Manuscript VII-E-ST@1 · VII-E-ST.tex · example · d28d9c5d

Example: An aggregate continuation can hide branch failure

Contract G: finite completely positive hybrid evolution › Intrinsic observation and histories · lines 1321–1339

VII-E-ST@1#ex:branch-failure

Let the input observation algebra be $span {I, Z}$ . Let

Ψ_{+} (X) = \frac{1}{2} H X H, Ψ_{-} (X) = \frac{1}{2} H Z X Z H,

where $H$ is the Hadamard matrix. The summed channel has zero output $Z$ -expectation on every input, so it preserves agreement for that terminal observation. But

Ψ_{+}^{*} (Z) = X /2, Ψ_{-}^{*} (Z) = - X /2.

The inputs $∣ + ⟩ ⟨ + ∣$ and $∣ - ⟩ ⟨ - ∣$ agree on the input diagonal algebra and have the same outcome probabilities, yet their conditioned $Z$ -expectations differ.

Displayed formulas in this passage

unlabeled · line 1325
$Ψ_{+} (X) = \frac{1}{2} H X H, Ψ_{-} (X) = \frac{1}{2} H Z X Z H,$
unlabeled · line 1333
$Ψ_{+}^{*} (Z) = X /2, Ψ_{-}^{*} (Z) = - X /2.$

Example: An aggregate continuation can hide branch failure

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