Can a single quantum experiment have multiple valid mathematical descriptions?

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bahamagreen
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Let's say Bob and Alice are passing through a lab on their way to lunch. In the center of the lab is a big table with some kind of experimental set up. They are curious about it and both silently examine the whole thing (there is an unrelated lecture going on nearby so they are silent and neither Bob nor Alice say a word while they look the experimental apparatus over very closely).

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)?
 

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Here is an entertaining argument that there can be more than one valid quantum description of an experimental situation.

http://arxiv.org/abs/quant-ph/0507214
Dialogue Concerning Two Views on Quantum Coherence: Factist and Fictionist
Stephen D. Bartlett, Terry Rudolph, Robert W. Spekkens
 
Wow, that's a long paper, but it looks like just what I'm asking, in my clumsy and uninformed way.
Thanks so much!
 
Well, after reading it and understanding every detail, maybe you can tell us whether it is right, or explain it at a level a layman can understand? I saw it some time ago, and have been meaning to read it carefully, but never got round to really understanding what's going on here.
 
If Alice and Bob made the same type of measurement, i.e. measured the same observable(s), then, according to QM, they must agree on the conclusions.