Manuscript Hawk05-B@1 · Hawk05-B.tex · proposition · f2e5d4f5

Proposition: Entropy after ensemble forgetting

Derivation of the finite Page curve › Two averages that must not be confused · lines 734–756 · proof included or linked

Hawk05-B@1#prop:averaging

For the fixed pure input,

E_{U} ρ_{R_{k}}^{U, ψ} = \frac{I _{R_{k}}}{r _{k}}, S (E_{U} ρ_{R_{k}}^{U, ψ}) = lo g_{2} r_{k} .

If the unitary is not known and the accessible channel itself is averaged over Haar measure, then

E_{U} N_{k}^{U} (X) = Tr (X) \frac{I _{R_{k}}}{r _{k}} .

Its recovery error is $a_{d}$ at every cut, including $k = N$ .

Proof. The Haar twirl of $J X J^{†}$ is $Tr (X) I_{D} / D$ . Take the residual partial trace. The recovery conclusion is Lemma lem:benchmarks.

Displayed formulas in this passage

unlabeled · line 737
$E_{U} ρ_{R_{k}}^{U, ψ} = \frac{I _{R_{k}}}{r _{k}}, S (E_{U} ρ_{R_{k}}^{U, ψ}) = lo g_{2} r_{k} .$
unlabeled · line 744
$E_{U} N_{k}^{U} (X) = Tr (X) \frac{I _{R_{k}}}{r _{k}} .$