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

The unsharp source instrument

An explicit resource-budgeted feedback witness › The unsharp source instrument · lines 1821–1856

VII-G@1#eq:unsharp, VII-G@1#eq:example-unitaries, VII-G@1#eq:example-instrument

Let $H = C^{2}$ , $P_{j} = ∣ j ⟩ ⟨ j ∣$ , and

Q = P_{1} - P_{0} .

Choose $0 \leq η \leq 1$ and define

E_{a} = \frac{1}{2} (I + a η Q), M_{a} = E_{a} .

Let

U_{k, q} = exp (- i ϑ_{k, q} Q), ϑ_{k, q} = π q /8.

For an active input $g = (k, q)$ , set

Ψ_{Γ_{a} (g) ∣ g} (X) = U_{Γ_{a} (g)} M_{a} X M_{a} U_{Γ_{a} (g)}^{†},

with all other branches zero. At $k = B$ , use $Ψ_{g ∣ g} = id$ .

This is an operational measurement–feedback prescription. Newtonian measurement-feedback models provide related motivation, not a gravitational validation of this finite construction TilloyDiosi2016.

Displayed formulas in this passage

VII-G@1#eq:unsharp · line 1828
$E_{a} = \frac{1}{2} (I + a η Q), M_{a} = E_{a} .$
VII-G@1#eq:example-unitaries · line 1835
$U_{k, q} = exp (- i ϑ_{k, q} Q), ϑ_{k, q} = π q /8.$
VII-G@1#eq:example-instrument · line 1842
$Ψ_{Γ_{a} (g) ∣ g} (X) = U_{Γ_{a} (g)} M_{a} X M_{a} U_{Γ_{a} (g)}^{†},$
unlabeled · line 1824
$Q = P_{1} - P_{0} .$

The unsharp source instrument

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