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When they get seated for lunch they talk about the experiment and find they disagree. Alice writes Dirac notation on a napkin, presents it to Bob, and claims it describes the states, operators, eigenstuff, etc... Bob writes on his napkin and describes a different situation...

Is there something in QM that prevents this (that is, between Alice and Bob, at least one of them must be incorrect), or do some QM experiments allow a sort of relative perspective on what is going on in the experiment?

I'm thinking there may be many ways to manifest the apparatus etc. for the intention described by a singular mathematical description, but can a single arraingment be described by multiple possibly contradictory mathematical descriptions (in terms of QM description)?