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bhobba
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It usually boils down to its associated observable commuting with the Hamiltonian - and is referred to as the commutativity criteria. See page 77 of Schlosshauer.atyy said:Is the predictability sieve the stability criterion? If it is not, please state the stability criterion exactly.
Its rare that this criteria doesn't work, but evidently it has been generalised, and that is where your predictability sieve comes in (see page 82 Schlosshauer)
Delving deeper into it really requires going through the book. Its been a while since I have delved into this detail, so if that's what you want to do I am afraid I will have to leave it up to you.
Thanks
Bill