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

Theorem: Accessible and complementary channels

Internally generated key and sequential logical transfer › Complete prefix channels · lines 848–904 · proof included or linked

Hawk01-B@1#thm:prefixchannels, Hawk01-B@1#eq:accessibleprefix, Hawk01-B@1#eq:complementprefix, Hawk01-B@1#eq:localkey, Hawk01-B@1#eq:localdata

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For $t \geq 1$ , with all emitted blocks accessible,

N_{t} (Y) = \frac{1}{d ^{2}} r \sum ∣ r ⟩ ⟨ r ∣_{R_{1}} \otimes [j = 2 \sum^{t} w_{j} J_{j} U_{r} Y U_{r}^{†} J_{j}^{†} + s_{t} Tr (Y) ∣ vac ⟩ ⟨ vac ∣] .

The complete residual complement is

C_{t} (Y) = \frac{1}{d ^{2}} r \sum [j = 2 \sum^{t} w_{j} Tr (Y) ∣ j, r ⟩ ⟨ j, r ∣ + s_{t} I_{f, r} U_{r} Y U_{r}^{†} I_{f, r}^{†}],

where $I_{f, r}$ embeds the pending logical system into its residual summand. Every individual emitted block is message-independent:

N_{R_{1}} (Y) N_{R_{j}} (Y) = Tr (Y) \frac{I _{d^{2}}}{d ^{2}}, = Tr (Y) [(1 - w_{j}) ∣ vac ⟩ ⟨ vac ∣ + w_{j} \frac{I _{d} ^{(1)}}{d}], j \geq 2.

Proof. Trace the orthogonal residual summands in (eq:accumulated). Different key labels also become diagonal because their residual supports are orthogonal. This gives (eq:accessibleprefix).

Conversely, tracing $R_{1}$ removes cross terms in the key. Tracing occupied/vacuum radiation removes cross terms between different transfer times and between transfer and pending branches. The resulting residual state is (eq:complementprefix).

The local formulas follow by tracing all other radiation blocks and applying Lemma lem:twirl.

Displayed formulas in this passage

Hawk01-B@1#eq:accessibleprefix · line 851
$N_{t} (Y) = \frac{1}{d ^{2}} r \sum ∣ r ⟩ ⟨ r ∣_{R_{1}} \otimes [j = 2 \sum^{t} w_{j} J_{j} U_{r} Y U_{r}^{†} J_{j}^{†} + s_{t} Tr (Y) ∣ vac ⟩ ⟨ vac ∣] .$
Hawk01-B@1#eq:complementprefix · line 863
$C_{t} (Y) = \frac{1}{d ^{2}} r \sum [j = 2 \sum^{t} w_{j} Tr (Y) ∣ j, r ⟩ ⟨ j, r ∣ + s_{t} I_{f, r} U_{r} Y U_{r}^{†} I_{f, r}^{†}],$
Hawk01-B@1#eq:localkey · line 877
$N_{R_{1}} (Y) N_{R_{j}} (Y) = Tr (Y) \frac{I _{d^{2}}}{d ^{2}}, = Tr (Y) [(1 - w_{j}) ∣ vac ⟩ ⟨ vac ∣ + w_{j} \frac{I _{d} ^{(1)}}{d}], j \geq 2.$
Hawk01-B@1#eq:localdata · line 877
$N_{R_{1}} (Y) N_{R_{j}} (Y) = Tr (Y) \frac{I _{d^{2}}}{d ^{2}}, = Tr (Y) [(1 - w_{j}) ∣ vac ⟩ ⟨ vac ∣ + w_{j} \frac{I _{d} ^{(1)}}{d}], j \geq 2.$

Theorem: Accessible and complementary channels

Displayed formulas in this passage

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