Manuscript VI-B@1 · VI-B-v1.tex · section · 33798723

A finite evaluation encoding

A finite evaluation encoding · lines 1471–1521

VI-B@1#eq:table, VI-B@1#eq:matrices, VI-B@1#eq:effects, VI-B@1#eq:score

The following construction makes the word “readout” precise. Its inputs are finite classical evaluations already specified by the preceding theorems.

Choose $λ_{a} = a + 1$ , $a = 0, 1, 2, 3$ . Use the source family of Corollary cor:scaling on the fixed named test $f_{s}$ , and the separate plateau family with scalar $λ_{a} ϕ_{w}$ . The four-by-four evaluation table is

b_{k a} = ⎩ ⎨ ⎧ λ_{a} a_{0}, 0, λ_{a}^{3} a_{2}, λ_{a} Δ, k = 0, k = 1, k = 2, k = 3.

Rows $0, 1, 2$ are descended solution-jet evaluations. Row $3$ is the representative gap of a different, explicitly named switched experiment; it is not itself asserted to be a descended observable of that switched identification.

Let

H_{X} = C^{4}, A_{k} = a = 0 \sum^{3} b_{k a} ∣ a ⟩ ⟨ a ∣_{X} .

These are finite Hermitian matrices. The labels $a$ identify classical sample configurations, not one-particle field states.

Choose

B > 2 k, a max ∣ b_{k a} ∣, E_{k, y} = \frac{1}{2} (I + y A_{k} / B), M_{k, y} = E_{k, y}^{1/2}, y = \pm 1.

The effects have spectrum in $(1/4, 3/4)$ , and

y \sum E_{k, y} = I, y \sum B y E_{k, y} = A_{k} .

Displayed formulas in this passage

VI-B@1#eq:table · line 1483
$b_{k a} = ⎩ ⎨ ⎧ λ_{a} a_{0}, 0, λ_{a}^{3} a_{2}, λ_{a} Δ, k = 0, k = 1, k = 2, k = 3.$
VI-B@1#eq:matrices · line 1499
$H_{X} = C^{4}, A_{k} = a = 0 \sum^{3} b_{k a} ∣ a ⟩ ⟨ a ∣_{X} .$
VI-B@1#eq:effects · line 1508
$B > 2 k, a max ∣ b_{k a} ∣, E_{k, y} = \frac{1}{2} (I + y A_{k} / B), M_{k, y} = E_{k, y}^{1/2}, y = \pm 1.$
VI-B@1#eq:score · line 1516
$y \sum E_{k, y} = I, y \sum B y E_{k, y} = A_{k} .$

A finite evaluation encoding

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