Manuscript XXII-C@1 · XXII-C.tex · example · b9a05a24

Example: A quotient is a substantive change of carrier

The frozen order-forming operation › Raw experiment-use relation · lines 682–712

XXII-C@1#ex:cycle

Mix, with probabilities $0 < λ < 1$ and $1 - λ$ , the qubit circuits $A \to B$ and $B \to A$ , and keep $C$ independent. At $A, B$ , use measure-and-prepare instruments

M_{u ∣ x} (X) = ⟨ u ∣ X ∣ u ⟩ ∣ x ⟩ ⟨ x ∣.

The $A, B$ outcome law is

P (u, v ∣ x, y) = λ δ_{u 0} δ_{v x} + (1 - λ) δ_{v 0} δ_{u y} .

Hence

P (v = 1∣ x, y) = λ x, P (u = 1∣ x, y) = (1 - λ) y .

Both direct signalling edges are present, even though the process is a mixture of definite-order circuits. The quotient identifies $A$ and $B$ , leaving that component and the isolated $C$ -component incomparable.

This is the phenomenon recorded in XXII-B CheungXXIIB (prop:signalling-cycle). Neither branch is a physical closed causal loop.

Displayed formulas in this passage

unlabeled · line 688
$M_{u ∣ x} (X) = ⟨ u ∣ X ∣ u ⟩ ∣ x ⟩ ⟨ x ∣.$
unlabeled · line 692
$P (u, v ∣ x, y) = λ δ_{u 0} δ_{v x} + (1 - λ) δ_{v 0} δ_{u y} .$
unlabeled · line 699
$P (v = 1∣ x, y) = λ x, P (u = 1∣ x, y) = (1 - λ) y .$

Example: A quotient is a substantive change of carrier

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