Manuscript En01-A@1 · En01-A.tex · corollary · 558b972b

Corollary: Exponential suppression with an accessible root

Cross operators, fidelity, screening, and noisy architecture › Independent noisy fragments · lines 1521–1555 · proof included or linked

En01-A@1#cor:noisy

Consider

∣Ψ ⟩ = p_{0} ∣0 ⟩_{R} a = 1 ⨂^{c - 1} ∣ e_{a}^{(0)} ⟩ + p_{1} ∣1 ⟩_{R} a = 1 ⨂^{c - 1} ∣ e_{a}^{(1)} ⟩ .

Dephase the perfect root. If $S$ contains that root and $m - 1$ fragments, then

δ_{J} (S) = p_{0} p_{1} a \in / S \prod γ_{a} .

For uniform $γ_{a} = γ$ ,

δ_{J} (S) = p_{0} p_{1} γ^{c - m} .

Complex overlap phases do not change this equality.

Proof. Tracing the inaccessible fragments yields

Γ^{S} = z ∣ u ⟩ ⟨ v ∣, ∣ z ∣ = a \in / S \prod γ_{a} .

The accessible vectors $u, v$ are orthogonal because $S$ retains the perfect root. The Hermitian cross sum therefore has trace norm $2∣ z ∣$ .

Displayed formulas in this passage

unlabeled · line 1524
$∣Ψ ⟩ = p_{0} ∣0 ⟩_{R} a = 1 ⨂^{c - 1} ∣ e_{a}^{(0)} ⟩ + p_{1} ∣1 ⟩_{R} a = 1 ⨂^{c - 1} ∣ e_{a}^{(1)} ⟩ .$
unlabeled · line 1533
$δ_{J} (S) = p_{0} p_{1} a \in / S \prod γ_{a} .$
unlabeled · line 1539
$δ_{J} (S) = p_{0} p_{1} γ^{c - m} .$
unlabeled · line 1547
$Γ^{S} = z ∣ u ⟩ ⟨ v ∣, ∣ z ∣ = a \in / S \prod γ_{a} .$