Manuscript En07-A@1 · En07-A.tex · example · 44229ce9

Example: A diary-independent record can assist correction

Classical records and exact destination examples › Record-only reconstruction · lines 1006–1052 · proof included or linked

En07-A@1#ex:assistance

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For a qubit, set

V ∣ ψ ⟩ = \frac{1}{2} (∣ ψ ⟩_{R} ∣ 0 ⟩_{B} + Z ∣ ψ ⟩_{R} ∣ 1 ⟩_{B}) .

Radiation has optimal recovery error $1/2$ . Computational readout of $B$ gives a uniform bit $x$ , independent of the logical input. Supplying that bit and applying $Z^{x}$ to $R$ gives exact quantum recovery. The bit alone is not a replacement for the diary.

Proof. The unnormalized branch is

ρ ⟼ \frac{1}{2} Z^{x} ρ Z^{x} .

Its probability is $1/2$ , and correction by $Z^{x}$ followed by summation returns $ρ$ . Without the record, the channel is $Δ_{Z} = (id + Ad_{Z}) /2$ . The two inputs $∣ + ⟩$ and $∣ - ⟩$ have identical dephased outputs, giving recovery error at least $1/2$ by the triangle inequality. The identity decoder attains $1/2$ by Lemma lem:unitary-mixture.

For a normalized input, the complete residual is

C (ρ) = \frac{1}{2} (1 Tr (Z ρ) Tr (Z ρ) 1) .

It is not constant. Only its computational record is diary independent.

Displayed formulas in this passage

unlabeled · line 1009
$V ∣ ψ ⟩ = \frac{1}{2} (∣ ψ ⟩_{R} ∣ 0 ⟩_{B} + Z ∣ ψ ⟩_{R} ∣ 1 ⟩_{B}) .$
unlabeled · line 1027
$ρ ⟼ \frac{1}{2} Z^{x} ρ Z^{x} .$
unlabeled · line 1041
$C (ρ) = \frac{1}{2} (1 Tr (Z ρ) Tr (Z ρ) 1) .$

Example: A diary-independent record can assist correction

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