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

Proposition: Exact fragment product law

Cross operators, fidelity, screening, and noisy architecture › Independent noisy fragments · lines 1491–1519 · proof included or linked

En01-A@1#prop:product-law

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Suppose the inaccessible branch states factorize as

σ_{r}^{S} = a ⨂ ∣ e_{a}^{(r)} ⟩ ⟨ e_{a}^{(r)} ∣ \otimes τ_{r}, σ_{r^{'}}^{S} = a ⨂ ∣ e_{a}^{(r^{'})} ⟩ ⟨ e_{a}^{(r^{'})} ∣ \otimes τ_{r^{'}} .

Then

v_{r r^{'}} (S) = a \prod ∣ ⟨ e_{a}^{(r^{'})} ∣ e_{a}^{(r)} ⟩ ∣ F (τ_{r}, τ_{r^{'}}) .

In particular, the overlap product alone is an upper bound, and is an equality when the residual fidelity is one.

Proof. Fidelity is multiplicative under tensor products: both the square root and the trace norm factorize. For pure factors, $F (∣ e ⟩ ⟨ e ∣, ∣ e^{'} ⟩ ⟨ e^{'} ∣) = ∣ ⟨ e^{'} ∣ e ⟩ ∣$ . Apply Theorem thm:fidelity.

Displayed formulas in this passage

unlabeled · line 1493
$σ_{r}^{S} = a ⨂ ∣ e_{a}^{(r)} ⟩ ⟨ e_{a}^{(r)} ∣ \otimes τ_{r}, σ_{r^{'}}^{S} = a ⨂ ∣ e_{a}^{(r^{'})} ⟩ ⟨ e_{a}^{(r^{'})} ∣ \otimes τ_{r^{'}} .$
unlabeled · line 1503
$v_{r r^{'}} (S) = a \prod ∣ ⟨ e_{a}^{(r^{'})} ∣ e_{a}^{(r)} ⟩ ∣ F (τ_{r}, τ_{r^{'}}) .$

Proposition: Exact fragment product law

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