Manuscript IV-B@1 · IV-B-v1.tex · proposition · dfb66408

Proposition: Edge convergence of complete instruments

Model 1: the fixed TT plane and removable edge › Edge removal and its topology · lines 683–722 · proof included or linked

IV-B@1#prop:edge, IV-B@1#eq:edge-complete, IV-B@1#eq:sector-edge

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For every $τ \in R$ ,

e^{i τ A_{ε}^{(1)}} ⟶ e^{i τ A^{(1)}} strongly.

For every trace-class $Z$ on $H_{1} \otimes H_{E}$ , with any untouched reference $E$ ,

((I_{A_{ε}^{(1)}} - I_{A^{(1)}}) \otimes id_{E}) (Z)_{1} ⟶ 0.

No diamond-norm convergence is asserted.

Proof. On the sector of at most $n$ particles, telescope the powers in the analytic series. For $2 C ∣ τ ∣ < 1$ , the preceding sector estimates give

(e^{i τ A_{ε}^{(1)}} - e^{i τ A^{(1)}}) Ψ \leq \frac{2 D ε ∣ τ ∣ ( n + 1 )}{( 1 - 2 C ∣ τ ∣ ) ^{n + 2}} ∥ Ψ ∥ .

Finite-particle vectors are dense and all exponentials are unitary, so convergence is strong for these small times. Subdivision of an arbitrary time and finite telescoping prove strong convergence for every $τ$ .

The corresponding Kraus operators and their adjoints converge strongly. For finite-rank $Z$ , this implies trace-norm convergence of each Kraus block, also after tensoring with $E$ . Approximate a general trace-class $Z$ by finite-rank operators and use the uniform channel bounds to obtain (eq:edge-complete).

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IV-B@1#eq:edge-complete · line 692
$((I_{A_{ε}^{(1)}} - I_{A^{(1)}}) \otimes id_{E}) (Z)_{1} ⟶ 0.$
IV-B@1#eq:sector-edge · line 705
$(e^{i τ A_{ε}^{(1)}} - e^{i τ A^{(1)}}) Ψ \leq \frac{2 D ε ∣ τ ∣ ( n + 1 )}{( 1 - 2 C ∣ τ ∣ ) ^{n + 2}} ∥ Ψ ∥ .$
unlabeled · line 686
$e^{i τ A_{ε}^{(1)}} ⟶ e^{i τ A^{(1)}} strongly.$

Proposition: Edge convergence of complete instruments

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