Manuscript Oai283-A@1 · Oai283-A.tex · proposition · 6fb75895

Proposition: Vanishing local errors can accumulate

Composition, workspace, and implementation allowances › Accumulation and a nonvanishing allowance · lines 624–650 · proof included or linked

Oai283-A@1#prop:accumulation

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For $d \geq 2$ , let

R_{d} (X) = Tr (X) I_{d} / d, P_{p} = (1 - p) id + p R_{d} .

For $p_{L} = λ / L$ , $λ > 0$ , each stage satisfies $d_{⋄} (P_{p_{L}}, id) \leq λ / L$ , but

L \to \infty lim inf d_{⋄} (P_{λ / L}^{L}, id) \geq (1 - e^{- λ}) (1 - d^{- 2}) > 0.

Proof. Since $R_{d}^{2} = R_{d}$ ,

P_{p}^{L} = (1 - p)^{L} id + (1 - (1 - p)^{L}) R_{d} .

On a maximally entangled input, the replacement part has overlap $d^{- 2}$ with $∣ Ω_{d} ⟩$ . Testing that projector proves the lower bound before taking the limit.

Displayed formulas in this passage

unlabeled · line 627
$R_{d} (X) = Tr (X) I_{d} / d, P_{p} = (1 - p) id + p R_{d} .$
unlabeled · line 633
$L \to \infty lim inf d_{⋄} (P_{λ / L}^{L}, id) \geq (1 - e^{- λ}) (1 - d^{- 2}) > 0.$
unlabeled · line 642
$P_{p}^{L} = (1 - p)^{L} id + (1 - (1 - p)^{L}) R_{d} .$

Proposition: Vanishing local errors can accumulate

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