Manuscript Em05-A@1 · Em05-A-v1.tex · theorem · 9cac0def

Theorem: Physicality and explicit channels

The complete signed extraction channel › Kraus operators and the complete complement · lines 357–398 · proof included or linked

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

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The map $V_{η}$ is an isometry for every $η \in [0, 1]$ . Both channels in Definition def:extractor 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 \in {00, 01, 10, 11} \sum Tr (A_{a} X A_{b}^{†}) ∣ a ⟩ ⟨ b ∣_{E} .

These formulas retain arbitrary external references.

Proof. The two columns of $V_{η}$ are orthogonal, and the occupied column has squared norm

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

Thus $V_{η}^{†} V_{η} = I_{2}$ . Reading its coefficients in the residual basis gives (eq:kraus). Direct calculation gives

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

The accessible map is therefore CPTP. The second partial trace of the same isometry is CPTP and gives (eq:complement). Tensoring the defining isometry with $I_{Q}$ proves the reference statement.

Displayed formulas in this passage

Em05-A@1#eq:kraus · line 363
$A_{00} A_{10} = ∣00 ⟩ ⟨ 0∣ + η ∣11 ⟩ ⟨ 1∣, = - η t ∣01 ⟩ ⟨ 1∣, A_{01} A_{11} = η t ∣10 ⟩ ⟨ 1∣, = t ∣00 ⟩ ⟨ 1∣.$
Em05-A@1#eq:complement · line 371
$C_{η} (X) = a, b \in {00, 01, 10, 11} \sum Tr (A_{a} X A_{b}^{†}) ∣ a ⟩ ⟨ b ∣_{E} .$
unlabeled · line 384
$η^{2} + 2 η t + t^{2} = (η + t)^{2} = 1.$
unlabeled · line 390
$a \sum A_{a}^{†} A_{a} = diag (1, η^{2} + 2 η t + t^{2}) = I_{2} .$

Nearby in this section

remark: Remark: Fixed spectator systems