Theorem: Quantum temporal certificate

Under Assumption ass:qcontract, identical subsequent inputs give

For different inputs and an implemented-channel error $e_t$ in half-diamond distance,

Consequently the temporal product-and-sum bound of Theorem thm:recurrence applies.

For equal-prior delayed labels stored only in two carrier states, with later inputs independent of the label,

Proof. Iterate contraction and insert a common-input channel for the perturbed recurrence. The half-diamond allowance controls each normalized input, including referenced inputs Watrous.

For binary equal-prior state discrimination, write an effect $F$ for declaring plus:

The maximal effect expectation difference is trace distance. Apply (eq:qforget).