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

Proposition: Exact source and disturbance formulas

An explicit resource-budgeted feedback witness › The unsharp source instrument · lines 1857–1898 · proof included or linked

VII-G@1#prop:unsharp, VII-G@1#eq:unsharp-p, VII-G@1#eq:unsharp-signal, VII-G@1#eq:unsharp-coherence, VII-G@1#eq:drift, VII-G@1#eq:variance

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The example is a complete hybrid channel. For a normalized active matter state $ρ$ ,

p_{a} p_{+} - p_{-} = \frac{1}{2} (1 + a η Tr (Qρ)), = η Tr (Qρ) .

Before the geometry-dependent unitary, its nonselective source map

M_{η} (X) = M_{+} X M_{+} + M_{-} X M_{-}

preserves diagonal blocks and satisfies

P_{0} M_{η} (X) P_{1} = 1 - η^{2} P_{0} X P_{1} .

Writing $x = δ q$ , the active geometric increment has

E [Δ x ∣ ρ] Var (Δ x ∣ ρ) = δ η Tr (Qρ), = δ^{2} (1 - η^{2} Tr (Qρ)^{2}) .

Proof. The positive effects sum to $I$ , so $\sum_{a} M_{a}^{†} M_{a} = I$ . Unitary feedback does not change branch traces, proving (eq:unsharp-p) and completeness.

On $P_{0}$ , the two square-root eigenvalues are $(1 - η) /2$ and $(1 + η) /2$ , in opposite order to their values on $P_{1}$ . Their two cross products sum to $1 - η^{2}$ ; their squared values sum to one on each sector. This proves (eq:unsharp-coherence). Finally, $Δ x = δ a$ , so its mean follows from (eq:unsharp-signal) and its second moment is $δ^{2}$ .

Displayed formulas in this passage

VII-G@1#eq:unsharp-p · line 1861
$p_{a} p_{+} - p_{-} = \frac{1}{2} (1 + a η Tr (Qρ)), = η Tr (Qρ) .$
VII-G@1#eq:unsharp-signal · line 1861
$p_{a} p_{+} - p_{-} = \frac{1}{2} (1 + a η Tr (Qρ)), = η Tr (Qρ) .$
VII-G@1#eq:unsharp-coherence · line 1870
$P_{0} M_{η} (X) P_{1} = 1 - η^{2} P_{0} X P_{1} .$
VII-G@1#eq:drift · line 1877
$E [Δ x ∣ ρ] Var (Δ x ∣ ρ) = δ η Tr (Qρ), = δ^{2} (1 - η^{2} Tr (Qρ)^{2}) .$
VII-G@1#eq:variance · line 1877
$E [Δ x ∣ ρ] Var (Δ x ∣ ρ) = δ η Tr (Qρ), = δ^{2} (1 - η^{2} Tr (Qρ)^{2}) .$
unlabeled · line 1866
$M_{η} (X) = M_{+} X M_{+} + M_{-} X M_{-}$

Proposition: Exact source and disturbance formulas

Displayed formulas in this passage

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