Manuscript En07-C@1 · En07-C.tex · example · 3f7e9ebb

Example: A constant selected residual is not a constant complement

Why the residual must include purifiers · lines 1186–1237 · proof included or linked

En07-C@1#ex:purifier, En07-C@1#eq:purifier-isometry, En07-C@1#eq:purifier-access, En07-C@1#eq:selected-residual, En07-C@1#eq:complete-purifier-complement

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Let $C = R = C^{2}$ , and let $E$ and $P$ have bases indexed by $a = (a_{1}, a_{2}) \in {0, 1}^{2}$ . Prepare

∣ Ω ⟩_{E P} = \frac{1}{2} a \sum ∣ a ⟩_{E} ∣ a ⟩_{P} .

Let $U_{a} = X^{a_{1}} Z^{a_{2}}$ be the four Pauli representatives and apply the controlled unitary to $C$ and $E$ . The complete isometry is

V ∣ ψ ⟩ = \frac{1}{2} a \sum U_{a} ∣ ψ ⟩_{R} ∣ a ⟩_{E} ∣ a ⟩_{P}, B = E \otimes P .

Then

N (X) Tr_{P} E (X) E (X) = Tr (X) I_{R} /2, = Tr (X) I_{E} /4, = \frac{1}{4} a, b \sum Tr (U_{b}^{†} U_{a} X) ∣ a, a ⟩ ⟨ b, b ∣ .

The selected residual $E$ is constant, but the complete residual $E P$ is not. The complete residual exactly recovers the unknown qubit.

Proof. Orthogonality of the key labels and unitarity of $U_{a}$ show $V^{†} V = I_{C}$ . Tracing $E P$ gives the Pauli twirl, which is (eq:purifier-access). Tracing $P$ first removes off-diagonal key terms; tracing $R$ then gives (eq:selected-residual). Tracing only $R$ gives (eq:complete-purifier-complement). For example, an off-diagonal key coefficient with $U_{b}^{†} U_{a} = Z$ distinguishes the two $Z$ eigenstates. Finally, (eq:purifier-access) and Corollary cor:nohiding give exact recovery from $E P$ .

Displayed formulas in this passage

En07-C@1#eq:purifier-isometry · line 1198
$V ∣ ψ ⟩ = \frac{1}{2} a \sum U_{a} ∣ ψ ⟩_{R} ∣ a ⟩_{E} ∣ a ⟩_{P}, B = E \otimes P .$
En07-C@1#eq:purifier-access · line 1207
$N (X) Tr_{P} E (X) E (X) = Tr (X) I_{R} /2, = Tr (X) I_{E} /4, = \frac{1}{4} a, b \sum Tr (U_{b}^{†} U_{a} X) ∣ a, a ⟩ ⟨ b, b ∣ .$
En07-C@1#eq:selected-residual · line 1207
$N (X) Tr_{P} E (X) E (X) = Tr (X) I_{R} /2, = Tr (X) I_{E} /4, = \frac{1}{4} a, b \sum Tr (U_{b}^{†} U_{a} X) ∣ a, a ⟩ ⟨ b, b ∣ .$
En07-C@1#eq:complete-purifier-complement · line 1207
$N (X) Tr_{P} E (X) E (X) = Tr (X) I_{R} /2, = Tr (X) I_{E} /4, = \frac{1}{4} a, b \sum Tr (U_{b}^{†} U_{a} X) ∣ a, a ⟩ ⟨ b, b ∣ .$
unlabeled · line 1190
$∣ Ω ⟩_{E P} = \frac{1}{2} a \sum ∣ a ⟩_{E} ∣ a ⟩_{P} .$

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section: Why the residual must include purifiers