Manuscript Em03-H@8 · Em03-H-v8.tex · theorem · 91e9e3bd

Theorem: Complete signed candidate dilation

Candidate dilation, decoder, and limitations · lines 1273–1298 · proof included or linked

Em03-H@8#thm:dilation

In bath-first order,

Z ∣ 0 ⟩ = ∣ 00 ⟩_{R} ∣ 00 ⟩_{E},

Z ∣ 1 ⟩ = q^{2} ∣ 11 ⟩_{R} ∣ 00 ⟩_{E} + q r ∣ 10 ⟩_{R} ∣ 01 ⟩_{E} - q r ∣ 01 ⟩_{R} ∣ 10 ⟩_{E} + r^{2} ∣ 00 ⟩_{R} ∣ 11 ⟩_{E} .

Unused modes are vacuum, and the evolved Schwarzian factor is common and pure.

Proof. Expand $(r e_{1}^{†} + q b_{1}^{†}) (r e_{2}^{†} + q b_{2}^{†})$ . CAR and bath-first regrouping give opposite mixed signs. The occupied column has norm squared $η^{2} + 2 η (1 - η) + (1 - η)^{2} = 1$ and is orthogonal to the vacuum column.

Displayed formulas in this passage

unlabeled · line 1276
$Z ∣ 0 ⟩ = ∣ 00 ⟩_{R} ∣ 00 ⟩_{E},$
unlabeled · line 1279
$Z ∣ 1 ⟩ = q^{2} ∣ 11 ⟩_{R} ∣ 00 ⟩_{E} + q r ∣ 10 ⟩_{R} ∣ 01 ⟩_{E} - q r ∣ 01 ⟩_{R} ∣ 10 ⟩_{E} + r^{2} ∣ 00 ⟩_{R} ∣ 11 ⟩_{E} .$

Theorem: Complete signed candidate dilation

Displayed formulas in this passage

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