Manuscript Em03-B@2 · Em03-B-v2.tex · proposition · c1a0977d

Proposition: Channel and observable identification

The B1–B3 mathematical interface › The saturating carrier · lines 491–519 · proof included or linked

Em03-B@2#prop:carrier

The map $C_{N}$ is CPTP. Its adjoint is

C_{N}^{*} (A) = P_{N} V_{N}^{†} A V_{N} P_{N} + ⟨ N ∣ A ∣ N ⟩_{C} Q_{N} .

For $D_{N} = \sum_{k = 0}^{N} k ∣ k ⟩ ⟨ k ∣_{C}$ ,

C_{N}^{*} (D_{N}) = min (N_{h}, N) .

The readout effects pulled back to the oscillator are

C_{N}^{*} (∣ k ⟩ ⟨ k ∣_{C}) = ⎩ ⎨ ⎧ ∣ k ⟩ ⟨ k ∣, n = N \sum^{\infty} ∣ n ⟩ ⟨ n ∣, k < N, k = N .

Proof. The Kraus maps are $K_{0} = V_{N} P_{N}$ and $K_{n} = ∣ N ⟩_{C} ⟨ n ∣$ , $n > N$ . Their effects sum strongly to $I$ . The positive output series converges in trace norm on positive trace-class inputs. This proves the normal CPTP extension. The adjoint and its specializations follow by substitution.

Displayed formulas in this passage

unlabeled · line 494
$C_{N}^{*} (A) = P_{N} V_{N}^{†} A V_{N} P_{N} + ⟨ N ∣ A ∣ N ⟩_{C} Q_{N} .$
unlabeled · line 499
$C_{N}^{*} (D_{N}) = min (N_{h}, N) .$
unlabeled · line 503
$C_{N}^{*} (∣ k ⟩ ⟨ k ∣_{C}) = ⎩ ⎨ ⎧ ∣ k ⟩ ⟨ k ∣, n = N \sum^{\infty} ∣ n ⟩ ⟨ n ∣, k < N, k = N .$

Proposition: Channel and observable identification

Displayed formulas in this passage

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