Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Fixed-block measurement predictions under independent shell preparation

Physical scope

Conditional predictions for a specified finite q-level noninteracting tensor model, adopted canonical and uniform-band preparations, a fixed finite subsystem and a declared finite POVM. No detector frequencies, thermalization, interacting or growing-block conclusion is asserted.

Statement

Assume the physical preparation postulate Adopted finite canonical and band preparation rules, and the exact independent-shell hypotheses and notation of Fixed-subsystem equivalence for independent finite energy shells. In the physical interpretation fixed by the preparation postulate, the primitive scalar levels $e_{j}$ and total label energies are in joules, the band widths $δ_{N}$ are in joules, $T$ is in kelvin, and $β = (k_{B} T)^{- 1} > 0$ is in joules $^{- 1}$ . For each fixed $k$ and each fixed $r$ -outcome POVM $(M_{a})_{a = 1}^{r}$ on the $k$ -site register, define the conditional model probabilities $p_{N, a}^{s h} : = Tr (σ_{N, k} M_{a})$ and $p_{N, a}^{c a n} : = Tr (γ_{k} M_{a})$ for $N \geq k$ . They satisfy $\sum_{a = 1}^{r} ∣ p_{N, a}^{s h} - p_{N, a}^{c a n} ∣ ⟶ 0.$ This is a conditional prediction for the adopted states and specified measurement; it supplies no empirical frequencies or sampling inference.

Facts & Assumptions

Given: An integer $q \geq 1$ , real finite additive levels, finite $β$ , widths satisfying $δ_{N} \geq q \sum_{j} ∣ e_{j} ∣$ and $δ_{N} / N \to 0$ , fixed $k, r$ , and a fixed finite POVM on the first $k$ sites.

[A1]

Adopted finite canonical and band preparation rules: under the adopted preparation rule the two states are the canonical density and the normalized uniform band density, and P2 assigns POVM probabilities by trace pairings.

[F1]

Fixed-subsystem equivalence for independent finite energy shells: the fixed- $k$ reduced states converge in trace norm.

[F2]

Finite tensor density, observable and measurement language: each POVM outcome is a positive effect, and the finite trace pairing is the expectation used here.

Proof

1.1givenA1F2algebra

By [A1], the local laws are $p_{N, a}^{s h} = Tr (σ_{N, k} M_{a})$ and $p_{N, a}^{c a n} = Tr (γ_{k} M_{a})$ . Put $D_{N} = σ_{N, k} - γ_{k}$ . The states are diagonal in the common product energy basis. Write $d_{N, x} = (D_{N})_{x x}$ and $m_{a, x} = (M_{a})_{x x}$ for its diagonal entries in the product basis. Positivity and completeness of the POVM imply $m_{a, x} \geq 0$ and $\sum_{a} m_{a, x} = 1$ for every basis label $x$ . Thus $Tr (D_{N} M_{a}) = \sum_{x} d_{N, x} m_{a, x}$ .

2.1A1F1step 1.1algebra∎

Apply the triangle inequality and sum over the finite outcomes: $\sum_{a} ∣ Tr (D_{N} M_{a}) ∣ \leq \sum_{x} ∣ d_{N, x} ∣ \sum_{a} m_{a, x} = \sum_{x} ∣ d_{N, x} ∣ = ∥ D_{N} ∥_{1} .$ By [F1] this last quantity tends to zero for fixed $k$ , proving the claim. The calculation compares model probabilities only. It makes no inference from finite observations, which would require separately declared sampling and apparatus assumptions.

Fixed-block measurement predictions under independent shell preparation

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources