  An example shows that weak decoherence is more restrictive than the minimal
logical decoherence structure that allows probabilities to be used consistently
for quantum histories. The probabilities in the sum rules that define minimal
decoherence are all calculated by using a projection operator to describe each
possibility for the state at each time. Weak decoherence requires more sum
rules. They bring in additional variables, that require different measurements
and a different way to calculate probabilities, and raise questions of
operational meaning. The example shows that extending the linearly positive
probability formula from weak to minimal decoherence gives probabilities that
are different from those calculated in the usual way using the Born and von
Neumann rules and a projection operator at each time.
