Manuscript Em03-G@1 · Em03-G-v1.tex · theorem · 0bb3a8a3

Theorem: All four matrix-unit characteristic functions

The complete scalar radiation channel · lines 670–714 · proof included or linked

Em03-G@1#thm:matrixunits, Em03-G@1#eq:fourchars

The exact source-to-field channel obeys

N_{G} (∣ 0 ⟩ ⟨ 0 ∣) (W (l)) N_{G} (∣ 1 ⟩ ⟨ 1 ∣) (W (l)) N_{G} (∣ 0 ⟩ ⟨ 1 ∣) (W (l)) N_{G} (∣ 1 ⟩ ⟨ 0 ∣) (W (l)) = q (l) e^{- i A_{l}} cos B_{l}, = q (l) e^{+ i A_{l}} cos B_{l}, = - i sin B_{l} q (l - 2 a_{1}), = - i sin B_{l} q (l + 2 a_{1}) .

The characteristic functional of every input follows by linearity. In particular, off-diagonal source dependence is generally nonzero.

If $a_{2} = 0$ , the result is the single- $Z$ -observable channel of Em03-F. If $a_{1} = 0$ , it is a single- $X$ -observable measure-and-prepare channel.

Proof. Let $r_{x z} = - z a_{1} - x a_{2}$ . Weyl multiplication gives

ω_{U} (B_{x w}^{†} W (l) B_{x z}) = exp [- \frac{i}{2} (z + w) A_{l} - i x B_{l}] \times q (l + (w - z) a_{1}) .

The phase from $s (r_{x w}, r_{x z}) = x (w - z) θ$ cancels the explicit branch-phase difference. For $z = w$ , the coefficient $m_{x z} m_{x w} = 1/2$ . For $z \neq = w$ , it is $x /2$ . Summing over $x$ proves all four formulas. The two reductions follow either from the endpoint unitary or from these characteristic functions.

Displayed formulas in this passage

Em03-G@1#eq:fourchars · line 673
$N_{G} (∣ 0 ⟩ ⟨ 0 ∣) (W (l)) N_{G} (∣ 1 ⟩ ⟨ 1 ∣) (W (l)) N_{G} (∣ 0 ⟩ ⟨ 1 ∣) (W (l)) N_{G} (∣ 1 ⟩ ⟨ 0 ∣) (W (l)) = q (l) e^{- i A_{l}} cos B_{l}, = q (l) e^{+ i A_{l}} cos B_{l}, = - i sin B_{l} q (l - 2 a_{1}), = - i sin B_{l} q (l + 2 a_{1}) .$
unlabeled · line 698
$ω_{U} (B_{x w}^{†} W (l) B_{x z}) = exp [- \frac{i}{2} (z + w) A_{l} - i x B_{l}] \times q (l + (w - z) a_{1}) .$

Theorem: All four matrix-unit characteristic functions

Displayed formulas in this passage

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