Manuscript Hawk02-A@1 · Hawk02-A.tex · proposition · 7fbae745

Proposition: Separate residual marginals are insufficient

Stage 6: endpoint, keys, and residual accounting › Stopping and completion are different claims · lines 1459–1476 · proof included or linked

Hawk02-A@1#prop:jointresidual

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Constant channels to each residual factor separately need not make the joint residual channel constant.

Proof. For a qubit input, define

K (X) = \frac{1}{4} r = 0 \sum^{3} ∣ r ⟩ ⟨ r ∣_{B_{1}} \otimes P_{r} X P_{r}^{†}_{B_{2}} .

The $B_{1}$ channel is $Tr (X) I_{B_{1}} /4$ and the $B_{2}$ channel is $Tr (X) I_{B_{2}} /2$ . The joint output is reversible: read $r$ and undo $P_{r}$ . It therefore depends on the input.

Displayed formulas in this passage

unlabeled · line 1467
$K (X) = \frac{1}{4} r = 0 \sum^{3} ∣ r ⟩ ⟨ r ∣_{B_{1}} \otimes P_{r} X P_{r}^{†}_{B_{2}} .$

Nearby in this section

section: Stopping and completion are different claims
proposition: Proposition: Empty complete residual
section: Stopping and completion are different claims
section: Stopping and completion are different claims