Manuscript VII-G@1 · VII-G.tex · lemma · 54f561e8

Lemma: Completely positive branches of the identity

Two limitations of informative classical feedback › Identity decompositions · lines 1147–1172 · proof included or linked

VII-G@1#lem:identity-branches

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If completely positive maps satisfy

a \sum Ψ_{a} = id_{L (H)},

then

Ψ_{a} = p_{a} id, p_{a} \geq 0, a \sum p_{a} = 1.

In particular, all record probabilities are state independent.

Proof. The Choi matrix of the identity is

J (id) = ∣ Ω ⟩ ⟨ Ω ∣, ∣ Ω ⟩ = i = 1 \sum^{d} ∣ i ⟩ ∣ i ⟩,

and has rank one. Each positive $J (Ψ_{a})$ lies below $J (id)$ , so its range is contained in the one-dimensional range of $J (id)$ . Thus $J (Ψ_{a}) = p_{a} J (id)$ . Choi injectivity gives the result.

Displayed formulas in this passage

unlabeled · line 1150
$a \sum Ψ_{a} = id_{L (H)},$
unlabeled · line 1154
$Ψ_{a} = p_{a} id, p_{a} \geq 0, a \sum p_{a} = 1.$
unlabeled · line 1163
$J (id) = ∣ Ω ⟩ ⟨ Ω ∣, ∣ Ω ⟩ = i = 1 \sum^{d} ∣ i ⟩ ∣ i ⟩,$

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section: Identity decompositions