Manuscript Em03-F@1 · Em03-F-v1.tex · theorem · 0d4d148b

Theorem: All source matrix units

The source-to-field channel and retained-source back-action · lines 737–778 · proof included or linked

Em03-F@1#thm:fieldchannel, Em03-F@1#eq:fieldchannel, Em03-F@1#eq:offdiagonalzero

The field-output map is the channel into field state functionals

N_{F} (ρ) = s = \pm 1 \sum ⟨ s ∣ ρ ∣ s ⟩ ω_{s} .

Its Heisenberg adjoint is the unital CP map

N_{F}^{*} (A) = s \sum ω_{s} (A) ∣ s ⟩ ⟨ s ∣ .

In particular,

N_{F} (∣ + ⟩ ⟨ + ∣) = ω_{+}, N_{F} (∣ - ⟩ ⟨ - ∣) = ω_{-},

N_{F} (∣ + ⟩ ⟨ - ∣) = N_{F} (∣ - ⟩ ⟨ + ∣) = 0.

For every reference-source input, its referenced output is

s \sum ⟨ s ∣ ρ_{R S} ∣ s ⟩ \otimes ω_{s} .

Proof. Expand the controlled unitary (eq:UF). Tracing the source keeps exactly its diagonal blocks. The global phase cancels. This proves the channel formula and all matrix-unit identities. Positivity of each state functional proves complete positivity of the displayed adjoint; their normalization proves unitality. The same block calculation retains $R$ .

Displayed formulas in this passage

Em03-F@1#eq:fieldchannel · line 740
$N_{F} (ρ) = s = \pm 1 \sum ⟨ s ∣ ρ ∣ s ⟩ ω_{s} .$
Em03-F@1#eq:offdiagonalzero · line 756
$N_{F} (∣ + ⟩ ⟨ - ∣) = N_{F} (∣ - ⟩ ⟨ + ∣) = 0.$
unlabeled · line 748
$N_{F}^{*} (A) = s \sum ω_{s} (A) ∣ s ⟩ ⟨ s ∣ .$
unlabeled · line 752
$N_{F} (∣ + ⟩ ⟨ + ∣) = ω_{+}, N_{F} (∣ - ⟩ ⟨ - ∣) = ω_{-},$
unlabeled · line 764
$s \sum ⟨ s ∣ ρ_{R S} ∣ s ⟩ \otimes ω_{s} .$

Theorem: All source matrix units

Displayed formulas in this passage

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