Manuscript Em05-B@1 · Em05-B-v1.tex · proposition · 6c050f5b

Proposition: Completeness, conditional states, and no signaling

The specified decoder and its flagged instrument › Visible off-code events · lines 1107–1164 · proof included or linked

Em05-B@1#prop:instrument, Em05-B@1#eq:codeBranch, Em05-B@1#eq:offBranch, Em05-B@1#eq:referencedCodeBranch, Em05-B@1#eq:referencedOffBranch

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The map $D_{⋆} : R \to K M^{'}$ is CPTP and

Tr_{K} D_{⋆} = D_{⋆} .

Writing $B_{η} = diag (1, η)$ , its logical branches are

I_{c} N_{η} (X) I_{o} N_{η} (X) = B_{η} X B_{η}^{†} + t^{2} ⟨ 1∣ X ∣1 ⟩ ∣0 ⟩ ⟨ 0∣, = η t ⟨ 1∣ X ∣1 ⟩ I_{M^{'}} .

For input (eq:input), their probabilities are

p_{o} = 2 p η t, p_{c} = 1 - 2 p η t .

For a referenced input $ρ_{QM}$ , put $H_{Q} = ⟨ 1 ∣_{M} ρ_{QM} ∣1 ⟩_{M}$ . The unnormalized branch outputs are

τ^{c}_{Q M^{'}} τ^{o}_{Q M^{'}} = (I_{Q} \otimes B_{η}) ρ_{QM} (I_{Q} \otimes B_{η}^{†}) + t^{2} H_{Q} \otimes ∣0 ⟩ ⟨ 0∣, = η t H_{Q} \otimes I_{M^{'}} .

Whenever $p_{y} > 0$ , the conditional state is $τ^{y} / p_{y}$ . The unconditional reference obeys

y \sum Tr_{M^{'}} τ^{y} = ρ_{Q} .

Proof. The branch Kraus operators are precisely the code and off-code groups used in Theorem thm:specified. Their effects sum to $I_{R}$ ; attaching orthogonal classical labels makes their sum CPTP. Equations (eq:codeBranch)–(eq:offBranch) follow by grouping (eq:rhoR). Applying the same operator identities with a reference gives (eq:referencedCodeBranch) and (eq:referencedOffBranch). Trace preservation yields the final identity.

Displayed formulas in this passage

Em05-B@1#eq:codeBranch · line 1114
$I_{c} N_{η} (X) I_{o} N_{η} (X) = B_{η} X B_{η}^{†} + t^{2} ⟨ 1∣ X ∣1 ⟩ ∣0 ⟩ ⟨ 0∣, = η t ⟨ 1∣ X ∣1 ⟩ I_{M^{'}} .$
Em05-B@1#eq:offBranch · line 1114
$I_{c} N_{η} (X) I_{o} N_{η} (X) = B_{η} X B_{η}^{†} + t^{2} ⟨ 1∣ X ∣1 ⟩ ∣0 ⟩ ⟨ 0∣, = η t ⟨ 1∣ X ∣1 ⟩ I_{M^{'}} .$
Em05-B@1#eq:referencedCodeBranch · line 1135
$τ^{c}_{Q M^{'}} τ^{o}_{Q M^{'}} = (I_{Q} \otimes B_{η}) ρ_{QM} (I_{Q} \otimes B_{η}^{†}) + t^{2} H_{Q} \otimes ∣0 ⟩ ⟨ 0∣, = η t H_{Q} \otimes I_{M^{'}} .$
Em05-B@1#eq:referencedOffBranch · line 1135
$τ^{c}_{Q M^{'}} τ^{o}_{Q M^{'}} = (I_{Q} \otimes B_{η}) ρ_{QM} (I_{Q} \otimes B_{η}^{†}) + t^{2} H_{Q} \otimes ∣0 ⟩ ⟨ 0∣, = η t H_{Q} \otimes I_{M^{'}} .$
unlabeled · line 1110
$Tr_{K} D_{⋆} = D_{⋆} .$
unlabeled · line 1126
$p_{o} = 2 p η t, p_{c} = 1 - 2 p η t .$
unlabeled · line 1148
$y \sum Tr_{M^{'}} τ^{y} = ρ_{Q} .$

Proposition: Completeness, conditional states, and no signaling

Displayed formulas in this passage

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