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Conditional prepared finite quantum control response

Physical scope

Exact prediction of the specified finite-dimensional isolated-control model under its adopted preparation, unitary-evolution, and Born-expectation postulates; it is not an observation, an apparatus validation, or evidence that the model describes nature.

Statement

Assume the adopted finite isolated model and fixed-preparation control convention of Adopted isolated prepared unitary model and Prepared control and expectation convention. Let $T > 0$ be finite in s, let $A = A^{*}$ be a bounded matrix observable, and set $α_{t} (A) = e^{i t H / ℏ} A e^{- i t H / ℏ}$ for time $t$ measured in s. Define $R_{A B} (u) = \frac{i}{ℏ} Tr ⁣ (σ [α_{u} (A), B]), C_{T} = \frac{2}{ℏ} \int_{0}^{T} ∣ h (s) ∣ ∥ B ∥ d s .$ Then for $0 \leq t \leq T$ the model expectation obeys $E_{ϵ, t} [A] = Tr (σ α_{t} (A)) + ϵ \int_{0}^{t} h (s) R_{A B} (t - s) d s + r_{ϵ} (t), ∣ r_{ϵ} (t) ∣ \leq ∥ A ∥ \frac{(∣ ϵ ∣ C_{T})^{2} e^{∣ ϵ ∣ C_{T}}}{2} .$ Here $H$ and $h B$ have unit J, $ℏ$ has unit J s, $t$ has unit s, $β$ has unit J $^{- 1}$ , and $ϵ, C_{T}$ are dimensionless. This is a conditional model prediction for the adopted preparation and dynamics, not a claim that an observed finite dataset must equal the prediction.

Facts & Assumptions

Given: The Axiom of Choice, the two adopted physical postulates, finite $T$ , the matrix observable $A$ , and the stated control $h, B$ .

[A1]

Adopted isolated prepared unitary model: the specified finite prepared state evolves as $ρ_{t} = U (t) ρ_{0} U (t)^{*}$ and the model expectation of bounded $A$ is $Tr (ρ_{t} A)$ .

[A2]

Prepared control and expectation convention: the controlled generator is $H - ϵ h (t) B$ on $[0, T]$ while the initial Gibbs matrix $σ$ remains fixed.

[F1]

Finite-time Kubo formula with norm remainder: for this fixed initial Gibbs density, the Heisenberg expectation has the displayed retarded commutator response and quadratic finite-time remainder.

[A3]

The Axiom of Choice: the full Choice convention is carried from the adopted normal-state measurement framework; no step below uses it to derive the finite matrix formula.

Proof

technique · conditional substitution of the adopted model inputs into the finite-time mathematical response theorem

Given: [A1]–[A3], [F1], and the exact finite-dimensional matrices and controls in the Statement.

1.1givenA1A2F1algebra

By [A1] and [A2], the initial state, controlled Hamiltonian, and model expectation are exactly the inputs used by [F1]; the control does not differentiate or replace the initial Gibbs matrix. The controlled density satisfies $Tr (ρ_{ϵ} (t) A) = Tr (ρ_{I} (t) α_{t} (A))$ by the finite trace identity in [F1]. Applying [F1] with this $A, B, h, ϵ, T$ gives the stated convolution and remainder bound.

2.1givenA1A2A3step 1.1∎

The equation is therefore a deduction within the adopted finite model, conditional on [A1] and [A2]. The full-Choice convention [A3] is inherited from the wider measurement framework, but the finite-matrix calculation in step 1.1 does not need it. The remainder is a mathematical bound; it supplies no measured response, apparatus reliability claim, or unconditional assertion about nature.

Conditional prepared finite quantum control response

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources