Manuscript Em03-E@1 · Em03-E-v1.tex · lemma · c4b07e6a

Lemma: Leakage of the dephasing environment

The full complementary channel and leakage constants · lines 1032–1061 · proof included or linked

Em03-E@1#lem:dephleak

For $e ˉ = (∣ e_{0} ⟩ ⟨ e_{0} ∣ + ∣ e_{1} ⟩ ⟨ e_{1} ∣) /2$ ,

d_{⋄} (C_{η}, R_{e ˉ}) = \frac{1}{2} 1 - η^{2} .

The Bell input attains this distance.

Proof. The channel difference is

X ⟼ \frac{tr ( Z X )}{2} (∣ e_{0} ⟩ ⟨ e_{0} ∣ - ∣ e_{1} ⟩ ⟨ e_{1} ∣) .

The stabilized norm of the functional $X \mapsto tr (Z X)$ is $∥ Z ∥ = 1$ . The trace norm of the difference of the two pure-state projectors is $2 1 - η^{2}$ . This gives the stated upper bound.

On the Bell input the difference is

\frac{1}{4} Z_{R} \otimes (∣ e_{0} ⟩ ⟨ e_{0} ∣ - ∣ e_{1} ⟩ ⟨ e_{1} ∣) .

Its half trace norm is the same value.

Displayed formulas in this passage

unlabeled · line 1036
$d_{⋄} (C_{η}, R_{e ˉ}) = \frac{1}{2} 1 - η^{2} .$
unlabeled · line 1045
$X ⟼ \frac{tr ( Z X )}{2} (∣ e_{0} ⟩ ⟨ e_{0} ∣ - ∣ e_{1} ⟩ ⟨ e_{1} ∣) .$
unlabeled · line 1056
$\frac{1}{4} Z_{R} \otimes (∣ e_{0} ⟩ ⟨ e_{0} ∣ - ∣ e_{1} ⟩ ⟨ e_{1} ∣) .$

Lemma: Leakage of the dephasing environment

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