Manuscript Hawk04-A@1 · Hawk04-A.tex · lemma · 526092ba

Lemma: Local isometric embeddings preserve leakage

Endpoints and the missing gravitational implications › Access-preserving output identifications · lines 1826–1844 · proof included or linked

Hawk04-A@1#lem:embedding

The leakage distance of $C_{t}$ is unchanged under the residual isometric embedding $J_{B, t}$ .

Proof. Embedding any replacement state proves one inequality. For the other, choose a fixed state $τ$ on $B_{t}$ and use the CPTP retraction

Y ⟼ J_{B, t}^{†} Y J_{B, t} + Tr ((I - J_{B, t} J_{B, t}^{†}) Y) τ .

It exactly reverses embedded outputs. Contraction applied to any candidate replacement state proves the reverse inequality.

Displayed formulas in this passage

unlabeled · line 1836
$Y ⟼ J_{B, t}^{†} Y J_{B, t} + Tr ((I - J_{B, t} J_{B, t}^{†}) Y) τ .$