Manuscript Oai276-A@1 · Oai276-A.tex · lemma · 215068af

Lemma: Complete channel specification

Stage 0: freeze the channel, units, and claims › The generalized amplitude-damping convention · lines 292–313 · proof included or linked

Oai276-A@1#lem:kraus

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The matrices in (eq:kraus) satisfy

r = 0 \sum^{3} L_{r}^{†} L_{r} = I_{2}

and implement (eq:channel). Thus the formula defines a CPTP map, including all parameter endpoints.

Proof. The two diagonal contributions to the completeness sum are

(1 - ν) diag (1, 1 - γ) + ν diag (1 - γ, 1) .

The jump contributions are $γ (1 - ν) ∣1 ⟩ ⟨ 1∣ + γ ν ∣0 ⟩ ⟨ 0∣$ . Their sum is $I_{2}$ . Direct multiplication gives excited-state population $(1 - γ) p + γ ν$ and coherence multiplier $k$ .

Displayed formulas in this passage

unlabeled · line 295
$r = 0 \sum^{3} L_{r}^{†} L_{r} = I_{2}$
unlabeled · line 304
$(1 - ν) diag (1, 1 - γ) + ν diag (1 - γ, 1) .$

Nearby in this section

section: The generalized amplitude-damping convention
section: The generalized amplitude-damping convention