Manuscript X-Enter@1 · X-Enter.tex · section · 484b71bc

The explicit convolution amplitude witness

The explicit convolution amplitude witness · lines 552–609

X-Enter@1#eq:read, X-Enter@1#eq:readlaw, X-Enter@1#eq:TVgap, X-Enter@1#eq:encdifference, X-Enter@1#eq:readderivative

The retained typed trial is

P_{read} = λ q : Q . let b = Enter (q) in ⟨ b, * ⟩,

of type $Q ⊸ Bit \otimes 1$ . It uses one live qubit.

Born readout gives

Pr (P_{read} = 1 ∣ a) = \frac{1 + tanh ( a / σ )}{2} .

On the convolution witness,

d_{TV} (Law_{read} (c_{ε}), Law_{read} (0)) = \frac{1}{2} tanh (c_{ε} / σ) > 0.

For a classical displacement $d$ , the actual encoding difference is

ι_{1} (a + d) - ι_{1} (a) = (1 - p_{1} (a + d) - 1 - p_{1} (a)) ∣ 0 ⟩ + (p_{1} (a + d) - p_{1} (a)) ∣ 1 ⟩ .

The induced record derivative is

\frac{d}{d λ} Pr (P_{read} = 1 ∣ λ c_{ε}) = \frac{c _{ε}}{2 σ} sech^{2} (λ c_{ε} / σ) > 0.

At the flat representative its coefficient is $c_{ε} / (2 σ)$ .

These calculations establish visibility for this encoding. Nonconstancy of a state encoding alone would not do so: $(∣ 0 ⟩ + e^{ia} ∣ 1 ⟩) / 2$ has constant direct Enter statistics. Likewise, a normalized pure-state encoding of all real inputs cannot be a faithful linear vector map $a \mapsto a v$ .

Displayed formulas in this passage

X-Enter@1#eq:read · line 556
$P_{read} = λ q : Q . let b = Enter (q) in ⟨ b, * ⟩,$
X-Enter@1#eq:readlaw · line 567
$Pr (P_{read} = 1 ∣ a) = \frac{1 + tanh ( a / σ )}{2} .$
X-Enter@1#eq:TVgap · line 574
$d_{TV} (Law_{read} (c_{ε}), Law_{read} (0)) = \frac{1}{2} tanh (c_{ε} / σ) > 0.$
X-Enter@1#eq:encdifference · line 582
$ι_{1} (a + d) - ι_{1} (a) = (1 - p_{1} (a + d) - 1 - p_{1} (a)) ∣ 0 ⟩ + (p_{1} (a + d) - p_{1} (a)) ∣ 1 ⟩ .$
X-Enter@1#eq:readderivative · line 592
$\frac{d}{d λ} Pr (P_{read} = 1 ∣ λ c_{ε}) = \frac{c _{ε}}{2 σ} sech^{2} (λ c_{ε} / σ) > 0.$