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

Proposition: Complete noisy channel

Population-preserving dephasing and its full environment · lines 1482–1525 · proof included or linked

Em05-B@1#prop:noisyChannel, Em05-B@1#eq:noisyComplement, Em05-B@1#eq:noisyPurity

A Kraus family is

L_{a, 0} = a_{λ} A_{a}, L_{a, 1} = b_{λ} S_{R} A_{a} .

The complete complement is

C_{η, λ} (X) = a, s, b, u \sum Tr (L_{a, s} X L_{b, u}^{†}) ∣ a, s ⟩ ⟨ b, u ∣_{E F}, s, u \in {0, 1} .

The accessible marginal is (eq:rhoR) with $η z$ replaced by $λ η z$ . Its occupation probabilities and $⟨ N_{R} ⟩$ are unchanged for every input. Its purity is

P_{R}^{(λ)} = (1 - p + p t^{2})^{2} + p^{2} η^{4} + 2 p^{2} η^{2} t^{2} + 2 λ^{2} η^{2} ∣ z ∣^{2} .

For a Bell input, the accessible marginal is independent of $λ$ for every $η$ .

Proof. The dilation has $U_{λ}^{†} U_{λ} = I_{R}$ ; the displayed Kraus effects therefore sum to $I_{M}$ . Taking both partial traces gives the channel and (eq:noisyComplement). Conjugation by $S_{R}$ preserves diagonal entries and negates the $00, 11$ coherence, so the mixture multiplies it by $a_{λ} - b_{λ} = λ$ . Squaring the entries gives (eq:noisyPurity). For a Bell input $z = 0$ .

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Em05-B@1#eq:noisyComplement · line 1491
$C_{η, λ} (X) = a, s, b, u \sum Tr (L_{a, s} X L_{b, u}^{†}) ∣ a, s ⟩ ⟨ b, u ∣_{E F}, s, u \in {0, 1} .$
Em05-B@1#eq:noisyPurity · line 1504
$P_{R}^{(λ)} = (1 - p + p t^{2})^{2} + p^{2} η^{4} + 2 p^{2} η^{2} t^{2} + 2 λ^{2} η^{2} ∣ z ∣^{2} .$
unlabeled · line 1485
$L_{a, 0} = a_{λ} A_{a}, L_{a, 1} = b_{λ} S_{R} A_{a} .$

Proposition: Complete noisy channel

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