Manuscript XVII-A@1 · XVII-A.tex · section · 711d7ee4

One analytical diagnostic

One analytical diagnostic · lines 830–900

XVII-A@1#eq:diagnostic-fields, XVII-A@1#eq:diagnostic-effect, XVII-A@1#eq:diagnostic-waveform, XVII-A@1#eq:diagnostic-agreement

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Specialize the declared model to $K = 1$ , $d = 2$ . Let

X = (0110), Y = (0 i - i 0), Z = (10 0 - 1) .

Then

Q_{bulk} = \frac{X}{π}, β (t) = \frac{X cos t + Y sin t}{π}, Q_{B} = \frac{X}{π} .

Since $λ_{*} / π = π /6$ and $X^{2} = 1$ ,

O_{bulk} = O_{B} = \frac{1}{2} 1 + \frac{1}{4} X .

Choose the explicitly prepared state

∣ ψ ⟩ = \frac{3}{2} ∣ 0 ⟩ + \frac{1}{2} ∣ 1 ⟩, ρ_{ψ} = ∣ ψ ⟩ ⟨ ψ ∣ .

Its Pauli expectations are

Tr (X ρ_{ψ}) = \frac{3}{2}, Tr (Y ρ_{ψ}) = 0.

Thus the boundary mean waveform is

ω_{B} (β (t)) = \frac{3}{2 π} cos t .

For this particular two-dimensional effect, the boundary algebra identity (eq:diagnostic-effect) also gives

ω_{B} (O_{B}) = \frac{1}{2} + \frac{π}{4} ω_{B} (β (0)) = \frac{1}{2} + \frac{3}{8} .

The direct bulk expression agrees:

Tr (O_{bulk} ρ_{ψ}) = \frac{1}{2} + \frac{3}{8} .

The complete reconstructed record is therefore

R_{B} (N_{B} (ρ_{ψ})) = (\frac{1}{2} - \frac{3}{8}) ∣ 0 ⟩ ⟨ 0 ∣ + (\frac{1}{2} + \frac{3}{8}) ∣ 1 ⟩ ⟨ 1 ∣ .

This is an exact analytical evaluation, not a numerical execution. The special first-moment formula used here follows from (eq:diagnostic-effect); it is not an assertion that arbitrary nonlinear field effects are determined by a first moment. VII-G's instrument, VII-H's susceptibility, and Em04-A's concurrence are not this diagnostic CheungVII,CheungEm04.

Displayed formulas in this passage

XVII-A@1#eq:diagnostic-fields · line 841
$Q_{bulk} = \frac{X}{π}, β (t) = \frac{X cos t + Y sin t}{π}, Q_{B} = \frac{X}{π} .$
XVII-A@1#eq:diagnostic-effect · line 850
$O_{bulk} = O_{B} = \frac{1}{2} 1 + \frac{1}{4} X .$
XVII-A@1#eq:diagnostic-waveform · line 868
$ω_{B} (β (t)) = \frac{3}{2 π} cos t .$
XVII-A@1#eq:diagnostic-agreement · line 881
$Tr (O_{bulk} ρ_{ψ}) = \frac{1}{2} + \frac{3}{8} .$
unlabeled · line 833
$X = (0110), Y = (0 i - i 0), Z = (10 0 - 1) .$
unlabeled · line 857
$∣ ψ ⟩ = \frac{3}{2} ∣ 0 ⟩ + \frac{1}{2} ∣ 1 ⟩, ρ_{ψ} = ∣ ψ ⟩ ⟨ ψ ∣ .$
unlabeled · line 862
$Tr (X ρ_{ψ}) = \frac{3}{2}, Tr (Y ρ_{ψ}) = 0.$
unlabeled · line 875
$ω_{B} (O_{B}) = \frac{1}{2} + \frac{π}{4} ω_{B} (β (0)) = \frac{1}{2} + \frac{3}{8} .$
unlabeled · line 886
$R_{B} (N_{B} (ρ_{ψ})) = (\frac{1}{2} - \frac{3}{8}) ∣ 0 ⟩ ⟨ 0 ∣ + (\frac{1}{2} + \frac{3}{8}) ∣ 1 ⟩ ⟨ 1 ∣ .$