Manuscript VII-G@1 · VII-G.tex · section · 54f561e8

Restricted unassisted comparison

Assisted stability and finite Choi certificates › Restricted unassisted comparison · lines 1697–1724

VII-G@1#eq:restricted-distance

For an observation space $O$ , define

d_{O} (F, F) = \frac{1}{2} ρ sup B = B^{†} \in O ∥ B ∥_{\infty, G} \leq 1 sup ∣⟨ B, (F - F) (ρ)⟩∣ .

Then

d_{O} (F, F) = \frac{1}{2} B = B^{†} \in O ∥ B ∥_{\infty, G} \leq 1 sup ∥(F^{*} - F^{*}) (B) ∥_{\infty, G} .

Indeed, interchange the suprema and optimize the Hermitian expectation by an eigenstate in a maximizing block. This is the hybrid form of VII-E's accessible-output distance CheungVIIE (Proposition prop:accessible). It is unassisted unless a reference is included explicitly.

Displayed formulas in this passage

VII-G@1#eq:restricted-distance · line 1709
$d_{O} (F, F) = \frac{1}{2} B = B^{†} \in O ∥ B ∥_{\infty, G} \leq 1 sup ∥(F^{*} - F^{*}) (B) ∥_{\infty, G} .$
unlabeled · line 1700
$d_{O} (F, F) = \frac{1}{2} ρ sup B = B^{†} \in O ∥ B ∥_{\infty, G} \leq 1 sup ∣⟨ B, (F - F) (ρ)⟩∣ .$