Manuscript Em05-A@1 · Em05-A-v1.tex · section · 9cac0def

Physical output operation and dilation

Population-preserving dephasing with its complete environment › Physical output operation and dilation · lines 855–897

Em05-A@1#eq:dephaser, Em05-A@1#eq:dephasingDilation

On the entire accessible space define

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

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

For $0 \leq λ \leq 1$ , put

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

and define

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

Let $F = C^{2}$ be an additional environmental register. The isometry

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

is a dilation of $Z^{R}_{λ}$ .

Define

V_{η, λ} = (U_{λ} \otimes I_{E}) V_{η},

with the fixed output reordering $R F E \to R E F$ . The new accessible channel is

N_{η, λ} = Z^{R}_{λ} N_{η} .

Its complete environment is $E F$ , not merely $E$ .

Displayed formulas in this passage

Em05-A@1#eq:dephaser · line 870
$Z^{R}_{λ} (Y) = a_{λ} Y + b_{λ} S_{R} Y S_{R} .$
Em05-A@1#eq:dephasingDilation · line 877
$U_{λ} x = a_{λ} x \otimes ∣0 ⟩_{F} + b_{λ} S_{R} x \otimes ∣1 ⟩_{F}$
unlabeled · line 858
$S_{R} = diag (1, 1, 1, - 1) .$
unlabeled · line 865
$a_{λ} = \frac{1 + λ}{2}, b_{λ} = \frac{1 - λ}{2},$
unlabeled · line 887
$V_{η, λ} = (U_{λ} \otimes I_{E}) V_{η},$
unlabeled · line 893
$N_{η, λ} = Z^{R}_{λ} N_{η} .$