Manuscript Em05-B@1 · Em05-B-v1.tex · section · 6c050f5b

Population-preserving dephasing and its full environment

Population-preserving dephasing and its full environment · lines 1439–1481

Em05-B@1#eq:dephaser, Em05-B@1#eq:noiseDilation

This section declares a different experiment: extraction followed by output noise.

On the entire accessible space, put

S_{R} = diag (1, 1, 1, - 1) .

It is Hermitian, unitary, parity preserving, and commutes with all occupation projectors and $N_{R}$ .

For $0 \leq λ \leq 1$ , define

a_{λ} = \frac{1 + λ}{2}, b_{λ} = \frac{1 - λ}{2},

and

Z^{R}_{λ} (Y) = a_{λ} Y + b_{λ} S_{R} Y S_{R} .

Its isometric dilation, with a fresh register $F = C^{2}$ , is

U_{λ} x = a_{λ} x \otimes ∣0 ⟩_{F} + b_{λ} S_{R} x \otimes ∣1 ⟩_{F} .

After the fixed reordering $R F E \to R E F$ , set

V_{η, λ} = (U_{λ} \otimes I_{E}) V_{η}, N_{η, λ} = Z^{R}_{λ} N_{η} .

The complete environment is $E F$ , not $E$ alone.

Displayed formulas in this passage

Em05-B@1#eq:dephaser · line 1458
$Z^{R}_{λ} (Y) = a_{λ} Y + b_{λ} S_{R} Y S_{R} .$
Em05-B@1#eq:noiseDilation · line 1465
$U_{λ} x = a_{λ} x \otimes ∣0 ⟩_{F} + b_{λ} S_{R} x \otimes ∣1 ⟩_{F} .$
unlabeled · line 1445
$S_{R} = diag (1, 1, 1, - 1) .$
unlabeled · line 1452
$a_{λ} = \frac{1 + λ}{2}, b_{λ} = \frac{1 - λ}{2},$
unlabeled · line 1473
$V_{η, λ} = (U_{λ} \otimes I_{E}) V_{η}, N_{η, λ} = Z^{R}_{λ} N_{η} .$