Manuscript Hawk01-B@1 · Hawk01-B.tex · theorem · cfa976f1

Theorem: Exact entropy and reference-information ledger

Exact recovery, leakage, and reference correlations › Entropy and mutual information · lines 1065–1107 · proof included or linked

Hawk01-B@1#thm:information, Hawk01-B@1#eq:radiationentropy, Hawk01-B@1#eq:residualentropy, Hawk01-B@1#eq:MIradiation, Hawk01-B@1#eq:MIresidual, Hawk01-B@1#eq:CMIincrement

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For an arbitrary code state $ρ$ and $t \geq 1$ ,

H (R_{\leq t}) H (B_{t}) = H (b_{t}) + 2 L_{d} + p_{t} H (ρ), = H (b_{t}) + 2 L_{d} + s_{t} H (ρ) .

For the input $∣ Φ_{d} ⟩^{S A}$ ,

I (S : R_{\leq t}) I (S : B_{t}) I (S : R_{t} ∣ R_{< t}) = 2 p_{t} L_{d}, = 2 s_{t} L_{d}, = 2 w_{t} L_{d}, t \geq 2.

The key emission has $I (S : R_{1}) = 0$ .

Proof. In (eq:accessibleprefix), completed blocks have conditional entropy $H (ρ)$ , pending blocks have entropy zero, and the classical key has entropy $2 L_{d}$ . The orthogonal branch labels contribute $H (b_{t})$ . This proves (eq:radiationentropy). The complementary channel has the opposite placement of the logical state, proving (eq:residualentropy).

For the Bell input, $H (ρ) = L_{d}$ and the global state $S R B_{t}$ is pure. Therefore

I (S : R) = L_{d} + H (R) - H (B_{t}) = 2 p_{t} L_{d},

and the complementary identity gives $I (S : B_{t}) = 2 s_{t} L_{d}$ . The conditional mutual information is the difference of successive reference–radiation mutual informations. Since $p_{t} - p_{t - 1} = w_{t}$ , (eq:CMIincrement) follows.

Displayed formulas in this passage

Hawk01-B@1#eq:radiationentropy · line 1068
$H (R_{\leq t}) H (B_{t}) = H (b_{t}) + 2 L_{d} + p_{t} H (ρ), = H (b_{t}) + 2 L_{d} + s_{t} H (ρ) .$
Hawk01-B@1#eq:residualentropy · line 1068
$H (R_{\leq t}) H (B_{t}) = H (b_{t}) + 2 L_{d} + p_{t} H (ρ), = H (b_{t}) + 2 L_{d} + s_{t} H (ρ) .$
Hawk01-B@1#eq:MIradiation · line 1077
$I (S : R_{\leq t}) I (S : B_{t}) I (S : R_{t} ∣ R_{< t}) = 2 p_{t} L_{d}, = 2 s_{t} L_{d}, = 2 w_{t} L_{d}, t \geq 2.$
Hawk01-B@1#eq:MIresidual · line 1077
$I (S : R_{\leq t}) I (S : B_{t}) I (S : R_{t} ∣ R_{< t}) = 2 p_{t} L_{d}, = 2 s_{t} L_{d}, = 2 w_{t} L_{d}, t \geq 2.$
Hawk01-B@1#eq:CMIincrement · line 1077
$I (S : R_{\leq t}) I (S : B_{t}) I (S : R_{t} ∣ R_{< t}) = 2 p_{t} L_{d}, = 2 s_{t} L_{d}, = 2 w_{t} L_{d}, t \geq 2.$
unlabeled · line 1100
$I (S : R) = L_{d} + H (R) - H (B_{t}) = 2 p_{t} L_{d},$

Theorem: Exact entropy and reference-information ledger

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