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

Theorem: Isometry, Kraus family, and complete complement

The complete signed extraction channel · lines 477–518 · proof included or linked

Em05-B@1#thm:channels, Em05-B@1#eq:kraus, Em05-B@1#eq:complement

The map $V_{η}$ is an isometry and both declared channels are CPTP. A Kraus family for $N_{η}$ , indexed by residual occupation, is

A_{00} A_{10} = ∣00 ⟩ ⟨ 0∣ + η ∣11 ⟩ ⟨ 1∣, = - η t ∣01 ⟩ ⟨ 1∣, A_{01} A_{11} = η t ∣10 ⟩ ⟨ 1∣, = t ∣00 ⟩ ⟨ 1∣.

The complete complement is

C_{η} (X) = a, b \sum Tr (A_{a} X A_{b}^{†}) ∣ a ⟩ ⟨ b ∣_{E}, a, b \in {00, 01, 10, 11} .

These channel identities apply with arbitrary external references.

Proof. The columns are orthogonal and the occupied column has squared norm

η^{2} + 2 η t + t^{2} = (η + t)^{2} = 1.

Hence $V_{η}^{†} V_{η} = I_{M}$ . Reading its residual coefficients gives (eq:kraus), with

a \sum A_{a}^{†} A_{a} = diag (1, η^{2} + 2 η t + t^{2}) = I_{M} .

Taking the other partial trace gives (eq:complement). Tensoring the isometry with a reference proves the referenced versions.

Displayed formulas in this passage

Em05-B@1#eq:kraus · line 482
$A_{00} A_{10} = ∣00 ⟩ ⟨ 0∣ + η ∣11 ⟩ ⟨ 1∣, = - η t ∣01 ⟩ ⟨ 1∣, A_{01} A_{11} = η t ∣10 ⟩ ⟨ 1∣, = t ∣00 ⟩ ⟨ 1∣.$
Em05-B@1#eq:complement · line 493
$C_{η} (X) = a, b \sum Tr (A_{a} X A_{b}^{†}) ∣ a ⟩ ⟨ b ∣_{E}, a, b \in {00, 01, 10, 11} .$
unlabeled · line 506
$η^{2} + 2 η t + t^{2} = (η + t)^{2} = 1.$
unlabeled · line 511
$a \sum A_{a}^{†} A_{a} = diag (1, η^{2} + 2 η t + t^{2}) = I_{M} .$