s) L_{0}} V (σ) d s + ϵ^{2} \int_{0 \leq u \leq s \leq t} h (s) h (u) e^{(t - s) L_{0}} V e^{(s - u) L_{0}} V (ρ_{u}^{ϵ}) d u d s .

The second integral is $O (ϵ^{2})$ uniformly on $[0, T]$ : $h$ is bounded, $V$ is a bounded linear map on the finite matrix space, $\sup_{0 \leq s \leq t \leq T} ∥ e^{(t - s) L_{0}} ∥_{1} < \infty$ , and $∥ ρ_{u}^{ϵ} ∥_{1} = 1$ because [F2] gives a channel propagator. Thus the first derivative exists and equals $\int_{0}^{t} h (s) e^{(t - s) L_{0}} V (σ) d s$ .

2.1F1F4step 1.1algebra

By the generator duality in [F1], the exponential power series gives $Tr (A e^{τ L_{0}} X) = Tr (e^{τ G} A X)$ for every finite matrix $X$ . Using [F4] and $V (σ) = i [B, σ] / ℏ$ ,

Tr ⁣ (A e^{τ L_{0}} V (σ)) = \frac{i}{ℏ} Tr ⁣ (σ [e^{τ G} A, B]) .

The scalar part of $A$ commutes with $B$ , so $A$ may be replaced by $A_{0}$ . Substituting this identity into step 1.1 gives the stated response formula.

3.1F3step 2.1algebra∎

Let $C = e^{τ G} A_{0}$ . The gap theorem gives $∥ C ∥_{σ} \leq e^{- γ τ} ∥ A_{0} ∥_{σ}$ . If $μ > 0$ is the smallest eigenvalue of $σ$ , then

∥ C ∥_{\infty} \leq \frac{∥ C σ^{1 / 2} ∥_{2}}{\sqrt{μ}} = \frac{∥ C ∥_{σ}}{\sqrt{μ}} .

The finite density trace satisfies $∣ Tr (σ M) ∣ \leq ∥ M ∥_{\infty}$ : diagonalize $σ$ , write its nonnegative eigenvalues summing to one, and bound each diagonal matrix element of $M$ by $∥ M ∥_{\infty}$ . Therefore $∣ Tr (σ [C, B]) ∣ \leq 2 ∥ B ∥_{\infty} ∥ C ∥_{\infty}$ , which proves the exponential bound and absolute integrability.

Remarks

If the bath rates or jump operators depended on the perturbing field, a derivative of the dissipator would appear in step 1.1. It is absent here by the explicit fixed-dissipator hypothesis. The formula is not a universal fluctuation-dissipation theorem and is not an empirical observation.

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Specified effective GKSL bath and instrument assumption

Physical scope

For a finite-dimensional subsystem represented by H=C^d (d>=1), the chosen thermal GKSL state law and energy-resolved instrument apply physically only on the declared finite interval [0,t_res], where t_res>0 is in seconds. All-time limits are mathematical continuations of the same fixed generator and instrument, with no physical-applicability claim beyond t_res. No microscopic bath limit, calibrated rate, or reported measurement is asserted.

Postulate

Let $d \geq 1$ be finite and let the modeled subsystem Hilbert space be $H = C^{d}$ . A state is a dimensionless matrix $ρ = ρ^{*} \in M_{d} (C)$ such that $x^{*} ρ x \geq 0$ for every $x \in C^{d}$ and $Tr ρ = 1$ . An initial preparation is a chosen state $ρ_{0}$ . Let $H_{s} = H_{s}^{*} \in M_{d} (C)$ have units joules, let $ℏ > 0$ have units joule-seconds, and let $L_{1}, \dots, L_{m} \in M_{d} (C)$ , $m \geq 0$ finite, have units seconds $^{- 1 / 2}$ . We adopt the time-independent finite GKSL generator of Finite GKSL generators and units,

$L (X) = - \frac{i}{ℏ} [H_{s}, X] + \sum_{j = 1}^{m} (L_{j} X L_{j}^{*} - \frac{1}{2} {L_{j}^{*} L_{j}, X}),$

where $[U, V] = U V - V U$ and ${U, V} = U V + V U$ . The exact state law is

$\frac{d}{d t} ρ_{t} = L (ρ_{t}), ρ_{t = 0} = ρ_{0}, ρ_{t} = e^{t L} ρ_{0}, 0 \leq t \leq t_{r e s} .$

Here $t_{r e s} \in (0, \infty)$ is the finite resolved duration, in seconds, adopted for this model. For mathematical long-time calculations, the same constant generator defines the continuation $e^{t L} ρ_{0}$ for every $t \geq 0$ ; this continuation makes no physical-applicability claim for $t > t_{r e s}$ . By Finite GKSL generates CPTP semigroups it is a density-state evolution on that mathematical continuation as well.

Let $k_{B} > 0$ denote Boltzmann's constant, with SI units joules per kelvin; it converts absolute temperature to an inverse-energy parameter. Let $T_{b a t h} > 0$ be the adopted absolute temperature parameter in kelvins and set $β : = (k_{B} T_{b a t h})^{- 1}$ , with units joules $^{- 1}$ . The temperature and rates are model inputs here; no measured calibration is asserted. The temperature parameter alone imposes no detailed-balance condition on an arbitrary choice of $L_{j}$ . When the generator is specialized to the connected Gibbs-jump model, its energies, $β$ , and rates must satisfy the hypotheses in Connected Gibbs jump generators satisfy detailed balance.

Within this effective formulation, the dimension $d$ , preparation $ρ_{0}$ , general coefficient matrices $H_{s}$ and $L_{j}$ , positive constants $ℏ$ and $k_{B}$ , temperature $T_{b a t h}$ , and resolved duration $t_{r e s}$ are primitive, adopted model inputs with the types and units stated above; none is derived here from a microscopic reservoir or inferred from a measurement. This status is local to this formulation, not a claim that each is fundamental in every physical theory. The space $C^{d}$ , inverse-energy parameter $β$ , generator $L$ , and state family $ρ_{t}$ are derived by their displayed constructions once those inputs and the physical state-law postulate are fixed. The selected Hermitian observable $A$ and its unit assignment are additional adopted observable data; its expectation is derived by the trace formula below.

In the connected Gibbs-jump parameterization, the orthonormal energy basis, real level energies $E_{a}$ in joules, and rates $w_{a b} \geq 0$ in seconds $^{- 1}$ are instead the adopted primitive spectral and rate data, subject to the Gibbs-balance and connectivity hypotheses of Connected Gibbs jump generators satisfy detailed balance. In that parameterization the operator coefficients are derived as $H_{s} = \sum_{a} E_{a} ∣ a ⟩ ⟨ a ∣$ and $L_{a b} = \sqrt{w_{a b}} ∣ b ⟩ ⟨ a ∣$ ; they replace, rather than supplement, the independent general matrix choices above. The Gibbs weights $π_{a} = e^{- β E_{a}} / \sum_{b} e^{- β E_{b}}$ are derived, while the energy-resolved instrument remains a separate adopted assumption. In particular, neither the temperature nor the Gibbs-balance relation derives the numerical rates.

For that Gibbs-jump specialization, record predictions additionally adopt the particular ideal energy-resolved instrument defined in Specified energy-jump instrument realizes a finite chain, and only on $0 \leq t \leq t_{r e s}$ as a physical model. For every allowed initial density $ρ_{0}$ , define $q_{a} : = ⟨ a ∣ ρ_{0} ∣ a ⟩ \geq 0$ , so $\sum_{a} q_{a} = 1$ . The full label path is initialized by $\Pr (a (0) = a) = q_{a}$ and, conditional on this label, follows the supplier's holding and destination laws. Before the first click this label is an auxiliary variable for the record law, not a claim that the conditional quantum state is an energy eigenstate or that an additional projective measurement has replaced $ρ_{0}$ by its diagonal part. More precisely, with the supplier's diagonal no-click matrix $K$ , the unnormalized no-click density is $N_{t} = e^{t K} ρ_{0} e^{t K^{*}}$ and its trace is $S (t) = \sum_{a} q_{a} e^{- r_{a} t} > 0$ ; conditional on no click the quantum state is $N_{t} / S (t)$ . The first click $a \to b$ has time density $w_{a b} q_{a} e^{- r_{a} t}$ and, when this density is positive, resets the quantum state to $∣ b ⟩ ⟨ b ∣$ , because $L_{a b} N_{t} L_{a b}^{*} = w_{a b} q_{a} e^{- r_{a} t} ∣ b ⟩ ⟨ b ∣$ . Zero-density branches are not normalized. Thus the no-click survival and every labelled click law agree with the stated initial-label mixture and its subsequent supplier chains, while the no-click quantum state retains any initial coherences. For a diagonal Gibbs preparation, $q = π$ . Its law for all $t \geq 0$ is used only as a mathematical continuation when deriving a long-time coefficient. A one-time channel family alone does not select this record law. Full The Axiom of Choice is retained as the formal set-theoretic premise in the monitored-chain supplier's construction of the joint path law; it is not a physical postulate about nature. A selected observable is represented by a Hermitian matrix $A \in M_{d} (C)$ with its assigned physical units, and its model expectation $Tr (ρ_{t} A)$ has those same units; no measurement outcome is reported.

Remarks

This postulate distinguishes the adopted effective bath model from derived finite-dimensional consequences. It does not provide measured rates or uncertainties from an experiment. If rates later come from a measurement, its apparatus, calibration, sampling conditions, and uncertainty must be stated as empirical premises in a separate appropriately scoped item.

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Stationary monitored-chain transport variance coefficient

Statement

Assume the full Axiom of Choice and the connected finite Gibbs jump data of Connected Gibbs jump generators satisfy detailed balance. Use the energy-resolved record path supplied by Specified energy-jump instrument realizes a finite chain, initialized with law $π$ . Let $v : E \to R$ be a real velocity observable with units metres per second and $π (v) : = \sum_{a} π_{a} v_{a} = 0$ . Define the record displacement

$Y_{T} : = \int_{0}^{T} v_{a (s)} d s, C_{v} (s) : = ⟨ v, e^{s W} v ⟩_{π},$

where $W f (a) = \sum_{b \neq a} w_{a b} (f_{b} - f_{a})$ , $⟨ f, g ⟩_{π} = \sum_{a} π_{a} f_{a} g_{a}$ , and $π^{⊥} : = {f : E \to R : \sum_{a} π_{a} f_{a} = 0}$ . Then for every $T \geq 0$ ,

$E_{π} [Y_{T}] = 0, {Var}_{π} (Y_{T}) = 2 \int_{0}^{T} (T - s) C_{v} (s) d s .$

If $γ_{c l} > 0$ is the reversible spectral gap of $W$ on $π$ -centered functions, then $∣ C_{v} (s) ∣ \leq e^{- γ_{c l} s} \sum_{a} π_{a} v_{a}^{2}$ , and

$D_{r e c} : = \lim_{T \to \infty} \frac{{Var}_{π} (Y_{T})}{2 T} = \int_{0}^{\infty} C_{v} (s) d s = ⟨ (- W ∣_{π^{⊥}})^{- 1} v, v ⟩_{π} \geq 0.$

The units of $D_{r e c}$ are square metres per second. This proves a variance coefficient for the selected finite record model; it does not prove a central-limit theorem or a generic quantum transport law.

Facts & Assumptions

Given: Full AC, the finite connected Gibbs jump model, the selected energy-resolved instrument, initial record law $π$ , and centered real $v$ .

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. This full axiom is required by the monitored-chain supplier's Ionescu–Tulcea path construction.

[F1]

Specified energy-jump instrument realizes a finite chain: the chosen instrument produces a nonexplosive finite continuous-time Markov path $a (t)$ with the prescribed rates and transition semigroup.

[F2]

Connected Gibbs jump generators satisfy detailed balance: $W$ is self-adjoint on $L^{2} (E, π)$ , has constants as its kernel, and has a strictly positive gap on centered functions.

[F3]

Dominated convergence: almost-everywhere convergence under one integrable majorant gives convergence of the integrals.

[F4]

Fubini's theorem for L^1 functions on a sigma-finite product: for two sigma-finite measure spaces and a product-measurable integrand with integrable absolute value, the product integral equals both iterated integrals.

Proof

1.1givenA1F1F2F4algebra

The path in [F1] has locally piecewise-constant sample paths with finitely many jumps on bounded intervals. For the supplied measurable jump times $S_{n}$ and labels $A_{n}$ , the event $a (s) = a$ is the countable union of ${(s, ω) : S_{n} (ω) \leq s < S_{n + 1} (ω), A_{n} (ω) = a}$ . Thus $(s, ω) \mapsto v_{a (s, ω)}$ is product-measurable and bounded by $M : = \max_{a} ∣ v_{a} ∣$ . On the probability space $(Ω, F, P_{π})$ times $[0, T]$ with Lebesgue measure, its absolute integral is at most $T M < \infty$ ; both measures are finite, so [F4] makes $Y_{T}$ a measurable bounded random variable and allows its expectation to pass through the time integral. It is stationary when initialized with $π$ , since [F2] gives $π W = 0$ . For $s, t \geq 0$ , stationarity and the transition semigroup give

E_{π} [v_{a (s)} v_{a (t)}] = ⟨ v, e^{∣ t - s ∣ W} v ⟩_{π} = C_{v} (∣ t - s ∣) .

For $t \geq s$ , this is the Markov property; for $s > t$ , the same scalar follows by self-adjointness of $e^{(s - t) W}$ . Since $v$ is centered, $E_{π} [Y_{T}] = 0$ .

2.1F4step 1.1algebra

On $Ω \times [0, T]^{2}$ , the product-measurable integrand $v_{a (s, ω)} v_{a (t, ω)}$ has absolute value at most $M^{2}$ , so its absolute product integral is at most $T^{2} M^{2}$ . The probability measure and the Lebesgue measures of $[0, T]$ and $[0, T]^{2}$ are finite and hence sigma-finite. Applying [F4] to exchange expectation with the square integral, and to the two time coordinates, therefore gives

{Var}_{π} (Y_{T}) = \int_{0}^{T} \int_{0}^{T} E_{π} [v_{a (s)} v_{a (t)}] d s d t = 2 \int_{0}^{T} (T - s) C_{v} (s) d s,

using the covariance identity from step 1.1 and splitting the square into its two triangles.

2.2F2step 1.1algebra

Expand $v$ in an orthonormal eigenbasis of the self-adjoint $W ∣_{π^{⊥}}$ . Its eigenvalues are at most $- γ_{c l} < 0$ , so the expansion gives $∣ C_{v} (s) ∣ \leq e^{- γ_{c l} s} ∥ v ∥_{π}^{2}$ . The finite matrix exponential makes $C_{v}$ continuous. On $[0, \infty)$ , set $f_{T} (s) = 1_{[0, T]} (s) (1 - s / T) C_{v} (s)$ . For every sequence $T_{n} \to \infty$ , $f_{T_{n}} (s) \to C_{v} (s)$ pointwise and $∣ f_{T_{n}} (s) ∣ \leq ∥ v ∥_{π}^{2} e^{- γ_{c l} s}$ , an integrable majorant.

3.1F2F3step 2.2algebra∎

Applying [F3] to step 2.2 gives $\int_{0}^{T_{n}} (1 - s / T_{n}) C_{v} (s) d s \to \int_{0}^{\infty} C_{v} (s) d s$ for every $T_{n} \to \infty$ , hence the real- $T$ limit exists and equals that integral. The same finite spectral expansion gives $\int_{0}^{\infty} e^{s W} v d s = (- W ∣_{π^{⊥}})^{- 1} v$ , so the integral is $⟨ (- W ∣_{π^{⊥}})^{- 1} v, v ⟩_{π}$ . Each eigencomponent contributes its squared coefficient divided by a positive gap eigenvalue, proving nonnegativity.

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Conditional specified thermal bath relaxation

Physical scope

Conditional predictions for the specified finite connected Gibbs-jump GKSL model, physically applicable only on the postulate's finite interval [0,t_res]. The all-time exponential estimate and qubit formulas are proved for the mathematical continuation of the fixed generator; no physical applicability for t>t_res, measured rates, microscopic bath limit, or universal thermalization law is claimed.

Statement

Assume full AC, the adopted finite effective-bath model Specified effective GKSL bath and instrument assumption on its physical interval $0 \leq t \leq t_{r e s}$ , and the connected finite Gibbs-jump hypotheses of Connected Gibbs jump generators satisfy detailed balance. For an initial density $ρ$ , let $ρ_{t} = e^{t L} ρ$ denote the mathematical all-time continuation of the fixed finite Schrödinger generator, and let $σ = Z_{β}^{- 1} e^{- β H_{s}}$ . If $d > 1$ , the finite detailed-balance gap $γ > 0$ gives the conditional model prediction for every $t \geq 0$ in that continuation

$∥ ρ_{t} - σ ∥_{1} \leq ∥ σ^{- 1 / 2} (ρ - σ) ∥_{2} e^{- γ t} .$

Under the adopted physical model this bound is applicable only for $0 \leq t \leq t_{r e s}$ ; its all-time form is a mathematical consequence of continuing the same fixed generator, not a claim about physical evolution for $t > t_{r e s}$ .

For the two-level specialization, let $H_{s} = Δ Z / 2$ , $Δ > 0$ joules, $Z = ∣ 0 ⟩ ⟨ 0 ∣ - ∣ 1 ⟩ ⟨ 1 ∣$ , and $X = ∣ 0 ⟩ ⟨ 1 ∣ + ∣ 1 ⟩ ⟨ 0 ∣$ , $Y = - i ∣ 0 ⟩ ⟨ 1 ∣ + i ∣ 1 ⟩ ⟨ 0 ∣$ . Adopt downward jump $\sqrt{a} ∣ 1 ⟩ ⟨ 0 ∣$ with $a > 0$ seconds $^{- 1}$ , upward jump $\sqrt{b} ∣ 0 ⟩ ⟨ 1 ∣$ with $b / a = e^{- β Δ}$ , and optionally a dephasing jump $\sqrt{γ_{ϕ} / 2} Z$ with $γ_{ϕ} \geq 0$ seconds $^{- 1}$ . Put

$ω : = Δ / ℏ, γ_{1} : = a + b, γ_{2} : = \frac{a + b}{2} + γ_{ϕ}, z_{e q} : = \frac{b - a}{a + b} = - \tanh (β Δ / 2) .$

For every $t \geq 0$ in the mathematical continuation, the exact model expectations and coherence satisfy

$⟨ Z ⟩_{t} = z_{e q} + e^{- γ_{1} t} (⟨ Z ⟩_{0} - z_{e q}), ρ_{01} (t) = e^{- (γ_{2} + i ω) t} ρ_{01} (0) .$

Equivalently, $e^{t G} X = e^{- γ_{2} t} (X \cos ω t - Y \sin ω t)$ and $e^{t G} Z = z_{e q} I + e^{- γ_{1} t} (Z - z_{e q} I)$ . For $d = 1$ , $ρ = σ$ and relaxation is trivial. These are conditional model deductions, not experimental findings or claims that a physical bath realizes the postulate.

The qubit formulas above are stated for the mathematical continuation; their physical model applicability is limited to $0 \leq t \leq t_{r e s}$ .

Facts & Assumptions

Given: Full AC, the adopted effective-bath postulate, finite connected Gibbs rates, $β > 0$ , and, for the qubit branch, the stated $H_{s}$ , jump matrices, and initial density.

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function. It is retained in the formal supplier chain for the selected monitoring instrument in Specified effective GKSL bath and instrument assumption; the density-decay calculation itself is finite-dimensional.

[A2]

Specified effective GKSL bath and instrument assumption: the specified subsystem follows the adopted finite GKSL state law and selected Gibbs-jump instrument physically only on $[0, t_{r e s}]$ ; its continuation for all $t \geq 0$ is mathematical, with no physical-applicability claim beyond $t_{r e s}$ , microscopic bath limit, or calibration data asserted.

[F1]

Connected Gibbs jump generators satisfy detailed balance: connected Gibbs-balanced energy jumps give a faithful stationary Gibbs density and a dissipator with scalar kernel.

[F2]

Faithful scalar-kernel detailed balance gives a relaxation gap: under [F1], all initial densities satisfy the stated trace-norm exponential relaxation bound.

Proof

1.1givenA1A2F1algebra

The rate relation $b / a = e^{- β Δ}$ is exactly $π_{0} a = π_{1} b$ for $π_{j} \propto e^{- β E_{j}}$ with $E_{0} = Δ / 2$ , $E_{1} = - Δ / 2$ . Since $a, b > 0$ , the two-vertex rate graph is connected. The postulate [A2] adopts this finite generator for the selected physical model; [F1] therefore supplies its faithful stationary Gibbs state and scalar dissipative kernel.

1.2A2F1F2algebra

Applying [F2] to this finite GNS detailed-balance generator proves the trace-norm estimate for $d > 1$ . For $d = 1$ there is only one density matrix, so the difference from $σ$ is zero.

1.3givenA2algebra

In the qubit basis, direct matrix multiplication gives ${\dot{ρ}}_{00} = - a ρ_{00} + b ρ_{11}$ , ${\dot{ρ}}_{11} = a ρ_{00} - b ρ_{11}$ , and ${\dot{ρ}}_{01} = - (γ_{2} + i ω) ρ_{01}$ . The dephasing jump contributes $- γ_{ϕ} ρ_{01}$ and vanishes on diagonal entries. With $z = ρ_{00} - ρ_{11}$ , the population equations become $\dot{z} = (b - a) - (a + b) z = - γ_{1} (z - z_{e q})$ .

2.1step 1.3algebra∎

Solving the scalar linear equations in step 1.3 yields $z (t) = z_{e q} + e^{- γ_{1} t} (z (0) - z_{e q})$ and $ρ_{01} (t) = e^{- (γ_{2} + i ω) t} ρ_{01} (0)$ . The dual observable generator acts as $G (X) = - γ_{2} X - ω Y$ , $G (Y) = ω X - γ_{2} Y$ , and $G (Z) = - γ_{1} (Z - z_{e q} I)$ ; exponentiating these two-dimensional and one-dimensional invariant subspaces gives the displayed formulas for $e^{t G}$ .

Remarks

The positive-temperature relation fixes the rate ratio, while the numerical values of $a$ , $b$ , and $γ_{ϕ}$ remain model inputs. No observation, calibration uncertainty, or empirical agreement is claimed here.

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Conditional instrument-record diffusion coefficient

Physical scope

Conditional finite-model prediction for the specified ideal energy-resolved monitoring instrument, Gibbs initial law, and assigned pointer velocities. Physical applicability is limited to the postulate's finite interval [0,t_res]; the T->infinity coefficient is mathematical for the continued finite-rate model. It is not a measured spatial diffusion coefficient or a generic quantum hydrodynamic limit.

Statement

Assume full AC, Specified effective GKSL bath and instrument assumption, and the finite connected Gibbs-jump model with its selected energy-resolved instrument. Prepare the system with diagonal Gibbs law $π$ . Assign a real pointer velocity $v_{a}$ in metres per second to each recorded energy label, with $\sum_{a} π_{a} v_{a} = 0$ , and define the model record displacement $Y_{T} = \int_{0}^{T} v_{a (s)} d s$ . The postulate's physical applicability is limited to $0 \leq t \leq t_{r e s}$ . For formulas at arbitrary $T \geq 0$ and their limit, continue the same finite-rate chain as an idealized mathematical model; this gives no physical-applicability claim for $T > t_{r e s}$ . Thus the displayed formula is a conditional physical-model prediction for $0 \leq T \leq t_{r e s}$ and a mathematical model identity for all $T \geq 0$ in the continuation:

$E_{π} [Y_{T}] = 0, {Var}_{π} (Y_{T}) = 2 \int_{0}^{T} (T - s) C_{v} (s) d s,$

where $C_{v} (s) = \sum_{a} π_{a} v_{a} (e^{s W} v)_{a}$ . The limit

$D_{r e c} : = \lim_{T \to \infty} \frac{{Var}_{π} (Y_{T})}{2 T} = \int_{0}^{\infty} C_{v} (s) d s = ⟨ (- W ∣_{π^{⊥}})^{- 1} v, v ⟩_{π} \geq 0$

exists and has units square metres per second. This long-time limit is a derived coefficient of the idealized mathematical continuation. It is not a measurement of a system's position, a calibrated material coefficient, or evidence that the microscopic bath obeys the effective model.

Facts & Assumptions

Given: Full AC, the effective-bath and monitoring postulate, the finite Gibbs rate model, the stationary initial law $π$ , and the specified pointer velocity $v$ .

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function; the exact formal use is the joint path construction inherited from Stationary monitored-chain transport variance coefficient.

[A2]

Specified effective GKSL bath and instrument assumption: the selected finite GKSL dynamics and energy-resolved instrument are physically adopted only on $[0, t_{r e s}]$ ; their continuation for all $t \geq 0$ is mathematical and has no physical-applicability claim beyond $t_{r e s}$ . No apparatus calibration or measured rate is supplied.

[F1]

Stationary monitored-chain transport variance coefficient: under full AC and the connected Gibbs-rate hypotheses, the stationary selected record has the stated finite-time variance formula and limit.

Proof

1.1givenA1A2

The assumptions [A1]–[A2] identify the selected, ideal energy-resolved instrument and its Gibbs initial law; the velocity labels are stipulated model observables. They introduce no empirical premise.

2.1givenF1step 1.1∎

Applying [F1] with these exact rates, initial law, and centered velocities yields the displayed finite-time variance, resolvent formula, nonnegativity, and limit for the mathematical continuation. Only its finite-time predictions with $0 \leq T \leq t_{r e s}$ have the postulate's physical applicability.

Remarks

The stochastic record is the selected instrument output. It is not a continuous position trajectory and does not follow uniquely from the one-time GKSL channel family.

5 · Examples, experiments and counterexamples

None yet.

Specified finite GNS detailed balance

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Specified finite GNS detailed balance

Definition

Facts & Assumptions

Proof

Remarks

Faithful scalar-kernel detailed balance gives a relaxation gap

Statement

Facts & Assumptions

Proof

Remarks

Connected Gibbs jump generators satisfy detailed balance

Statement

Facts & Assumptions

Proof

Remarks

Specified energy-jump instrument realizes a finite chain

Definition

Facts & Assumptions

Proof

Remarks

Gibbs jump commuting relative entropy production

Statement

Facts & Assumptions

Proof

Gap-controlled correlation resolvent and positive coefficients

Statement

Facts & Assumptions

Proof

Fixed-dissipator finite response and integrable kernel

Statement

Facts & Assumptions

Proof

Remarks

Specified effective GKSL bath and instrument assumption

Postulate

Remarks

Stationary monitored-chain transport variance coefficient

Statement

Facts & Assumptions

Proof

Conditional specified thermal bath relaxation

Statement

Facts & Assumptions

Proof

Remarks

Conditional instrument-record diffusion coefficient

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, experiments and counterexamples

Sources